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quantum error correction procedure for a generic +quantum noise acting on a quantum circuit. In general, there is no analytic universal procedure +to obtain the encoding and correction unitary gates, and the problem is even harder if the noise +is unknown and has to be reconstructed. The existing procedures rely on variational quantum +algorithms and are very difficult to train since the size of the gradient of the cost function decays +exponentially with the number of qubits. We address this problem using a cost function based on +the quantum Wasserstein distance of order 1. Our results show that such cost function significantly +increases both the probability of a successful training and the fidelity of the recovered state. +I. +INTRODUCTION +Performing reliable computations on physical imperfect +hardware is something that has become usual nowadays, +given the current state of classical computers, which can +produce perfect results without any software-side miti- +gation of the imperfections of the physical media where +the computation happens. Error correction is based on +the fact that these machines perform automatically, on +the hardware side, procedures that allow errors to hap- +pen and to be fixed without any intervention of the end +user. This kind of setting is even more crucial in a quan- +tum scenario where the current noisy intermediate-scale +quantum computers (NISQ) have a much larger error +rate than their classical counterparts [1]. Performing re- +liable computations with a trustworthy error correction +procedure has direct implications not only in quantum +computation [2, 3], but also in quantum key distribution +and cryptography [4–8]. +In the typical Quantum Error Correction (QEC) +scheme, the quantum state that has to be protected is +stored in a subspace of a larger Hilbert space, using an +encoding procedure. Stabilizer codes [9], which are within +the best analytical results in this field, are not universal +because they are tailored for a generic noise acting on a +small but unknown subset of qubits. Several attempts +have already been made to create a numerical optimiza- +tion procedure to find an error correction code for specific +noise models [10–13], but these studies are not universal +because they rely heavily on the type of noise on the +specific quantum circuit and this is a problem because +real quantum devices are not characterized by a single +kind of quantum noise. Some attempts have been made +to characterize the noise of the current and near-term +devices [14, 15], but these methods will become very diffi- +cult to implement soon because classical computers are +not able to simulate efficiently quantum circuits when +the number of qubit increases. Near-term devices with +approximately 50 qubits may already be intractable to +simulate for supercomputers [16]. +If we define a figure of merit of the quality of the state +after the action of the noise and its corresponding cor- +rection, the obvious choice for the kind of maximization +algorithm is a Variational Quantum Algorithm [17]. These +are hybrid algorithms that couple a quantum computer +with a classical one. In this kind of algorithms, usually, a +parametric quantum circuit is applied to some reference +state, some measurements are performed on the system, +and the outcomes are given to the classical computer to +perform a minimization procedure of a given cost function. +Some interesting examples of this class of algorithms are +the variational quantum eigensolver [18] and the Quan- +tum Approximate Optimization Algorithm (QAOA) [19]. +The optimization procedure in a variational quantum al- +gorithm can be seen as the training phase in machine +learning, for example, to train a neural network. +Some variational quantum algorithms applied to quan- +tum error correction are already known in literature [20]. +However, VQAs usually suffer from the phenomenon of +barren plateaus [21, 22], namely the gradient of the cost +function decays exponentially with respect to the number +of qubits of the system, leading to an untrainable model. +Some general results have already been found about this +topic [22], which justifies the presence of barren plateaus +when the cost function is a global function of the quantum +circuit, namely the measurement of a highly non-local op- +erator. For this reason, we compared the performance of +an algorithm inspired by [20] using two different cost func- +tions: the fidelity and an approximation of the quantum +Wasserstein distance. +The quantum Wasserstein distance is a generalization of +the classical Wasserstein distance between probability dis- +tributions [23–27]. Several quantum generalizations of the +Wasserstein distance have been proposed [28–62]. This +arXiv:2301.05273v1 [quant-ph] 12 Jan 2023 + +2 +work is based on the quantum Wasserstein distance of or- +der 1 (or quantum W1 distance) proposed in Refs. [63, 64], +which is not unitarily invariant and recovers the Hamming +distance [65] for the states of the computational basis. We +expect this new distance to improve the barren plateau +phenomenon and we can give an intuitive reason. If we +use a unitarily invariant distance as the trace distance or +the distances derived from the fidelity, all the states of the +computational basis are equally orthogonal and thus have +all maximum distance one with respect to the other. The +quantum W1 distance instead measures how many qubits +are different between the two states and this allows the +gradient to be less flat in the regions that are not already +very close to a local minimum. Indeed, the quantum W1 +distance has succesfully been employed as cost function +of quantum Generative Adversarial Networks [64, 66–70] +The manuscript is organized as follows: in Sec. II we +present some basic notion on conventional QEC proce- +dures which allow us to set the notation and the theo- +retical background; in Sec. III we introduce our VQA +discussing the different choices of cost functions that can +be used in order to guide it; in Sec. IV we present our nu- +merical results where comparing the performances of the +VQA implemented with different types of cost functions. +Conclusions are given in Sec. V. +II. +PRELIMINARIES ON QEC +Let Q be a quantum register we wish to protect (at least +in part) from the action of some external noise source. +In a typical QEC scenario [71] this problem is addressed +through the following three-step procedure: +i) Before the action the noise, a unitary encoding gate +ˆVQA is used to distribute the information originally +contained in Q on the larger system QA. Here A +is an auxiliary quantum register that is assumed to +be intialized in a fiduciary quantum state, and that +is affected by the same noise that tampers with Q; +ii) After the action of the noise a measurement on QA +is performed to reveal the nature of the latter and, +based on the associated outcome, a unitary recovery +operation is applied to the system. Equivalently this +step can be described by introducing yet an extra +quantum register B (also intialized on a fiduciary +state but not affected by the noise) that is coupled +with QA trough a recovering unitary transformation +ˆWQAB which effectively mimics the measurement +and the recovery operation; +iii) The inverse of the gate ˆVQA is finally used on QA +to refocus the recovered information in Q. +Denoting with |ψ⟩Q the input state of Q, the corre- +sponding output state of QA that emerges from the pro- +cess at the end of the step iii) can be expressed as the +density matrix +ˆρ(V,W ) +QA +(ψ) := trB +� +V† +QA ◦ WQAB ◦ ΦQA +(1) +◦VQA +� +|ψ⟩Q⟨ψ| ⊗ |Ø⟩A⟨Ø| ⊗ |Ø⟩B⟨Ø| +�� +:= V† +QA ◦ Φ(R) +QA ◦ ΦQA ◦ VQA +� +|ψ⟩Q⟨ψ| ⊗ |Ø⟩A⟨Ø| +� +where |Ø⟩X represents the fiduciary state of the X register, +trB{· · · } is the partial trace over B, and given a unitary +ˆUX on X we adopted the symbol UX(· · · ) := ˆUX · · · ˆU † +X +to denote its action as super-operator. +In the above +expressions ΦQA is the LCPT quantum channel [71] +describing the noise on Q and A, while Φ(R) +QA(· · · ) := +trB{WQAB(· · ·⊗|Ø⟩B⟨Ø|)} is the LCPT (recovery) quan- +tum channel on QA originating from the interaction with +B, that attempts to undo the action of ΦQA. +An ideal QEC procedure capable to completely remove +the noise from the system will make sure that ˆρ(V,W ) +QA +(ψ) +corresponds to |ψ⟩Q|Ø⟩A, irrespectively from the specific +choice of |ψ⟩Q. A bona-fide figure of merit to character- +ize the effectiveness of a generic QEC scheme is hence +provided by the average input-output fidelity +F(V, W) := +� +dµψ Q⟨ψ|A⟨Ø|ˆρ(V,W ) +QA +(ψ)|ψ⟩Q|Ø⟩A , (2) +where dµψ is the uniform measure on the set of the in- +put states of Q originated from the Haar measure on +the associated unitary group [72] or from an exact or +approximate unitary 2-design S [71, 73] that simulates +the latter1. Notice that by expressing |ψ⟩Q = ˆUQ|Ø⟩Q, +Eq. (2) can equivalently be casted in the more compact +form +F(V, W) = QA⟨Ø|ˆρ(V,W ) +QA +|Ø⟩QA , +(3) +with |Ø⟩QA := |Ø⟩Q ⊗ |Ø⟩A and where the state +ˆρ(V,W ) +QA +:= 1 +|S| +� +ˆUQ∈S +U† +Q ◦ V† +QA ◦ Φ(R) +QA ◦ ΦQA +◦ VQA ◦ UQ +� +|Ø⟩QA⟨Ø| +� +, +(4) +now includes the average over all possible inputs. An +ideal QEC procedure will enable one to get F(V, W) = 1. +A natural benchmark for lowest admissible F(V, W) is +represented instead by the value one would get if one +decides not to perform corrections on the register that we +1 We remind that a unitary 2-design is a probability distribution +over the set of unitary operators which can duplicate properties of +the probability distribution over the Haar measure for polynomials +of degree 2 or less. When Q is a single qubit, a 2-design can +be realized by an uniform sampling over a set S composed by +only 6 elements ˆ1, ˆσ1, e±iπ/4ˆσ1, e±iπ/4ˆσ2 that maps its logical +state |0⟩Q into the vectors |0⟩Q , |1⟩Q , (|0⟩Q ±i |1⟩Q)/ +√ +2, (|0⟩Q ∓ +|1⟩Q)/ +√ +2. + +3 +Yα1 +Yα7 +Xα2 +Xα8 +Xα9 +Yα5 +Xα6 +Xα10 +Xα3 +Xα4 +Q +A +Yα11 Xα12 +Yα13 Xα14 Xα15 Xα16 +Yα17 Xα18 +Y /2 +π +Xθ′ ′ +k +Xθ′ ′ ′ +k +Zθ′ k += +Xθk +Y /2 +π +B +Q +A +Yβ1 +Yβ11 +Yβ21 +Yβ7 +Yβ17 +Yβ27 +Yβ31 +Yβ35 +Yβ33 +Yβ37 +Xβ2 +Xβ12 +Xβ22 +Xβ3 +Xβ13 +Xβ23 +Xβ5 +Xβ15 +Xβ25 +Xβ8 +Xβ18 +Xβ28 +Xβ32 +Xβ36 +Xβ34 +Xβ38 +Zβ4 +Zβ14 +Zβ24 +Zβ6 +Zβ16 +Zβ26 +Xβ10 +Xβ20 +Xβ30 +Zβ9 +Zβ19 +Zβ29 += +Zθk +Y /2 +π +Zθ′ ′ +k +Zθ′ ′ ′ +k +Zθ′ k +Y /2 +π +Figure 1. +Parametric gates ˆVQA(⃗α) (green element) and +ˆ +WQAB(⃗β) (red element) used for case of quantum registers Q, +A, and B with k = 1, n − k = 2, and r = 2 qubits respectively. +Indicating with ˆσ1, ˆσ2, and ˆσ3 the Pauli operators, the Xθ, Yθ, +and Zθ elements of the figure represent single qubit rotations +e−iθˆσ1, e−iθˆσ2, and e−iθˆσ3 with the angles θ determined to +the components of the vectors ⃗α, ⃗β, respectively. Vertical +lines indicate instead quantum control operations which are +activated when the control qubits (indicated by the full or +empty circles) are in the logical state |1⟩ (full circle) or in |0⟩ +(empty circle). As shown on the inset, each one of those gates +depend parametrically upon elements of the control vectors ⃗α +and ⃗β through single qubit operations. +compute by setting ˆVQA and ˆWQAB equal to the identity +operators i.e.2 +F 0 := QA⟨Ø|ˆρ(11,11) +QA +|Ø⟩QA . +(5) +III. +VARIATIONAL QUANTUM ALGORITHM +While the enormous progress has been made in the +study of QEC procedures, identifying efficient choices for +the operations that leads to (non trivial) high values of +F(V, W) for a specific noise model, is still a challenging +open problem. A possible solution in this case is to employ +variational quantum algorithms to run numerical searches. +Our approach follows a training strategy inspired by the +work of Johnson et al. [20]. Assuming hence Q, A, and +B to be formed by collections of independent qubits (k +for Q, n − k for A, and r for B), we introduce a manifold +2 Equation (5) accounts for the noise effects both on Q and A. A +more conservative estimation of F 0 can be obtained by focusing +directly on the noise on Q alone, i.e. tracing out the A component +of ˆρ(11,11) +QA +and studying its fidelity with |Ø⟩Q, i.e. F (strong) +0 +:= +Q⟨Ø|ˆρ(11,11) +Q +|Ø⟩Q ≥ F 0, with ˆρ(11,11) +Q +:= trA ˆρ(11,11) +QA +. Notice that +for the noise model of Sec. III C the two are directly connected +via the identity F 0 = F (strong) +0 +− n−1 +n p(1 − |⟨0|ˆσ|0⟩|2). +ï +ï +ï +ï +ï +ï +|0ï +|0ï +|0ï +|0ï +|0ï +|0ï +|0ï +|0ï +|0ï +Q +A +B +UQ(j) +ï +ï +ï +M +M +M +M +C( +÷³ , +÷ +´ ) +M +M +U† +Q(j) +WQAB( +÷ +´ ) +VQA( +÷³ ) +V† +QA( +÷³ ) +j +÷³ , +÷ +´ +n aˆK +ic +r t +ovth +er +e w +qu +we += 1 +F +cor +NOISE +Figure 2. (Color online) Sketch of the variational quantum +algorithm: Q, A and B are quantum registers formed respec- +tively by k, n−k and r qubits. The initial information we wish +to protect is written in Q by the unitary gate ˆUQ(j) extracted +from a 2-design set S; A and B are two auxiliary elements +(containing respectively n − k and r qubits) that are used to +implement the QEC procedure described by the parametric +gates ˆVQA(⃗α), ˆ +WQAB(⃗β), and ˆV † +QA(⃗α) of Fig. 1. The patterned +element in the central part of the scheme represents the noise +on Q and A (no noise is assumed to be active on B). Lastly, +the D-shaped measurements at the end of the circuit represent +local measurements on QA whose outcomes over the entire +collection of the possible inputs generated by the entire set S, +are processed by a classical computer which, evaluating the +cost function C(⃗α, ⃗β) defined in Section III B, decides how to +update the values of the parameters ⃗α and ⃗β. Thick grey lines +in the figure represent classical control lines. +of transformations ˆVQA(⃗α), ˆWQAB(⃗β) parametrized by +classical controls vectors ⃗α, ⃗β (see Fig. 1), and construct +the quantum circuit of Fig. 2. The method then proceeds +along the following stages: +1. Having selected the values of ⃗α and ⃗β, the regis- +ter Q is prepared into a collection of known quan- +tum state {|ψ(1)⟩Q, · · · , |ψ(m)⟩Q} operating on the +vector |Ø⟩Q = |0⟩⊗k through action of the con- +trol gates ˆUQ(1), · · · , ˆUQ(m) (first cyan element of +the figure) which define the 2-design S entering +in Eq. (4). Each of such inputs is hence evolved +via a circuit (pale-orange area of the figure) that +emulates both the effect of the noise (patterned +square of the figure, see Section III C and Fig. 4), +and the transformations ˆVQA(⃗α), ˆWQAB(⃗β), and +ˆV † +QA(⃗α) that are meant to implement the steps ii) +and iii) of the QEC procedure (green and red ele- +ments of the figure). Notice that in the ideal case +(i.e. if ˆVQA(⃗α) and ˆWQAB(⃗β) manage to completely +suppress the noise) then in correspondence with + +4 +the input |ψ(j)⟩Q the registers QA should emerge +in the state |ψ(j)⟩Q ⊗ |Ø⟩A := |ψ(j)⟩Q ⊗ |0⟩⊗n−k, +which will be hence mapped into the final configu- +ration |Ø⟩QA := |0⟩⊗n by the inverse ˆU † +Q(j) of the +state preparation gate (second cyan element of the +figure). +2. For each choice of the index j ∈ {1, · · · , m} a mea- +surement on the system is performed at the end of +the transformations described in the stage 1 and the +resulting m collected outcomes used to compute a +cost function C(⃗α, ⃗β) which evaluates the effective- +ness of the adopted QEC strategy in leading large +values of the average input-output fidelity. The spe- +cific choice of the cost function is very important +and is discussed in Section III A. +3. A classical computer decides, given the results of +the measurement, how to change the value of the +parameters ⃗α and ⃗β to be used in the subsequent +run in order to minimize the cost function C(⃗α, ⃗β). +This is discussed in detail in Section III B. +A. +Cost function +The natural choice for the cost function at the stage 2 +of our algorithm is provided by the expectation value of +the self-adjoint operator +ˆH(fid) +QA := ˆ1QA − |Ø⟩QA ⟨Ø| , +(6) +computed on the mean state of system QA which emerges +at the output of the quantum circuit of Fig. 2, i.e. the +quantity +C(fid)(⃗α, ⃗β) := tr{ˆρ(V (⃗α),W (⃗β)) +QA +ˆH(fid) +QA } , +(7) +where ˆρ(V (⃗α),W (⃗β)) +QA +is the density matrix (4) evaluated +for ˆVQA = ˆVQA(⃗α) and ˆWQAB = ˆWQAB(⃗β). This choice +has two main advantages. First of all, the expectation +value C(fid)(⃗α, ⃗β) can be evaluated by performing (sim- +ple) local measurement on the qubits of Q and A (in- +deed it can be computed by simply checking whether of +not each one of them is in the logical state |0⟩). Most +importantly, since by explicit evaluation one has that +C(fid)(⃗α, ⃗β) = 1 − F(V (⃗α), W(⃗β)), it is clear that by us- +ing (7) the algorithm will be forced to luck for values +of ⃗α, ⃗β that yield higher average input-output fidelities. +Despite all this, the use of C(fid)(⃗α, ⃗β) as a cost function, +has a major drawback associated with the fact that the +spectrum of the Hamiltonian ˆH(fid) +QA exhibits maximum +degeneracy with respect to space orthogonal to the target +state |Ø⟩QA (see Fig. 3). Due do this fact a numerical +search based on a training procedure that simply target +the minimization of C(fid)(⃗α, ⃗β), has non trivial chances +to get stuck somewhere in the large flat plateau associated +with the eigenvalue 1 of ˆH(fid) +QA without finding any good +direction. in the large flat plateau A possible way to avoid +this problem is to introduce new cost-functions Hamilto- +nians which, while maintaining the target vector |Ø⟩QA +as unique ground state and still being easy to compute, +manage to remove the huge degeneracy of the excited +part of the spectra of ˆH(fid) +QA . Our choice is based on the +quantum Wasserstein distance of order 1 (W1) introduced +Ref. [63] which, even though it lacks some interesting +properties that the fidelity has, is less likely to be affected +by the barren plateaus phenomena [22]. A good estima- +tion of the W1 distance that separate ˆρ(V (⃗α),W (⃗β)) +QA +from +the target state, is provided by the following quantity +C(wass)(⃗α, ⃗β) := tr{ˆρ(V (⃗α),W (⃗β)) +QA +ˆH(wass) +QA +} , +(8) +ˆH(wass) +QA +:= +n +� +j=1 +j ˆΠ(j) +QA , +(9) +where ˆΠ(j) +QA represents the sub-space of the register QA +in which we have j qubits in |1⟩ and the remaining one in +|0⟩. Observe that ˆH(wass) +QA +is nothing but the sum of the +number operators acting on the individual qubits of the +register QA, (i.e. ˆH(wass) +QA += �n +ℓ=1 ˆπℓ with ˆπℓ the projector +on the logical state |1⟩ of the ℓ-th qubit): accordingly, +as C(fid)(⃗α, ⃗β), C(wass)(⃗α, ⃗β) can be computed from local +measurement. What C(wass)(⃗α, ⃗β) does is to count the +total number of logical ones present in the system. To +understand why using (8) could in principle lead to a +more efficient numerical search than the one obtained by +using (7), notice that Eq. (6) can be equivalently written +as ˆH(fid) +QA = +n +� +j=1 +ˆΠ(j) +QA. A comparison with (9) reveals hence +that indeed while both ˆH(fid) +QA and ˆH(wass) +QA +admit |Ø⟩QA +as unique ground state, the Wasserstein Hamiltonian +removes large part of the degeneracy of the high energy +spectrum of the fidelity Hamiltonian. Accordingly it is +reasonable to expect that a numerical search that uses +ˆH(wass) +QA +, has less chances to get trapped into regions of +constant energy (barren plateau) than a search based on +ˆH(fid) +QA ,3. +3 It goes without mentioning that alternative choices for the cost +function Hamiltonians are also available. For instance one can +use operators that also remove the residual degeneracies that +affect ˆH(wass) +QA +– e.g. using the operator ˆH(full) +QA += �n +ℓ=1 wℓˆπℓ +with ωℓ positive weights selected so that different allocation of |1⟩ +states inside the eigenspaces of ˆH(wass) +QA +get an assigned ordering. +Our numerical analysis however seems to indicate that these +refinement do not contribute significantly in improving numerical +search of the algorithm. + +5 +eigenvalues of +̂H(fid) +QA +eigenvalues of +̂H(wass) +QA +N = 3 +N = 3 +N = 2 +N = 1 +N = 2 +N = 0 +1 +2 +3 +0 +N = 1 +N = 2 +N = 3 +N = 0 +0 +1 +Figure 3. Pictorial rendering of the spectra of the Hamiltonians +ˆH(fid) +QA +(top panel) and ˆH(wass) +QA +(lower panel). While ˆH(fid) +QA +is characterized by a unique, flat plateau that includes all +the excited state, ˆH(wass) +QA +partially removes the associated +degeneracy assigning higher energy to subspaces that have +higher number of qubits in the logical state |1⟩. +B. +Descent algorithm +The algorithm that we used for this work is a gradient +descent algorithm with momentum [74]. To overcome the +numerical difficulties of using finite differences to estimate +the gradients of the cost function C(⃗α, ⃗β), we exploit a +variation of the parameter-shift rule introduced in [75] +which reduces the problem to compute linear combina- +tions of the function itself evaluated in different points +that are not infinitesimally close. Specifically we observe +that, irrespectively from the choice of the operator ˆHQA, +the functional dependence of C(⃗α, ⃗β) upon the j-th com- +ponent of the vector ⃗β is of the form +C(⃗α, ⃗β) = f(βj) := +� +k +tr +�ˆΩ(k) +1 eiβj ˆσ ˆΩ(k) +2 e−iβj ˆσ� +, +(10) +with ˆΩ(k) +1,2 being multi-qubits operators which do not de- +pend upon βj, and with e−iβj ˆσ a single qubit rotation +generated by an element ˆσ of the Pauli set. Therefore its +gradient can be written as +∂C(⃗α, ⃗β) +∂βj += i +� +k +tr +�ˆΩ(k) +1 eiβj ˆσ[ˆσ, ˆΩ(k) +2 ]e−iβj ˆσ� += f(βj + π +4 ) − f(βj − π +4 ) , +(11) +where in the last passage we used the identity +i[ˆσ, ˆΩ(k) +2 ] = ei π +4 ˆσ ˆ +Ω2 +(k)e−i π +4 ˆσ − e−i π +4 ˆσ ˆ +Ω2 +(k)ei π +4 ˆσ. +(12) +The gradient with respect the vector ⃗α can be computed +similarly. In this case however we observe that, due to +the fact that ˆρ(V (⃗α),W (⃗β)) +QA +(ψ) depends upon the parame- +ters ⃗α via ˆVQA(⃗α) and through its adjoint ˆV † +QA(⃗α), the +dependence of C(⃗α, ⃗β) upon the j-th component of ⃗α is +slightly more complex. Indeed in this case we have +C(⃗α, ⃗β) = g(αj, αj) , +(13) +where g(α(1) +j , α(2) +j ) is the function +g(α(1) +j , α(2) +j ) := +� +k +tr +�ˆΩ(k) +1 eiα(1) +j +ˆσ ˆΩ(k) +2 e−iα(1) +j +ˆσ +(14) +׈Ω(k) +3 eiα(2) +j +ˆσ ˆΩ(k) +4 e−iα(2) +j +ˆσ� +, +with ˆΩ(k) +1,2,3,4 representing multi-qubits operators which +do not depend neither upon α(1) +j +nor α(2) +j . It is important +to stress that g(α(1) +j , α(2) +j ) can be computed using the +same circuit of Fig. 2, by simply replacing the phases αj +of ˆVQA(⃗α) and ˆV † +QA(⃗α) with α(1) +j +and α(2) +j +respectively. +Notice finally that exploiting the identity Eq. (12) we can +write +∂C(⃗α, ⃗β) +∂αj += +∂g(α(1) +j , αj) +∂α(1) +j +����� +α(1) +j +=αj ++ +∂g(αj, α(2) +j ) +∂α(2) +j +����� +α(2) +j +=αj +(15) += g(αj + π +4 , αj) − g(αj − π +4 , αj) ++ g(αj, αj + π +4 ) − g(αj, αj − π +4 ) , +which shows that computing the gradient of C(⃗α, ⃗β) with +respect to αj simply accounts to evaluate the circuit +that express g(α(1) +j , α(2) +j ) for four distinct values of the +parameters. +C. +Noise model +The scheme presented so far can in principle be applied +to arbitrary classes of noises. In our research however +we focused on a specific model that has been extensively +studied in the literature producing explicit examples of +efficient QEC solutions which can be used as a theoretical +benchmark for our variational search. Specifically we +assume Q and A to be respectively a single qubit register +(k = 1) and a two qubit register (n = 3), globally affected +by a given species of single-qubit noise [76, 77]. These +transformations can be represented in terms of a LCPT +map of the form +ΦQA(· · · ) = +n +� +ℓ=0 +ˆK(ℓ) +QA · · · ˆK(ℓ)† +QA , +(16) + +6 +Q +NOISE +io +d l +est +, t +noise, +ep +tsmth +A += +3 +K(3) +QA +Figure 4. Circuital implementation of the noise element of +Fig. 2: here ˆK(ℓ) +QA are weighted unitaries of Eq. (17). +Xπ/2 +Xπ/4 +Q +A +Xπ/2 +B +Q +A +Xπ/2 +Xπ/4 +Xπ/4 +Zπ/2 +Xπ/4 +Xπ/4 +Xπ/4 +Xπ/4 +Xπ/2 +Xπ/2 +Xπ/2 +Zπ/2 +Zπ/2 +Figure 5. Circuital implementations of the ideal transforma- +tions ˆVQA(⃗α) (left) and ˆ +WQAB(⃗β) (right) which allow for exact +noise suppression of a single-qubit bit-flip noise model [i.e. (16) +with ˆσ(ℓ) = ˆσ(ℓ) +1 ] using a quantum register B with r = 2 qubit. +Here H represent Hadamard gates, while the control-element +are C-NOT gates. +with Kraus operators [71] +ˆK(0) +QA := +� +1 − p ˆ1QA , +ˆK(ℓ) +QA := +� p +n ˆσ(ℓ) , +(17) +where for ℓ ∈ {1, · · · , n}, ˆσ(ℓ) is the Pauli operator acting +on the ℓ-th qubit of QA which defines the noise species +we have selected. +For instance in the case we choose +to describe phase-flip noise then ˆσ(ℓ) = ˆσ(ℓ) +3 , while for +describing bit-flip we have ˆσ(ℓ) = ˆσ(ℓ) +1 . Explicit exam- +ples of ˆVQA, ˆWQAB which allow for exact suppression +of the noise (F(V, W) = 1) are shown in Fig. 5. No- +tice that by construction the circuit parametrization of +ˆVQA(⃗α), ˆWQAB(⃗β) given in Fig. 2 include such gates as +special solution: accordingly if properly guided by an +efficient cost function, our numerical VQA search has a +chance to find the solution of Fig. 5. +IV. +RESULTS +In this section we study the impact of the cost func- +tion on the efficiency of the optimization algorithm of +Sec. III. Assuming the single-qubit noise model detailed +in Sec. III C and taking B to be a r = 2 qubit register, +we run two distinct numerical searches: the first obtained +7 +0 +500 +1,000 +1,500 +2,000 +0 +10 +20 +30 +40 +Iterations +1 � F +W1 +0.82 +0.83 +0.84 +0.85 +0.86 +0.87 +0 +50 +100 +150 +Average fidelity +1 � F +W1 +Figure 7. +Comparison of the the input-output average fi- +delity (3) attainable by running our optimization algorithm us- +ing the cost function C(fid)(~↵, ~�) (blue data) and C(wass)(~↵, ~�) +(orange data). Here the error model is a single-qubit bit-flip +noise (ˆ� = ˆ�1 in (16)) with p = 0.8. The no error correction +threshold (5) of this scheme is F 0 ⇡ 0.822 – orange peak in +the fidelity plot, up to numerical precision. Only the runs +that produced a fidelity of at least F 0 have been included. +For the C(fid)(~↵, ~�) this is 0.2%, while for C(wass)(~↵, ~�) this +corresponds to 29.6%. +to begin with it succeeds in overcoming the threshold F 0 +in one third of the simulations (specifically 40.6% for the +phase-flip noise model and 29.6% for the bit-flip noise +model). Furthermore the algorithm reach convergency +with a number of iterations which are typically smaller +than those required by C(fid)(~↵, ~�). +V. +CONCLUSIONS +TO BE REWRITTEN To summarize, we have +shown a variational quantum algorithm that allows finding +the most suitable error correction procedure for a specific +noise on quantum hardware. We compared the perfor- +mance of two di↵erent versions of this algorithm using two +di↵erent cost functions, the fidelity and an approximation +of the Wasserstein distance of order one. We compared +the di↵erence in speed and the ability to obtain a useful +solution between the two algorithms, finding really di↵er- +ent trends between the two optimization procedures. The +optimization process based on the fidelity su↵ers greatly +from the phenomenon of the barren plateaus, leading to +very slow convergence or no convergence at all, while the +algorithm based on the W1 approximation allows us to +find the configurations that correct the errors, at least in +the examples that we explored. The results obtained are +still not enough to use this method as a silver bullet to +handle this problem, but show a clear improvement and +allow us to explore further improvements of these meth- +ods, like using di↵erent algorithms for the minimization +process, like stochastic gradient descent or higher-order +algorithms like Newton or pseudo-Newton algorithms. +Given that the gradient can be expressed only with the +cost function evaluated in a small number of circuits that +di↵er only for the parameter choice, allows us to compute +the gradient of the cost function on the same hardware +that will be used for the correction procedure. Moreover, +simulating this circuit may be di�cult because of the +exponential scaling of the dimension of the Hilbert space +of a set of qubits, but this problem does not apply when +all the circuit is built on hardware, gaining a quantum +advantage. +Moreover, given that the gradient of the cost function +can be expressed in terms of the same cost function, the +same procedure can be iterated to compute the exact Hes- +sian of the cost function and then apply a second-order +method like the Newton method as a descent algorithm. +However, this has not been done because the circuits that +we marked as useful have a relatively big number of pa- +rameters, and computing the hessian scales quadratically +with this number, leading to intractable computations. A +second-order pseudo-newton method may improve conver- +gence speed once the algorithm has got near convergence +and this is a good idea for future developments. +Acknowledgments +FZ and VG acknowledge financial support by MIUR +(Ministero dell’ Istruzione, dell’ Universit`a della Ricerca) +by PRIN 2017 Taming complexity via Quantum Strate- +gies: a Hybrid Integrated Photonic approach (QUSHIP) +Id. +2017SRN-BRK, and via project PRO3 Quantum +Pathfinder. GDP is a member of the “Gruppo Nazionale +per la Fisica Matematica (GNFM)” of the “Istituto +Nazionale di Alta Matematica “Francesco Severi” (IN- +(M) +Number of simulations +Number of simulations +C(fid) +C(wass) +C(fid) +C(wass) +F0 +Figure 6. +Comparison of the the input-output average fi- +delity (3) attainable by running our optimization algorithm us- +ing the cost function C(fid)(⃗α, ⃗β) (blue data) and C(wass)(⃗α, ⃗β) +(orange data). Here the error model is a single-qubit bit-flip +noise (ˆσ = ˆσ1 in (16)) with p = 0.8. The no error correction +threshold (5) of this scheme is F 0 ≈ 0.822 – orange peak in +the fidelity plot, up to numerical precision. Only the runs +that produced a fidelity of at least F 0 have been included. +For the C(fid)(⃗α, ⃗β) this is 0.2%, while for C(wass)(⃗α, ⃗β) this +corresponds to 29.6%. +by identifying C(⃗α, ⃗β) with C(fid)(⃗α, ⃗β) and the second +choosing instead C(wass)(⃗α, ⃗β). Results are reported in +Figs. 6 and 7 for two different choices of the noise mod- +els (16), i.e. phase-flip and bit-flip. For both we compare +the input-output average fidelity (3) at the end of the +procedure obtained with the two different cost functions, +and the number of iterations M needed for convergence. +Regarding this last quantity we set a maximum value +Mmax equal to 2000 before convergence and we chose this + +7 +limit mainly with practical choices like the maximum time +for the simulation, enforcing that a single run does not +require more than a few hours of computational time: +in case the algorithm fails to reach the convergency we +simply stop the numerical search (this is the reason for +the peak at the end of the upper orange plot in Fig. 7). +The plots report only the simulations that manage to +achieve an average fidelity that is greater or equal than +no-correction threshold bound F 0. +The first thing to observe is that for both noise mod- +els, C(fid)(⃗α, ⃗β) has problem in reaching the do-nothing +threshold F 0: the probability of success being 2.6% for +the phase-flip case of Fig. 7 and only 0.2% for the bit-flip +case of Fig. 6 (for both noise models the total number of +simulations analyzed was 500). Observe also that in this +last case the algorithm never yields average input-output +fidelity values strictly larger than F 0 and that, even in +those cases, it requires a number M of iterations which +saturate the maximum allow value Mmax (blue peak in the +upper plot of Fig. 7). C(was)(⃗α, ⃗β) performs definitely bet- +ter: to begin with it succeeds in overcoming the threshold +F 0 in one third of the simulations (specifically 40.6% for +the phase-flip noise model and 29.6% for the bit-flip noise +model). Furthermore the algorithm reach convergency +with a number of iterations which are typically smaller +than those required by C(fid)(⃗α, ⃗β). +To better enlighten the differences between the two cost +functions, we proceeded with further simulations, whose +results are summarized in Fig. 8. The idea here is to run a +two-step optimization process composed by two sequences +of runs: in the first run we start the optimization proce- +dure from a random point in the parameter space (⃗α, ⃗β) +with one of the two cost functions (say C(fid)(⃗α, ⃗β)), up +to convergence; after that we start a second optimization +run using the other cost function (say C(wass)(⃗α, ⃗β)) but +assuming as initial condition for the parameters the final +point reached by the first run. The plots report the dif- +ference in fidelity between the second and the first run: +when we start using the C(wass)(⃗α, ⃗β) in the first run, the +fidelity cannot further improve the result that is already +found, and this is represented by the fact that the best +improvement is of the order of 10−5; on the contrary if +we started employing C(fid)(⃗α, ⃗β) in the first run, the use +of C(wass)(⃗α, ⃗β) in the second run typically yields sub- +stantial improvements of the performance4. Moreover, we +sampled some single descent processes and plotted the +cost in function of the iteration. When we move from +y w +cte +vec +0 +500 +1,000 +1,500 +2,000 +0 +20 +40 +60 +Iterations +1 � F +W1 +0.85 +0.9 +0.95 +1 +0 +50 +100 +150 +Average fidelity +1 � F +W1 +(M) +Number of simulations +Number of simulations +C(fid) +C(wass) +C(fid) +C(wass) +F0 +Figure 7. +Comparison of the the input-output average fi- +delity (3) attainable by running our optimization algorithm us- +ing the cost function C(fid)(⃗α, ⃗β) (blue data) and C(wass)(⃗α, ⃗β) +(orange data). Here the error model is a single-qubit phase-flip +noise (ˆσ = ˆσ3 in (16) with p = 0.8. The no error correction +threshold (5) of this scheme is F 0 ≈ 0.822 – orange peak in +the fidelity plot, up to numerical precision. Only the runs +that produced a fidelity of at least F 0 have been included. +For the C(fid)(⃗α, ⃗β) this is 2.6%, while for C(wass)(⃗α, ⃗β) this +corresponds to 40.6%. +fidelity to W1, the descent part after the change of cost +function is qualitatively indistinguishable from starting +from a random point. +4 It has to be said that in few cases the figure of merit is worse +after the second optimization – see the negative bar in right panel +of Fig. 8. This is due to the fact that when using C(wass)(⃗α, ⃗β) +we are not maximizing the fidelity but minimizing a function +whose stationary point corresponds to the maximum of the latter: +accordingly the final point of convergence for C(wass)(⃗α, ⃗β) can +be slightly off mark in terms of fidelity. This is not a problem +because these two functions do not have a constant ratio, and we +checked that the inequalities between them are still satisfied. + +8 +0 +1 +2 +3 +4 +5 +·10−5 +0 +100 +200 +300 +400 +500 +Average fidelity +Number of simulations +0 +0.2 +0.4 +0.6 +0.8 +0 +10 +20 +30 +40 +50 +Average fidelity +Number of simulations +Figure 8. Improvement of simulations when changing the cost function in a two run optimization process that uses different cost +functions to drive the descent algorithm. In the left plot, we started the descent on a random initial point, ran the optimization +using C(wass)(⃗α, ⃗β) as cost function until convergence and then we started the descent algorithm again but using C(fid)(⃗α, ⃗β) +as cost function, starting from the final point of the previous descent. In the right part, the roles of the two cost functions +are inverted (we start using C(fid)(⃗α, ⃗β) and then we use C(wass)(⃗α, ⃗β)). The histograms represent the difference in average +input-output fidelity (2) after the change of cost function, namely the difference between the fidelity achieved after the second +descent and the fidelity after the first descent (positive values correspond to improved performances).Please notice the scale +difference on the x-axis between the left and right plot. +V. +CONCLUSIONS +To summarize, we have shown a variational quantum al- +gorithm that allows finding the most suitable error correc- +tion procedure for a specific noise on quantum hardware. +We compared the performance of two different versions +of this algorithm using two different cost functions, the +fidelity and an approximation of the quantum Wasser- +stein distance of order one. We compared the difference +in speed and the ability to obtain a useful solution be- +tween the two algorithms, finding really different trends +between the two optimization procedures. The optimiza- +tion process based on the fidelity suffers greatly from the +phenomenon of the barren plateaus, leading to very slow +convergence or no convergence at all, while the algorithm +based on the quantum W1 distance allows us to find the +configurations that correct the errors in the examples +that we explored. The obtained results show a clear im- +provement and allow us to explore further improvements +of these methods, as using different algorithms for the +minimization process, e.g. stochastic gradient descent or +higher-order algorithms like Newton or pseudo-Newton +algorithms. +Given that the gradient can be expressed only with +the cost function evaluated in a small number of circuits +that differ only in the parameter choice, the gradient of +the cost function can be computed on the same hardware +that will be used for the correction procedure. Moreover, +simulating this circuit may be difficult because of the +exponential scaling of the dimension of the Hilbert space +of a set of qubits, but this problem does not apply when +all the circuit is built on hardware, gaining a quantum +advantage. For the same reason, the same procedure +can be iterated to compute the exact Hessian of the cost +function and then apply a second-order method like the +Newton method as a descent algorithm. However, this +has not been done because the circuits that we marked +as useful have a relatively big number of parameters, +and computing the hessian scales quadratically with this +number, leading to intractable computations. +Acknowledgments +FZ and VG acknowledge financial support by MIUR +(Ministero dell’ Istruzione, dell’ Universit`a della Ricerca) +by PRIN 2017 Taming complexity via Quantum Strate- +gies: a Hybrid Integrated Photonic approach (QUSHIP) +Id. +2017SRN-BRK, and via project PRO3 Quantum +Pathfinder. 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Negrevergne, +Benchmarking quantum computers: The five-qubit error +correcting code, Physical Review Letters 86, 5811 (2001). + diff --git a/1dE4T4oBgHgl3EQfzQ2-/content/tmp_files/load_file.txt b/1dE4T4oBgHgl3EQfzQ2-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c9e99e0514cc040c3acd9f1e3ff487ec19a4dc5f --- /dev/null +++ b/1dE4T4oBgHgl3EQfzQ2-/content/tmp_files/load_file.txt @@ -0,0 +1,731 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf,len=730 +page_content='Improving the speed of variational quantum algorithms for quantum error correction Fabio Zoratti Scuola Normale Superiore, I-56126 Pisa, Italy Giacomo De Palma Department of Mathematics, University of Bologna, 40126 Bologna, Italy Vittorio Giovannetti NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy We consider the problem of devising a suitable quantum error correction procedure for a generic quantum noise acting on a quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' In general, there is no analytic universal procedure to obtain the encoding and correction unitary gates, and the problem is even harder if the noise is unknown and has to be reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The existing procedures rely on variational quantum algorithms and are very difficult to train since the size of the gradient of the cost function decays exponentially with the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' We address this problem using a cost function based on the quantum Wasserstein distance of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Our results show that such cost function significantly increases both the probability of a successful training and the fidelity of the recovered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' INTRODUCTION Performing reliable computations on physical imperfect hardware is something that has become usual nowadays, given the current state of classical computers, which can produce perfect results without any software-side miti- gation of the imperfections of the physical media where the computation happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Error correction is based on the fact that these machines perform automatically, on the hardware side, procedures that allow errors to hap- pen and to be fixed without any intervention of the end user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' This kind of setting is even more crucial in a quan- tum scenario where the current noisy intermediate-scale quantum computers (NISQ) have a much larger error rate than their classical counterparts [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Performing re- liable computations with a trustworthy error correction procedure has direct implications not only in quantum computation [2, 3], but also in quantum key distribution and cryptography [4–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' In the typical Quantum Error Correction (QEC) scheme, the quantum state that has to be protected is stored in a subspace of a larger Hilbert space, using an encoding procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Stabilizer codes [9], which are within the best analytical results in this field, are not universal because they are tailored for a generic noise acting on a small but unknown subset of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Several attempts have already been made to create a numerical optimiza- tion procedure to find an error correction code for specific noise models [10–13], but these studies are not universal because they rely heavily on the type of noise on the specific quantum circuit and this is a problem because real quantum devices are not characterized by a single kind of quantum noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Some attempts have been made to characterize the noise of the current and near-term devices [14, 15], but these methods will become very diffi- cult to implement soon because classical computers are not able to simulate efficiently quantum circuits when the number of qubit increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Near-term devices with approximately 50 qubits may already be intractable to simulate for supercomputers [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' If we define a figure of merit of the quality of the state after the action of the noise and its corresponding cor- rection, the obvious choice for the kind of maximization algorithm is a Variational Quantum Algorithm [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' These are hybrid algorithms that couple a quantum computer with a classical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' In this kind of algorithms, usually, a parametric quantum circuit is applied to some reference state, some measurements are performed on the system, and the outcomes are given to the classical computer to perform a minimization procedure of a given cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Some interesting examples of this class of algorithms are the variational quantum eigensolver [18] and the Quan- tum Approximate Optimization Algorithm (QAOA) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The optimization procedure in a variational quantum al- gorithm can be seen as the training phase in machine learning, for example, to train a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Some variational quantum algorithms applied to quan- tum error correction are already known in literature [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' However, VQAs usually suffer from the phenomenon of barren plateaus [21, 22], namely the gradient of the cost function decays exponentially with respect to the number of qubits of the system, leading to an untrainable model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Some general results have already been found about this topic [22], which justifies the presence of barren plateaus when the cost function is a global function of the quantum circuit, namely the measurement of a highly non-local op- erator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' For this reason, we compared the performance of an algorithm inspired by [20] using two different cost func- tions: the fidelity and an approximation of the quantum Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The quantum Wasserstein distance is a generalization of the classical Wasserstein distance between probability dis- tributions [23–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Several quantum generalizations of the Wasserstein distance have been proposed [28–62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' This arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='05273v1 [quant-ph] 12 Jan 2023 2 work is based on the quantum Wasserstein distance of or- der 1 (or quantum W1 distance) proposed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' [63, 64], which is not unitarily invariant and recovers the Hamming distance [65] for the states of the computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' We expect this new distance to improve the barren plateau phenomenon and we can give an intuitive reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' If we use a unitarily invariant distance as the trace distance or the distances derived from the fidelity, all the states of the computational basis are equally orthogonal and thus have all maximum distance one with respect to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The quantum W1 distance instead measures how many qubits are different between the two states and this allows the gradient to be less flat in the regions that are not already very close to a local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Indeed, the quantum W1 distance has succesfully been employed as cost function of quantum Generative Adversarial Networks [64, 66–70] The manuscript is organized as follows: in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' II we present some basic notion on conventional QEC proce- dures which allow us to set the notation and the theo- retical background;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' III we introduce our VQA discussing the different choices of cost functions that can be used in order to guide it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' IV we present our nu- merical results where comparing the performances of the VQA implemented with different types of cost functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Conclusions are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' PRELIMINARIES ON QEC Let Q be a quantum register we wish to protect (at least in part) from the action of some external noise source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' In a typical QEC scenario [71] this problem is addressed through the following three-step procedure: i) Before the action the noise, a unitary encoding gate ˆVQA is used to distribute the information originally contained in Q on the larger system QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Here A is an auxiliary quantum register that is assumed to be intialized in a fiduciary quantum state, and that is affected by the same noise that tampers with Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' ii) After the action of the noise a measurement on QA is performed to reveal the nature of the latter and, based on the associated outcome, a unitary recovery operation is applied to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Equivalently this step can be described by introducing yet an extra quantum register B (also intialized on a fiduciary state but not affected by the noise) that is coupled with QA trough a recovering unitary transformation ˆWQAB which effectively mimics the measurement and the recovery operation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' iii) The inverse of the gate ˆVQA is finally used on QA to refocus the recovered information in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Denoting with |ψ⟩Q the input state of Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' the corre- sponding output state of QA that emerges from the pro- cess at the end of the step iii) can be expressed as the density matrix ˆρ(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='W ) QA (ψ) := trB � V† QA ◦ WQAB ◦ ΦQA (1) VQA � |ψ⟩Q⟨ψ| ⊗ |Ø⟩A⟨Ø| ⊗ |Ø⟩B⟨Ø| �� := V† QA ◦ Φ(R) QA ◦ ΦQA ◦ VQA � |ψ⟩Q⟨ψ| ⊗ |Ø⟩A⟨Ø| � where |Ø⟩X represents the fiduciary state of the X register,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' trB{· · · } is the partial trace over B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' and given a unitary ˆUX on X we adopted the symbol UX(· · · ) := ˆUX · · · ˆU † X to denote its action as super-operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' In the above expressions ΦQA is the LCPT quantum channel [71] describing the noise on Q and A, while Φ(R) QA(· · · ) := trB{WQAB(· · ·⊗|Ø⟩B⟨Ø|)} is the LCPT (recovery) quan- tum channel on QA originating from the interaction with B, that attempts to undo the action of ΦQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' An ideal QEC procedure capable to completely remove the noise from the system will make sure that ˆρ(V,W ) QA (ψ) corresponds to |ψ⟩Q|Ø⟩A, irrespectively from the specific choice of |ψ⟩Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' A bona-fide figure of merit to character- ize the effectiveness of a generic QEC scheme is hence provided by the average input-output fidelity F(V, W) := � dµψ Q⟨ψ|A⟨Ø|ˆρ(V,W ) QA (ψ)|ψ⟩Q|Ø⟩A , (2) where dµψ is the uniform measure on the set of the in- put states of Q originated from the Haar measure on the associated unitary group [72] or from an exact or approximate unitary 2-design S [71, 73] that simulates the latter1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Notice that by expressing |ψ⟩Q = ˆUQ|Ø⟩Q, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' (2) can equivalently be casted in the more compact form F(V, W) = QA⟨Ø|ˆρ(V,W ) QA |Ø⟩QA , (3) with |Ø⟩QA := |Ø⟩Q ⊗ |Ø⟩A and where the state ˆρ(V,W ) QA := 1 |S| � ˆUQ∈S U† Q ◦ V† QA ◦ Φ(R) QA ◦ ΦQA VQA ◦ UQ � |Ø⟩QA⟨Ø| � , (4) now includes the average over all possible inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' An ideal QEC procedure will enable one to get F(V, W) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' A natural benchmark for lowest admissible F(V, W) is represented instead by the value one would get if one decides not to perform corrections on the register that we 1 We remind that a unitary 2-design is a probability distribution over the set of unitary operators which can duplicate properties of the probability distribution over the Haar measure for polynomials of degree 2 or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' When Q is a single qubit, a 2-design can be realized by an uniform sampling over a set S composed by only 6 elements ˆ1, ˆσ1, e±iπ/4ˆσ1, e±iπ/4ˆσ2 that maps its logical state |0⟩Q into the vectors |0⟩Q , |1⟩Q , (|0⟩Q ±i |1⟩Q)/ √ 2, (|0⟩Q ∓ |1⟩Q)/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 3 Yα1 Yα7 Xα2 Xα8 Xα9 Yα5 Xα6 Xα10 Xα3 Xα4 Q A Yα11 Xα12 Yα13 Xα14 Xα15 Xα16 Yα17 Xα18 Y /2 π Xθ′ ′ k Xθ′ ′ ′ k Zθ′ k = Xθk Y /2 π B Q A Yβ1 Yβ11 Yβ21 Yβ7 Yβ17 Yβ27 Yβ31 Yβ35 Yβ33 Yβ37 Xβ2 Xβ12 Xβ22 Xβ3 Xβ13 Xβ23 Xβ5 Xβ15 Xβ25 Xβ8 Xβ18 Xβ28 Xβ32 Xβ36 Xβ34 Xβ38 Zβ4 Zβ14 Zβ24 Zβ6 Zβ16 Zβ26 Xβ10 Xβ20 Xβ30 Zβ9 Zβ19 Zβ29 = Zθk Y /2 π Zθ′ ′ k Zθ′ ′ ′ k Zθ′ k Y /2 π Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Parametric gates ˆVQA(⃗α) (green element) and ˆ WQAB(⃗β) (red element) used for case of quantum registers Q, A, and B with k = 1, n − k = 2, and r = 2 qubits respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Indicating with ˆσ1, ˆσ2, and ˆσ3 the Pauli operators, the Xθ, Yθ, and Zθ elements of the figure represent single qubit rotations e−iθˆσ1, e−iθˆσ2, and e−iθˆσ3 with the angles θ determined to the components of the vectors ⃗α, ⃗β, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Vertical lines indicate instead quantum control operations which are activated when the control qubits (indicated by the full or empty circles) are in the logical state |1⟩ (full circle) or in |0⟩ (empty circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' As shown on the inset, each one of those gates depend parametrically upon elements of the control vectors ⃗α and ⃗β through single qubit operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' compute by setting ˆVQA and ˆWQAB equal to the identity operators i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='2 F 0 := QA⟨Ø|ˆρ(11,11) QA |Ø⟩QA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' (5) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' VARIATIONAL QUANTUM ALGORITHM While the enormous progress has been made in the study of QEC procedures, identifying efficient choices for the operations that leads to (non trivial) high values of F(V, W) for a specific noise model, is still a challenging open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' A possible solution in this case is to employ variational quantum algorithms to run numerical searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Our approach follows a training strategy inspired by the work of Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Assuming hence Q, A, and B to be formed by collections of independent qubits (k for Q, n − k for A, and r for B), we introduce a manifold 2 Equation (5) accounts for the noise effects both on Q and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' A more conservative estimation of F 0 can be obtained by focusing directly on the noise on Q alone, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' tracing out the A component of ˆρ(11,11) QA and studying its fidelity with |Ø⟩Q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' F (strong) 0 := Q⟨Ø|ˆρ(11,11) Q |Ø⟩Q ≥ F 0, with ˆρ(11,11) Q := trA ˆρ(11,11) QA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Notice that for the noise model of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' III C the two are directly connected via the identity F 0 = F (strong) 0 − n−1 n p(1 − |⟨0|ˆσ|0⟩|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' ï ï ï ï ï ï |0ï |0ï |0ï |0ï |0ï |0ï |0ï |0ï |0ï Q A B \x03UQ(j) ï ï ï M M M M C( ÷³ , ÷ ´ ) M M \x03U† Q(j) \x03WQAB( ÷ ´ ) \x03VQA( ÷³ ) \x03V† QA( ÷³ ) j ÷³ , ÷ ´ n aˆK ic r t ovth er e w qu we = 1 F cor NOISE Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' (Color online) Sketch of the variational quantum algorithm: Q, A and B are quantum registers formed respec- tively by k, n−k and r qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The initial information we wish to protect is written in Q by the unitary gate ˆUQ(j) extracted from a 2-design set S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' A and B are two auxiliary elements (containing respectively n − k and r qubits) that are used to implement the QEC procedure described by the parametric gates ˆVQA(⃗α), ˆ WQAB(⃗β), and ˆV † QA(⃗α) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The patterned element in the central part of the scheme represents the noise on Q and A (no noise is assumed to be active on B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Lastly, the D-shaped measurements at the end of the circuit represent local measurements on QA whose outcomes over the entire collection of the possible inputs generated by the entire set S, are processed by a classical computer which, evaluating the cost function C(⃗α, ⃗β) defined in Section III B, decides how to update the values of the parameters ⃗α and ⃗β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Thick grey lines in the figure represent classical control lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' of transformations ˆVQA(⃗α), ˆWQAB(⃗β) parametrized by classical controls vectors ⃗α, ⃗β (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 1), and construct the quantum circuit of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The method then proceeds along the following stages: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Having selected the values of ⃗α and ⃗β, the regis- ter Q is prepared into a collection of known quan- tum state {|ψ(1)⟩Q, · · · , |ψ(m)⟩Q} operating on the vector |Ø⟩Q = |0⟩⊗k through action of the con- trol gates ˆUQ(1), · · · , ˆUQ(m) (first cyan element of the figure) which define the 2-design S entering in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Each of such inputs is hence evolved via a circuit (pale-orange area of the figure) that emulates both the effect of the noise (patterned square of the figure, see Section III C and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 4), and the transformations ˆVQA(⃗α), ˆWQAB(⃗β), and ˆV † QA(⃗α) that are meant to implement the steps ii) and iii) of the QEC procedure (green and red ele- ments of the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Notice that in the ideal case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' if ˆVQA(⃗α) and ˆWQAB(⃗β) manage to completely suppress the noise) then in correspondence with 4 the input |ψ(j)⟩Q the registers QA should emerge in the state |ψ(j)⟩Q ⊗ |Ø⟩A := |ψ(j)⟩Q ⊗ |0⟩⊗n−k, which will be hence mapped into the final configu- ration |Ø⟩QA := |0⟩⊗n by the inverse ˆU † Q(j) of the state preparation gate (second cyan element of the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' For each choice of the index j ∈ {1, · · · , m} a mea- surement on the system is performed at the end of the transformations described in the stage 1 and the resulting m collected outcomes used to compute a cost function C(⃗α, ⃗β) which evaluates the effective- ness of the adopted QEC strategy in leading large values of the average input-output fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The spe- cific choice of the cost function is very important and is discussed in Section III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' A classical computer decides, given the results of the measurement, how to change the value of the parameters ⃗α and ⃗β to be used in the subsequent run in order to minimize the cost function C(⃗α, ⃗β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' This is discussed in detail in Section III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Cost function The natural choice for the cost function at the stage 2 of our algorithm is provided by the expectation value of the self-adjoint operator ˆH(fid) QA := ˆ1QA − |Ø⟩QA ⟨Ø| , (6) computed on the mean state of system QA which emerges at the output of the quantum circuit of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' the quantity C(fid)(⃗α, ⃗β) := tr{ˆρ(V (⃗α),W (⃗β)) QA ˆH(fid) QA } , (7) where ˆρ(V (⃗α),W (⃗β)) QA is the density matrix (4) evaluated for ˆVQA = ˆVQA(⃗α) and ˆWQAB = ˆWQAB(⃗β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' This choice has two main advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' First of all, the expectation value C(fid)(⃗α, ⃗β) can be evaluated by performing (sim- ple) local measurement on the qubits of Q and A (in- deed it can be computed by simply checking whether of not each one of them is in the logical state |0⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Most importantly, since by explicit evaluation one has that C(fid)(⃗α, ⃗β) = 1 − F(V (⃗α), W(⃗β)), it is clear that by us- ing (7) the algorithm will be forced to luck for values of ⃗α, ⃗β that yield higher average input-output fidelities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Despite all this, the use of C(fid)(⃗α, ⃗β) as a cost function, has a major drawback associated with the fact that the spectrum of the Hamiltonian ˆH(fid) QA exhibits maximum degeneracy with respect to space orthogonal to the target state |Ø⟩QA (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Due do this fact a numerical search based on a training procedure that simply target the minimization of C(fid)(⃗α, ⃗β), has non trivial chances to get stuck somewhere in the large flat plateau associated with the eigenvalue 1 of ˆH(fid) QA without finding any good direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' in the large flat plateau A possible way to avoid this problem is to introduce new cost-functions Hamilto- nians which, while maintaining the target vector |Ø⟩QA as unique ground state and still being easy to compute, manage to remove the huge degeneracy of the excited part of the spectra of ˆH(fid) QA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Our choice is based on the quantum Wasserstein distance of order 1 (W1) introduced Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' [63] which, even though it lacks some interesting properties that the fidelity has, is less likely to be affected by the barren plateaus phenomena [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' A good estima- tion of the W1 distance that separate ˆρ(V (⃗α),W (⃗β)) QA from the target state, is provided by the following quantity C(wass)(⃗α, ⃗β) := tr{ˆρ(V (⃗α),W (⃗β)) QA ˆH(wass) QA } , (8) ˆH(wass) QA := n � j=1 j ˆΠ(j) QA , (9) where ˆΠ(j) QA represents the sub-space of the register QA in which we have j qubits in |1⟩ and the remaining one in |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Observe that ˆH(wass) QA is nothing but the sum of the number operators acting on the individual qubits of the register QA, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' ˆH(wass) QA = �n ℓ=1 ˆπℓ with ˆπℓ the projector on the logical state |1⟩ of the ℓ-th qubit): accordingly, as C(fid)(⃗α, ⃗β), C(wass)(⃗α, ⃗β) can be computed from local measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' What C(wass)(⃗α, ⃗β) does is to count the total number of logical ones present in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' To understand why using (8) could in principle lead to a more efficient numerical search than the one obtained by using (7), notice that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' (6) can be equivalently written as ˆH(fid) QA = n � j=1 ˆΠ(j) QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' A comparison with (9) reveals hence that indeed while both ˆH(fid) QA and ˆH(wass) QA admit |Ø⟩QA as unique ground state, the Wasserstein Hamiltonian removes large part of the degeneracy of the high energy spectrum of the fidelity Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Accordingly it is reasonable to expect that a numerical search that uses ˆH(wass) QA , has less chances to get trapped into regions of constant energy (barren plateau) than a search based on ˆH(fid) QA ,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 3 It goes without mentioning that alternative choices for the cost function Hamiltonians are also available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' For instance one can use operators that also remove the residual degeneracies that affect ˆH(wass) QA – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' using the operator ˆH(full) QA = �n ℓ=1 wℓˆπℓ with ωℓ positive weights selected so that different allocation of |1⟩ states inside the eigenspaces of ˆH(wass) QA get an assigned ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Our numerical analysis however seems to indicate that these refinement do not contribute significantly in improving numerical search of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 5 eigenvalues of ̂H(fid) QA eigenvalues of ̂H(wass) QA N = 3 N = 3 N = 2 N = 1 N = 2 N = 0 1 2 3 0 N = 1 N = 2 N = 3 N = 0 0 1 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Pictorial rendering of the spectra of the Hamiltonians ˆH(fid) QA (top panel) and ˆH(wass) QA (lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' While ˆH(fid) QA is characterized by a unique, flat plateau that includes all the excited state, ˆH(wass) QA partially removes the associated degeneracy assigning higher energy to subspaces that have higher number of qubits in the logical state |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Descent algorithm The algorithm that we used for this work is a gradient descent algorithm with momentum [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' To overcome the numerical difficulties of using finite differences to estimate the gradients of the cost function C(⃗α, ⃗β), we exploit a variation of the parameter-shift rule introduced in [75] which reduces the problem to compute linear combina- tions of the function itself evaluated in different points that are not infinitesimally close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Specifically we observe that, irrespectively from the choice of the operator ˆHQA, the functional dependence of C(⃗α, ⃗β) upon the j-th com- ponent of the vector ⃗β is of the form C(⃗α, ⃗β) = f(βj) := � k tr �ˆΩ(k) 1 eiβj ˆσ ˆΩ(k) 2 e−iβj ˆσ� , (10) with ˆΩ(k) 1,2 being multi-qubits operators which do not de- pend upon βj, and with e−iβj ˆσ a single qubit rotation generated by an element ˆσ of the Pauli set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Therefore its gradient can be written as ∂C(⃗α, ⃗β) ∂βj = i � k tr �ˆΩ(k) 1 eiβj ˆσ[ˆσ, ˆΩ(k) 2 ]e−iβj ˆσ� = f(βj + π 4 ) − f(βj − π 4 ) , (11) where in the last passage we used the identity i[ˆσ, ˆΩ(k) 2 ] = ei π 4 ˆσ ˆ Ω2 (k)e−i π 4 ˆσ − e−i π 4 ˆσ ˆ Ω2 (k)ei π 4 ˆσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' (12) The gradient with respect the vector ⃗α can be computed similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' In this case however we observe that, due to the fact that ˆρ(V (⃗α),W (⃗β)) QA (ψ) depends upon the parame- ters ⃗α via ˆVQA(⃗α) and through its adjoint ˆV † QA(⃗α), the dependence of C(⃗α, ⃗β) upon the j-th component of ⃗α is slightly more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Indeed in this case we have C(⃗α, ⃗β) = g(αj, αj) , (13) where g(α(1) j , α(2) j ) is the function g(α(1) j , α(2) j ) := � k tr �ˆΩ(k) 1 eiα(1) j ˆσ ˆΩ(k) 2 e−iα(1) j ˆσ (14) ׈Ω(k) 3 eiα(2) j ˆσ ˆΩ(k) 4 e−iα(2) j ˆσ� , with ˆΩ(k) 1,2,3,4 representing multi-qubits operators which do not depend neither upon α(1) j nor α(2) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' It is important to stress that g(α(1) j , α(2) j ) can be computed using the same circuit of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 2, by simply replacing the phases αj of ˆVQA(⃗α) and ˆV † QA(⃗α) with α(1) j and α(2) j respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Notice finally that exploiting the identity Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' (12) we can write ∂C(⃗α, ⃗β) ∂αj = ∂g(α(1) j , αj) ∂α(1) j ����� α(1) j =αj + ∂g(αj, α(2) j ) ∂α(2) j ����� α(2) j =αj (15) = g(αj + π 4 , αj) − g(αj − π 4 , αj) + g(αj, αj + π 4 ) − g(αj, αj − π 4 ) , which shows that computing the gradient of C(⃗α, ⃗β) with respect to αj simply accounts to evaluate the circuit that express g(α(1) j , α(2) j ) for four distinct values of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Noise model The scheme presented so far can in principle be applied to arbitrary classes of noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' In our research however we focused on a specific model that has been extensively studied in the literature producing explicit examples of efficient QEC solutions which can be used as a theoretical benchmark for our variational search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Specifically we assume Q and A to be respectively a single qubit register (k = 1) and a two qubit register (n = 3), globally affected by a given species of single-qubit noise [76, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' These transformations can be represented in terms of a LCPT map of the form ΦQA(· · · ) = n � ℓ=0 ˆK(ℓ) QA · · · ˆK(ℓ)† QA , (16) 6 Q NOISE io d l est , t noise, ep tsmth A = 3 \x03K(3) QA Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Circuital implementation of the noise element of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 2: here ˆK(ℓ) QA are weighted unitaries of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Xπ/2 Xπ/4 Q A Xπ/2 B Q A Xπ/2 Xπ/4 Xπ/4 Zπ/2 Xπ/4 Xπ/4 Xπ/4 Xπ/4 Xπ/2 Xπ/2 Xπ/2 Zπ/2 Zπ/2 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Circuital implementations of the ideal transforma- tions ˆVQA(⃗α) (left) and ˆ WQAB(⃗β) (right) which allow for exact noise suppression of a single-qubit bit-flip noise model [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' (16) with ˆσ(ℓ) = ˆσ(ℓ) 1 ] using a quantum register B with r = 2 qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Here H represent Hadamard gates, while the control-element are C-NOT gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' with Kraus operators [71] ˆK(0) QA := � 1 − p ˆ1QA , ˆK(ℓ) QA := � p n ˆσ(ℓ) , (17) where for ℓ ∈ {1, · · · , n}, ˆσ(ℓ) is the Pauli operator acting on the ℓ-th qubit of QA which defines the noise species we have selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' For instance in the case we choose to describe phase-flip noise then ˆσ(ℓ) = ˆσ(ℓ) 3 , while for describing bit-flip we have ˆσ(ℓ) = ˆσ(ℓ) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Explicit exam- ples of ˆVQA, ˆWQAB which allow for exact suppression of the noise (F(V, W) = 1) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' No- tice that by construction the circuit parametrization of ˆVQA(⃗α), ˆWQAB(⃗β) given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 2 include such gates as special solution: accordingly if properly guided by an efficient cost function, our numerical VQA search has a chance to find the solution of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' RESULTS In this section we study the impact of the cost func- tion on the efficiency of the optimization algorithm of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Assuming the single-qubit noise model detailed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' III C and taking B to be a r = 2 qubit register, we run two distinct numerical searches: the first obtained 7 0 500 1,000 1,500 2,000 0 10 20 30 40 Iterations 1 � F W1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='87 0 50 100 150 Average fidelity 1 � F W1 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Comparison of the the input-output average fi- delity (3) attainable by running our optimization algorithm us- ing the cost function C(fid)(~↵, ~�) (blue data) and C(wass)(~↵, ~�) (orange data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Here the error model is a single-qubit bit-flip noise (ˆ� = ˆ�1 in (16)) with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The no error correction threshold (5) of this scheme is F 0 ⇡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='822 – orange peak in the fidelity plot, up to numerical precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Only the runs that produced a fidelity of at least F 0 have been included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' For the C(fid)(~↵, ~�) this is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='2%, while for C(wass)(~↵, ~�) this corresponds to 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' to begin with it succeeds in overcoming the threshold F 0 in one third of the simulations (specifically 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='6% for the phase-flip noise model and 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='6% for the bit-flip noise model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Furthermore the algorithm reach convergency with a number of iterations which are typically smaller than those required by C(fid)(~↵, ~�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' CONCLUSIONS TO BE REWRITTEN To summarize, we have shown a variational quantum algorithm that allows finding the most suitable error correction procedure for a specific noise on quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' We compared the perfor- mance of two di↵erent versions of this algorithm using two di↵erent cost functions, the fidelity and an approximation of the Wasserstein distance of order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' We compared the di↵erence in speed and the ability to obtain a useful solution between the two algorithms, finding really di↵er- ent trends between the two optimization procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The optimization process based on the fidelity su↵ers greatly from the phenomenon of the barren plateaus, leading to very slow convergence or no convergence at all, while the algorithm based on the W1 approximation allows us to find the configurations that correct the errors, at least in the examples that we explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The results obtained are still not enough to use this method as a silver bullet to handle this problem, but show a clear improvement and allow us to explore further improvements of these meth- ods, like using di↵erent algorithms for the minimization process, like stochastic gradient descent or higher-order algorithms like Newton or pseudo-Newton algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Given that the gradient can be expressed only with the cost function evaluated in a small number of circuits that di↵er only for the parameter choice, allows us to compute the gradient of the cost function on the same hardware that will be used for the correction procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Moreover, simulating this circuit may be di�cult because of the exponential scaling of the dimension of the Hilbert space of a set of qubits, but this problem does not apply when all the circuit is built on hardware, gaining a quantum advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Moreover, given that the gradient of the cost function can be expressed in terms of the same cost function, the same procedure can be iterated to compute the exact Hes- sian of the cost function and then apply a second-order method like the Newton method as a descent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' However, this has not been done because the circuits that we marked as useful have a relatively big number of pa- rameters, and computing the hessian scales quadratically with this number, leading to intractable computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' A second-order pseudo-newton method may improve conver- gence speed once the algorithm has got near convergence and this is a good idea for future developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Acknowledgments FZ and VG acknowledge financial support by MIUR (Ministero dell’ Istruzione, dell’ Universit`a della Ricerca) by PRIN 2017 Taming complexity via Quantum Strate- gies: a Hybrid Integrated Photonic approach (QUSHIP) Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 2017SRN-BRK, and via project PRO3 Quantum Pathfinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' GDP is a member of the “Gruppo Nazionale per la Fisica Matematica (GNFM)” of the “Istituto Nazionale di Alta Matematica “Francesco Severi” (IN- (M) Number of simulations Number of simulations C(fid) C(wass) C(fid) C(wass) F0 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Comparison of the the input-output average fi- delity (3) attainable by running our optimization algorithm us- ing the cost function C(fid)(⃗α, ⃗β) (blue data) and C(wass)(⃗α, ⃗β) (orange data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Here the error model is a single-qubit bit-flip noise (ˆσ = ˆσ1 in (16)) with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The no error correction threshold (5) of this scheme is F 0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='822 – orange peak in the fidelity plot, up to numerical precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Only the runs that produced a fidelity of at least F 0 have been included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' For the C(fid)(⃗α, ⃗β) this is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='2%, while for C(wass)(⃗α, ⃗β) this corresponds to 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' by identifying C(⃗α, ⃗β) with C(fid)(⃗α, ⃗β) and the second choosing instead C(wass)(⃗α, ⃗β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Results are reported in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 6 and 7 for two different choices of the noise mod- els (16), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' phase-flip and bit-flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' For both we compare the input-output average fidelity (3) at the end of the procedure obtained with the two different cost functions, and the number of iterations M needed for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Regarding this last quantity we set a maximum value Mmax equal to 2000 before convergence and we chose this 7 limit mainly with practical choices like the maximum time for the simulation, enforcing that a single run does not require more than a few hours of computational time: in case the algorithm fails to reach the convergency we simply stop the numerical search (this is the reason for the peak at the end of the upper orange plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The plots report only the simulations that manage to achieve an average fidelity that is greater or equal than no-correction threshold bound F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The first thing to observe is that for both noise mod- els, C(fid)(⃗α, ⃗β) has problem in reaching the do-nothing threshold F 0: the probability of success being 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='6% for the phase-flip case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 7 and only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='2% for the bit-flip case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 6 (for both noise models the total number of simulations analyzed was 500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Observe also that in this last case the algorithm never yields average input-output fidelity values strictly larger than F 0 and that, even in those cases, it requires a number M of iterations which saturate the maximum allow value Mmax (blue peak in the upper plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' C(was)(⃗α, ⃗β) performs definitely bet- ter: to begin with it succeeds in overcoming the threshold F 0 in one third of the simulations (specifically 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='6% for the phase-flip noise model and 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='6% for the bit-flip noise model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Furthermore the algorithm reach convergency with a number of iterations which are typically smaller than those required by C(fid)(⃗α, ⃗β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' To better enlighten the differences between the two cost functions, we proceeded with further simulations, whose results are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The idea here is to run a two-step optimization process composed by two sequences of runs: in the first run we start the optimization proce- dure from a random point in the parameter space (⃗α, ⃗β) with one of the two cost functions (say C(fid)(⃗α, ⃗β)), up to convergence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' after that we start a second optimization run using the other cost function (say C(wass)(⃗α, ⃗β)) but assuming as initial condition for the parameters the final point reached by the first run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The plots report the dif- ference in fidelity between the second and the first run: when we start using the C(wass)(⃗α, ⃗β) in the first run, the fidelity cannot further improve the result that is already found, and this is represented by the fact that the best improvement is of the order of 10−5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' on the contrary if we started employing C(fid)(⃗α, ⃗β) in the first run, the use of C(wass)(⃗α, ⃗β) in the second run typically yields sub- stantial improvements of the performance4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Moreover, we sampled some single descent processes and plotted the cost in function of the iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' When we move from y w cte vec 0 500 1,000 1,500 2,000 0 20 40 60 Iterations 1 � F W1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='95 1 0 50 100 150 Average fidelity 1 � F W1 (M) Number of simulations Number of simulations C(fid) C(wass) C(fid) C(wass) F0 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Comparison of the the input-output average fi- delity (3) attainable by running our optimization algorithm us- ing the cost function C(fid)(⃗α, ⃗β) (blue data) and C(wass)(⃗α, ⃗β) (orange data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Here the error model is a single-qubit phase-flip noise (ˆσ = ˆσ3 in (16) with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The no error correction threshold (5) of this scheme is F 0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='822 – orange peak in the fidelity plot, up to numerical precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Only the runs that produced a fidelity of at least F 0 have been included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' For the C(fid)(⃗α, ⃗β) this is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='6%, while for C(wass)(⃗α, ⃗β) this corresponds to 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' fidelity to W1, the descent part after the change of cost function is qualitatively indistinguishable from starting from a random point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 4 It has to be said that in few cases the figure of merit is worse after the second optimization – see the negative bar in right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' This is due to the fact that when using C(wass)(⃗α, ⃗β) we are not maximizing the fidelity but minimizing a function whose stationary point corresponds to the maximum of the latter: accordingly the final point of convergence for C(wass)(⃗α, ⃗β) can be slightly off mark in terms of fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' This is not a problem because these two functions do not have a constant ratio, and we checked that the inequalities between them are still satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 8 0 1 2 3 4 5 10−5 0 100 200 300 400 500 Average fidelity Number of simulations 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='8 0 10 20 30 40 50 Average fidelity Number of simulations Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Improvement of simulations when changing the cost function in a two run optimization process that uses different cost functions to drive the descent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' In the left plot, we started the descent on a random initial point, ran the optimization using C(wass)(⃗α, ⃗β) as cost function until convergence and then we started the descent algorithm again but using C(fid)(⃗α, ⃗β) as cost function, starting from the final point of the previous descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' In the right part, the roles of the two cost functions are inverted (we start using C(fid)(⃗α, ⃗β) and then we use C(wass)(⃗α, ⃗β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The histograms represent the difference in average input-output fidelity (2) after the change of cost function, namely the difference between the fidelity achieved after the second descent and the fidelity after the first descent (positive values correspond to improved performances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='Please notice the scale difference on the x-axis between the left and right plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' CONCLUSIONS To summarize, we have shown a variational quantum al- gorithm that allows finding the most suitable error correc- tion procedure for a specific noise on quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' We compared the performance of two different versions of this algorithm using two different cost functions, the fidelity and an approximation of the quantum Wasser- stein distance of order one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' We compared the difference in speed and the ability to obtain a useful solution be- tween the two algorithms, finding really different trends between the two optimization procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The optimiza- tion process based on the fidelity suffers greatly from the phenomenon of the barren plateaus, leading to very slow convergence or no convergence at all, while the algorithm based on the quantum W1 distance allows us to find the configurations that correct the errors in the examples that we explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' The obtained results show a clear im- provement and allow us to explore further improvements of these methods, as using different algorithms for the minimization process, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' stochastic gradient descent or higher-order algorithms like Newton or pseudo-Newton algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Given that the gradient can be expressed only with the cost function evaluated in a small number of circuits that differ only in the parameter choice, the gradient of the cost function can be computed on the same hardware that will be used for the correction procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Moreover, simulating this circuit may be difficult because of the exponential scaling of the dimension of the Hilbert space of a set of qubits, but this problem does not apply when all the circuit is built on hardware, gaining a quantum advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' For the same reason, the same procedure can be iterated to compute the exact Hessian of the cost function and then apply a second-order method like the Newton method as a descent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' However, this has not been done because the circuits that we marked as useful have a relatively big number of parameters, and computing the hessian scales quadratically with this number, leading to intractable computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' Acknowledgments FZ and VG acknowledge financial support by MIUR (Ministero dell’ Istruzione, dell’ Universit`a della Ricerca) by PRIN 2017 Taming complexity via Quantum Strate- gies: a Hybrid Integrated Photonic approach (QUSHIP) Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' 2017SRN-BRK, and via project PRO3 Quantum Pathfinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE4T4oBgHgl3EQfzQ2-/content/2301.05273v1.pdf'} +page_content=' GDP is a member of the “Gruppo Nazionale per la Fisica Matematica (GNFM)” of the “Istituto Nazionale di Alta Matematica “Francesco Severi” (IN- dAM)”.' metadata={'source': 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of Theorem 1.6 +8 +5. +Further remarks +9 +References +9 +Abstract. In this short paper, we prove Strichartz estimates for N-body Schr¨odinger +equations in the waveguide manifold setting (i.e. on semiperiodic spaces Rm ×Tn +where m ≥ 3), provided that interaction potentials are small enough (depending +on the number of the particles and the universal constants, not on the initial +data). The proof combines both the ideas of Tzvetkov-Visciglia [29] and Hong +[17]. As an immediate application, the scattering asymptotics for this model is +also obtained. This result extends Hong [17] to the waveguide case. +Keywords: Strichartz estimate, many body Schr¨odinger equations, scattering, waveg- +uide manifolds +Mathematics Subject Classification (2020) Primary: 35Q55; Secondary: 35R01, +37K06, 37L50. +1. introduction +1.1. Background and Motivations. Let d = m + n, m ≥ 3, n ≥ 1 and N ≥ 1. We +consider the many body Schr¨odinger equations in the waveguide setting as follows, +(1.1) +(i∂t + HN)u(x1, ...xN) = 0, +u(0, x1, ...xN) = u0(x1, ...xN) ∈ L2 +x1,...xN, +where HN = ∆x − VN = �N +α=1 ∆xα − � +1≤α<β≤N V (xα − xβ), +and α-th particle xα ∈ Rm × Tn for any α ∈ [1, ..., N]. +From physical explanations, N ≥ 1 indicates the number of particles in a quantum +system (which is often very large) and the interacting potentials of form V (xα − +xβ) indicates the interactions of any two particles, which depends on their relative +distance. Moreover, the product spaces of form Rm × Tn is known as semi-periodic +space or waveguide manifold. +d = m + n is the whole dimension while m is the +dimension for the Euclidean component and n is the dimension for the tori component. +When N = 1, initial value problem(1.1) is exactly the standard nonlinear Schr¨odinger +equation (NLS) with a potential, which has been well studied (in the Euclidean case, +i.e. replacing Rm×Tn by Rd). It is also known as ‘the one-body case’ and the research +on the decay properties has a long history (see the Introduction of [17], the survey +[25] and the references therein). In this paper, we mainly concern the general case +(N ≥ 1 can be arbitrarily large), i.e. the many body Schr¨odinger case, which will +involve some new difficulties than the single body case such as the issue of interacting +potentials. +1 + +2 +ZEHUA ZHAO +The purpose of this paper is to investigate time decay properties of solutions to +the N-body Schr¨odinger equation (1.1) in the waveguide setting. In particular, we +discuss the Strichartz-type estimate and the scattering behavior for (1.1). We note +that the Euclidean case of (1.1) has been studied in [17]. (See also [7] for a recently +result which deals with the two body case via the scheme of [20].) +We intend to +generalize [17] to the waveguide case. That is one main motivation of this paper. +Another motivation is the recent developments for the topic: ‘Long time behavior for +NLS on waveguides’ so the author is interested in combining both of ‘waveguides’ and +‘many body Schr¨odinger equations’ together, i.e. studying the estimates and the long +time behavior for many body Schr¨odinger equations on waveguides. We will briefly +mention the background for ‘NLS on waveguides’ in the next paragraph. +Waveguide manifolds of form Rm ×Tn are of particular interest in nonlinear optics +of telecommunications. Generally, well-posedness theory and long time behavior of +NLS are hot topics in the area of dispersive equations and have been studied widely +in recent decades. Naturally, the Euclidean case is first treated and the theory, at +least in the defocusing setting, has been well established. +We refer to [9, 10, 21] +for some typical Euclidean results. Moreover, we refer to [5, 4, 14, 15, 16, 18, 19, +22, 31, 32, 34, 35, 36] with regard to the torus and waveguide settings. (See also +[24, 27, 33] for other dispersive equations on waveguides.) One may roughly think +of the waveguide case as the “intermediate point” between the Euclidean case and +the torus case since the waveguide manifold is a product of Euclidean spaces and the +tori. The techniques used in Euclidean and torus settings are frequently combined +and applied to the waveguides problems. At last, we refer to [2, 11, 28] for some +classical textbooks on the study of NLS. +Since the current paper concerns the estimates and the PDE-level of (1.1) rather +than the mathematical physics level, we will not mention too much for the background +of the many body problems/equations from physical perspectives. We refer to the +Introductions of [3, 6, 7, 8, 12, 26] and the references therein for more information. +To the authors’ best knowledge, the current paper is the first result towards un- +derstanding long time dynamics for the many body Schr¨odinger equations within the +context of waveguides. +As last, we note that, as in [17], we need to assume some smallness for the potential +V and this smallness does not depend on the initial data (only depends on the particle +number N and the universal constant). +1.2. The statement of main results. Now we are ready to state the two main +results of this paper. We start with the Strichartz estimate as follows since the other +one is an application of it. +Theorem 1.1 (Strichartz estimate). Let m ≥ 3, n ≥ 1 and 1 < p < 2. There exists +a small number ǫ such that if ∥V ∥ +L +d +2 ,∞ +y +L2z +≤ +ǫ +N 2 , then +(1.2) +∥1[0,+∞)e−itHN u0∥V p +∆x ≲ ∥u0∥L2x. +Remark 1.2. Here V p +∆x-norm (known as variation spaces) is introduced by Koch- +Tataru [23] (see Section 2 for discussions). See also [15, 16, 18, 19] for more information +and some other applications. +In viewing of the properties of V p +∆x-type spaces, Theorem 1.1 directly implies +Corollary 1.3. Let m ≥ 3 and n ≥ 1. There exists a small number ǫ such that if +∥V ∥ +L +d +2 ,∞ +y +L2z +≤ +ǫ +N 2 , then for any m-dimensional admissible pair (q, r) and 1 ≤ α ≤ N, +we have +(1.3) +∥e−itHNu0∥Lq +tLryαL2zαL2 +ˆxα ≲ ∥u0∥L2x, + +MANY BODY SCHR ¨ODINGER EQUATION ON WAVEGUIDE MANIFOLDS +3 +where ˆxα is the N − 1 spatial variables except the α-th variable xα, i.e., +(1.4) +ˆxα = (x1, ...xα−1, xα+1..., xN) ∈ Rd(N−1), +and xα is the α-th variable with Euclidean component yα and tori component zα, i.e., +(1.5) +xα = (yα, zα) ∈ Rm × Tn. +Moreover, for any mN-dimensional admissible pair (q, r), we have +(1.6) +∥e−itHNu0∥Lq +tLryL2z ≲ ∥u0∥L2x, +where y is for the whole Euclidean component (mN-dimensional) and z is for the +whole tori component nN-dimensional. +Remark 1.4. See Theorem 1.1 and Theorem 1.2 in [17] for the Euclidean case. We will +give the proof for Corollary 1.3 after the proof of Theorem 1.1 in the end of Section +3. +Remark 1.5. As shown above, the formulation of the Strichartz estimates for (1.1) +combines both the ideas of [29] and [17]. As in [29] (see also [30]), we fix the tori +component by using L2-norm. (In other words, one decomposes the function along +the tori direction and derive the Strichartz estimate using the dispersion from the Eu- +clidean direction.) As in [17], we fix other particles by only considering the dispersion +of one certain particle. Thus, we consider the dispersion of the Euclidean component +of one particle; fixing other particles and the tori component of this particle by using +L2-norm. +As a direct application of Theorem 1.1, we show the scattering behavior for an +N-body Schr¨odinger operator with rough small interactions in the following sense, +Theorem 1.6 (Scattering). Let m ≥ 3, n ≥ 1 and 1 < p < 2. let ǫ be a small +constant given in Theorem 1.1. If ∥V ∥ +L +m +2 ,∞ +y +L2 +z ≤ +ǫ +N 2 , then for each u0 ∈ L2 +x, there +exist scattering states u± such that +(1.7) +lim +t→±∞ +��e−itHN u0 − eit∆xu± +�� +L2x = 0. +Remark 1.7. We note that for the tori case of (1.1), the scattering behavior is not +expected due to the lack of dispersion, though a Strichartz estimate can still be +possibly obtained with suitable modifications. We leave it for interested readers. +Remark 1.8. For the above results, the dimension of the tori component n ≥ 1 does +not matter. (When n = 0, it is exactly the Euclidean case [17]). However, if one +considers the long time dynamics of a nonlinear problem on waveguide manifolds, +the dimension of the tori component often matters a lot. In general, the difficulty of +the critical NLS problem on waveguide manifolds increases if the whole dimension is +increased or if the Euclidean component is decreased. See the Introductions in [18, 19] +for more information. +Remark 1.9. To be more general, the tori component Tn in (1.1) can be generalized +to a compact Riemannian manifold M such that Theorem 1.1, Corollary 1.3 and +Theorem 1.6 still hold. +Next, we briefly introduce the main strategy of the proofs for Theorem 1.1, Corol- +lary 1.3 and Theorem 1.6. In fact, the proofs for Corollary 1.3 and Theorem 1.6 are +standard and less complicated. Corollary 1.3 follows from Theorem 1.1 (see Section +3) according to the transfer principle of the function space V p +∆x. Theorem 1.6 also +follows from Theorem 1.1 (see Section 4), together with some other basic estimates +like in [17]. Thus we will focus on the proof of Theorem 1.1 as follows. +The proof of Theorem 1.1 (Strichartz estimate) is based on the properties of func- +tion space V p +∆x and a perturbation method (see Section 3). The main idea is: one + +4 +ZEHUA ZHAO +establishes nonlinear estimate for one arbitrary interacting potential (treating it as +a perturbation) and then sum them up. The key estimate is Proposition 3.1 which +deals with one arbitrary interacting potential by regarding it as a forcing term. With +the help of it, one can handle all of the interacting potentials by treating them as +perturbations. Eventually, according to the smallness assumption, one can use per- +turbation method to show the Strichartz estimate as desired. Compared with the +single potential case (N = 1), the interacting potentials (involves rotations) cause dif- +ficulties thus the ‘rotation flexible’ function space V p +∆x is needed; compared with the +Euclidean analogue ([17]), the new difference is the appearance of the tori component. +1.3. Structure of this paper. The rest of the article is organized as follows. In +Section 2, we discuss function spaces and some estimates for this model; in Section +3, we give the proof for Theorem 1.1 (Strichartz estimate); in Section 4, we give the +proof for Theorem 1.6 (scattering asymptotics); in Section 5, we give a few further +remarks on this research line. +1.4. Notations. We write A ≲ B to say that there is a constant C such that A ≤ CB. +We use A ≃ B when A ≲ B ≲ A. Particularly, we write A ≲u B to express that +A ≤ C(u)B for some constant C(u) depending on u. We use C for universal constants +and N for the number of particles. +We say that the pair (p, q) is d-(Strichartz) admissible if +(1.8) +2 +p + d +q = d +2, +2 ≤ p, q ≤ ∞ +(p, q, d) ̸= (2, ∞, 2). +Throughout this paper, we regularly refer to the spacetime norms +(1.9) +∥u∥Lp +t Lq +z(It×Rm×Tn) = +�� +It +�� +Rm×Tn |u(t, z)|qdz +� p +q +dt +� 1 +p +. +Similarly we can define the composition of three Lp-type norms like Lp +tLq +xL2 +y. As in +Theorems 1.1, 1.6 and Corollary 1.3, we use Lr,s for the Lorentz norm (see [1]). One +can define the composition of norms in a similar way. +As stated in the above Theorems, in general, we refer to x for the whole spatial +variable; y for the whole Euclidean spatial variable; z for the whole tori spatial vari- +able; xα for the α-th spatial variable; yα for the α-th Euclidean spatial variable; zα +for the α-th tori spatial variable for convenience. +Similar to the Euclidean case, function spaces such as V p +∆ are also tightly involved. +we will discuss them in Section 2. (See also [17].) +To deal with the interacting potentials, we define the rotation operator Rαβ by +(1.10) +Rαβ(f(x1, ...xα−1, xα − xβ +√ +2 +, xα+1...xβ−1, xα + xβ +√ +2 +, xβ+1, ...xN)) = f(x1, ...xN). +Acknowledgment. The author was supported by the NSF grant of China (No. +12101046, 12271032), Chinese overseas high-level young talents program (2022) and +the Beijing Institute of Technology Research Fund Program for Young Scholars. The +author has learned many body Schr¨odinger model and related background during his +postdoc career at University of Maryland (2019-2021). Thus he highly appreciates +Prof. M. Grillakis, Prof. M. Machedon and their group (Dr. J. Chong and Dr. X. +Huang) for related discussions, especially the paper of Hong [17]. +2. Preliminaries +In this section, we discuss function spaces and some estimates for the model (1.1). +See Section 2 to Section 4 in [17] for the Euclidean analogue. + +MANY BODY SCHR ¨ODINGER EQUATION ON WAVEGUIDE MANIFOLDS +5 +First, similar to the Euclidean case, one can easily show: if the potential V is +small enough, then the Strichartz estimate for operator eit(∆x−V ) also holds for the +waveguide case as follows. +Lemma 2.1. Let m ≥ 3, n ≥ 1, and let c0 be the implicit constant given in Proposi- +tion 2.2. If ∥V ∥ +L +d +2 ,∞ +y +L2z +< +1 +c0 , then +(2.1) +∥eit(∆x−V )u0∥Lq +tLr +yL2 +z(R×Rm×Tn) ≤ +c0 +1 − c0∥V ∥ +L +d +2 ,∞ +y +L2z +∥u0∥L2y,z(Rm×Tn), +for all m-admissible pair (q, r). +As in the Euclidean case, to finish the proof of Lemma 2.1, recall the Strichartz +estimates in the waveguide setting as follows (see Proposition 2.1 in [29], in fact, this +result is more general since it concerns the compact Riemannian manifold case). +Proposition 2.2. For every n ≥ 1 and for every compact Riemannian manifold +M k +y , one considers functions f(x, y), F(x, y) on Rn ×M k +y , then the following estimate +holds: +(2.2) ∥eit∆x,yf∥Lp +t Lq +xL2y+ +�� +� t +0 +ei(t−s)∆x,yF(s, x, y)ds +�� +Lp +t Lq +xL2 +y ≲ ∥f∥L2x,y+∥F∥L ˜ +p +t L˜ +q +xL2y, +where (p, q) and (˜p, ˜q) are Strichartz admissible pairs. +Proof of Lemma 2.1. Since the potential is small, the proof is purely perturbative. +One can just use waveguide Strichartz estimate Proposition 2.2 to treat the potential +as a perturbation term provided the potential is small (the Duhamel’s formula and +the H¨older inequality are also used). Thus we omit the proof. See Theorem 2.1 in +[17] for the Euclidean analogue. +□ +Next, we discuss Strichartz estimates with frozen spatial variables. (See Propo- +sition 1 in [17] and Theorem 3.1 in [3] for the Euclidean analogue.) The difference +is that: now we fix both of the tori component of a certain particle and the other +particles by using L2-norm. In other words, the ‘frozen spatial variables’ are the tori +component and the other particles. Standard dispersive estimate and an important +lemma in [20] which ‘lifts’ dispersive estimates to Strichartz estimates are used. +Proposition 2.3. Let m ≥ 3 and n ≥ 1. Then for any m-dimensional admissible +pair (q, r), (˜q, ˜r) and 1 ≤ α ≤ N, we have +(2.3) +∥eit∆xu0∥Lq +tLr,2 +yα L2zαL2 +ˆxα ≲ ∥u0∥L2x, +(2.4) +∥ +� +R +e−is∆xF(s)ds∥L2x ≲ ∥F∥ +L˜ +q′ +t L˜r′ ,2 +yα L2zαL2 +ˆxα +, +and +(2.5) +∥ +� t +0 +e−i(t−s)∆xF(s)ds∥Lq +t Lr,2 +yα L2zαL2 +ˆxα ≲ ∥F∥ +L˜ +q′ +t L˜r′ ,2 +yα L2zαL2 +ˆxα +, +where +(2.6) +ˆxα = (x1, ...xα−1, xα+1..., xN) ∈ Rd(N−1), +and xα is the α-th variable with Euclidean component yα and tori component zα, i.e., +(2.7) +xα = (yα, zα) ∈ Rm × Tn. +Proof. We consider a complex-valued function f(x) : RdN +x +→ C in Lr,2 +yα L2 +zαL2 +ˆxα with +the function-valued function f(yα; zα, ˆxα) in Lr,2 +yα . We note that r ≥ 2. Using unitarity +property, +(2.8) +∥eit∆xu0∥LryαL2zαL2 +ˆxα = ∥eit∆yαu0∥LryαL2zαL2 +ˆxα = ∥eit∆yαu0∥L2zαL2 +ˆxαLryα. + +6 +ZEHUA ZHAO +Then, by the standard dispersive estimate (for yα-direction which is m-dimensional) +(2.9) +∥eit∆yαu0∥Lryα ≲ +1 +|t|m( 1 +2 − 1 +r ) ∥f∥Lr′ +yα, +we obtain (the Minkowski allows one change the order of norms) +(2.10) ∥eit∆xu0∥LryαL2zαL2 +ˆxα ≲ +1 +|t|m( 1 +2 − 1 +r ) ∥u0∥L2 +zαL2 +ˆxαLr′ +yα ≲ +1 +|t|m( 1 +2 − 1 +r ) ∥u0∥Lr′ +yαL2 +zαL2 +ˆxα. +The proposition follows from Theorem 10.1 in [20]. +□ +Now we briefly discuss the function spaces and corresponding estimates. They will +be essentially used in the following two sections. As mentioned in the end of Section +3.3 in [17], Strichartz estimates with frozen spatial variables are still not sufficient to +complete the proof of Theorem 1.1 (Strichartz estimate) because of the interacting +potentials. That is why a space-time norm that plays the role of the rotated space- +time norm is needed. This part is almost the same as Section 4.1 in [17] with natural +modifications. We also refer to [15, 16, 23] for more details. +We note that the definitions and properties in Subsection 4.1. of [17] are general +enough which can be applied for our model in the waveguide setting naturally. They +construct function spaces with nice properties for a separable Hilbert space H and +self-adjoint operator S. In this paper, we can just choose H to be L2 +x and S to be ∆x in +the waveguide setting, where x = (x1, ..., xN) and xα ∈ Rm × Tn for α ∈ {1, ..., N} as +in (1.1). Then the definitions and associated properties for our case will hold as well. +Thus we refer to Subsection 4.1. of [17] for the function spaces and corresponding +estimates/properties. For instance, we will use the following property of V p +∆-space. +(It follows from the definition. See Proposition 2 in [17].) +(2.11) +∥1[0,∞)]eit∆xu0∥V p +∆x = ∥u0∥L2 +x. +Moreover, the duality, the inclusion properties and the transference principle of V p +∆- +space are also often used. (See Subsection 4.1. of [17]) +3. The proof of Theorem 1.1 +In this section, we discuss the proof of Theorem 1.1 (Strichartz estimate). Corollary +1.3 will also be obtained using the properties of function space V (xα−xβ). Like in [17], +we handle the potential terms by treating them as perturbations. The key estimate +is as follows, +Proposition 3.1. Let m ≥ 3, n ≥ 1 and 1 < p < 2. Consider u in the waveguide +setting as in Theorem 1.1. Then, we have +(3.1) +��1[0,+∞) +� t +0 +ei(t−s)∆x(V (xα − xβ)u(s))ds +�� +V p +∆ ≤ C∥V ∥ +L +m +2 ,∞ +y +L2z∥u∥V p +∆, +where C is for the universal constant. +Remark 3.2. Proposition 3.1 indicates that one can regard the potential terms as +perturbations. As we can see from the proof below, it suffices to consider one arbitrary +interacting potential V (xα − xβ) since the V p +∆-norm is rotation-flexible. +Remark 3.3. See Proposition 4 in [17] for the Euclidean analogue. The main new +difference for the waveguide case is the appearance of the tori component. +Proof. For notational convenience, we denote +(3.2) +w = 1[0,∞) +� t +0 +ei(t−s)∆x(F(s))ds, +where F = V (xα −xβ)u(s) is treated as the forcing term (or say a perturbative term). + +MANY BODY SCHR ¨ODINGER EQUATION ON WAVEGUIDE MANIFOLDS +7 +We will estimate w by the duality argument. Since we only expect w ∈ V p +−, not +w ∈ V p, we consider ˜w(t) = w(−t). +Similar to Proposition 4 of [17], using duality, it suffices to show that +(3.3) +J +� +j=1 +⟨a(tj−1), ˜w(j) − ˜w(tj−1)⟩L2x ≲ ∥V ∥ +L +m +2 ,∞ +y +L2z∥u∥V p +∆ +for any fine partition of unity t = {tj}J +j=0 and any U p +′ +-atom a(t) = �K +k=1 1(sk−1,sk)φk−1. +(We note that the U p +′ +-space is the dual of the V p +∆-space.) +Doing some standard simplifications as in Proposition 4 of [17] (expanding atoms +a in terms of φk), one can get a simpler sum +(3.4) +K +� +k=1 +⟨φk−1, ˜w(sk) − ˜w(sk−1)⟩L2x. +We further write it as +K +� +k=1 +⟨φk−1, ˜w(sk) − ˜w(sk−1)⟩L2x +(3.5) += − +K +� +k=1 +� −sk−1 +−sk +⟨φk−1, e−is∆x(F(s))⟩L2xds +(3.6) += − +K +� +k=1 +� −sk−1 +−sk +⟨eis∆xRφk−1, R(F(s))⟩L2xds +(3.7) += − +K +� +k=1 +� +R +⟨eis∆xRφk−1, 1[−sk,−sk−1]R(F(s))⟩L2 +xds, +(3.8) +where R denotes any rotation operator. +(It is just Rαβ for interacting potential +V (xα − xβ).) We want to control it by ∥V ∥ +L +m +2 ,∞ +y +L2 +z∥u∥V p +∆. +Then, applying the H¨older inequality and the Strichartz estimate Proposition 2.3, +we estimate it by +K +� +k=1 +⟨φk−1, ˜w(sk) − ˜w(sk−1)⟩L2x +(3.9) +≲ +K +� +k=1 +∥eit∆Rφk−1∥ +L2 +tL +2m +m−2 ,2 +yα +L2zαL2 +ˆxα +∥1[−sk,−sk−1]R(F(s))∥ +L2 +tL +2m +m+2 ,2 +yα +L2zαL2 +ˆxα +(3.10) +≲ +K +� +k=1 +∥φk−1∥L2 +x∥V ∥ +L +m +2 ,∞ +y +L2z∥1[−sk,−sk−1]R(u)∥ +L2 +tL +2m +m−2 ,2 +yα +L2zαL2 +ˆxα +(3.11) +≲ +K +� +k=1 +∥φk−1∥L2 +x∥V ∥ +L +m +2 ,∞ +y +L2z∥1[−sk,−sk−1](u)∥V p +∆x +(3.12) +≲ ∥V ∥ +L +m +2 ,∞ +y +L2z +��∥φk−1∥L2 +x +�� +lp′ · +��∥1[−sk,−sk−1](u)∥V p +∆x +�� +lp +(3.13) +≲ ∥V ∥ +L +m +2 ,∞ +y +L2z +��∥1[−sk,−sk−1](u)∥V p +∆x +�� +lp. +(3.14) +We note that we have used the inclusion property of discrete Lp spaces (i.e. lp-spaces). +(1 < p < 2 implies p +′ > 2.) +To close the argument, now it remains to show that +(3.15) +��∥1[−sk,−sk−1](u)∥V p +∆x +�� +lp = +� K +� +k=1 +∥1[−sk,−sk−1)u∥p +V p +∆x +� 1 +p ≤ ∥u∥V p +∆x. + +8 +ZEHUA ZHAO +This estimate follows exactly as the Euclidean case (using the definition of V p +∆x). +There is no difference in the waveguide setting. Thus the proof of Proposition 3.1 is +complete. +□ +With the help of Proposition 3.1, we give the proof of Theorem 1.1 as follows. We +can now treat the potential terms as perturbations. +Proof. Applying Proposition 3.1 to the Duhamel formula for u = e−itHN u0, we have, +(3.16) +∥1[0,+∞)u(t)∥V p +∆ ≤ ∥u0∥L2x + N(N − 1) +2 +C∥V ∥ +L +d +2 ,∞ +y +L2 +z +∥u∥V p +∆. +Theorem 1.1 now follows from the smallness assumption of potential V . ( N(N−1) +2 +is +the number of interacting potentials.) +□ +Corollary 1.3 follows from Theorem 1.1 in viewing of the following lemma: +Lemma 3.4 (Transference principle). Let d ≥ 1, 1 < p < 2, q ≥ 2 and X be a +Banach space. If a function u : R → X satisfies the bound +(3.17) +∥e∆xu0∥Lq +tX ≲ ∥u0∥L2x, +then +(3.18) +∥u∥Lq +tX ≲ ∥u∥V p +∆x. +Remark 3.5. We note that the Bourgain spaces Xs,b (also known as Fourier restriction +space) enjoy the similar transfer principle (see [28] for more info.). As summarized in +[17], the Strichartz estimates in the V p +∆x sharpen the bounds in Xs,b by 0+ in that +Strichartz estimates in the Xs,b space do not cover the endpoint Strichartz estimates, +while those in the V p +∆x-space do. +See Proposition 3 in [17] for the proof. As a direct consequence, it shows that +the V p +∆x-norm dominates the two Strichartz-type space-time norms in Corollary 1.3. +Thus, Corollary 1.3 follows from Theorem 1.1. +4. The proof of Theorem 1.6 +Now we are ready to discuss the proof of Theorem 1.6, i.e. +the scattering for +(1.1). Since we have established proper Strichartz-type estimate, the proof will follow +similarly as in [17]. For the sake of completeness, we include it as follows. +Without loss of generality, we only consider for the positive time. It suffices to +show that +(4.1) +u+ = +lim +t→+∞ e−it∆xe−itHN u0 +exists in L2 +x as t → ∞. Indeed, by the Duhamel formula +∥e−it2∆e−it2HN u0 − e−it1∆e−it1HN u0∥L2x +(4.2) +≤ +� +1≤α<β≤N +�� +� t2 +t1 +e−is∆x((V (xα − xβ))e−isHN u0)ds +�� +L2x. +(4.3) + +MANY BODY SCHR ¨ODINGER EQUATION ON WAVEGUIDE MANIFOLDS +9 +It suffices to consider one single potential. According to Theorem 1.1 and Corollary +1.3, we have that +�� +� t2 +t1 +e−is∆x((V (xα − xβ))e−isHN u0)ds +�� +L2x +(4.4) += +��Rαβ +� t2 +t1 +e−is∆x((V (xα − xβ))e−isHN u0)ds +�� +L2x +(4.5) += +�� +� t2 +t1 +e−is∆x(V ( +√ +2xα)(Rαβe−isHN u0))ds +�� +L2x +(4.6) +≤ c0 +��V ( +√ +2xα)(Rαβe−isHN u0) +�� +L2 +t∈[t1,t2]L +2d +d+2 ,2 +yα +L2zαL2 +ˆxα +(4.7) +≤ c0 +2 ∥V ∥ +L +d +2 ,∞ +z +L2z +��(Rαβe−isHN u0) +�� +L2 +t∈[t1,t2]L +2d +d−2 ,2 +yα +L2zαL2 +ˆxα +→ 0 +(4.8) +as t1, t2 → ∞. Then we can see that the limit exists. +5. Further remarks +In this section, we make a few more remarks for many body model (1.1) and +Theorems 1.1, 1.6 as follows. +1. The main results in this paper and [17] are based on perturbative scheme which +are tightly dependent on the smallness assumption of the potentials. One may consider +removing the smallness assumption to prove Strichartz estimates like Theorem 1.1 or +Corollary 1.3. It may be hard to consider the general case thus the two body case +may be a good model to start with. (See [7] for the Euclidean case.) +2. +It is also interesting to consider many body equation with a nonlinearity +F(t, x1, ..., xN) and study the long time behavior. +There are few general theories +and results regarding this topic, especially the scattering-type results. Also, it may +be hard to consider the general case thus the two body case may still be a good +model to start with. (The Λ-equation in the Hartree–Fock–Bogoliubov equations is +an example for the two body case, though it is in a coupled system which makes it +more complicated. See [6, 7].) +We also note that via the standard T -T ∗ argument and the Christ-Kiselev lemma, +one can obtain the inhomogeneous Strichartz analogue of Corollary 1.3 (excluding +the double endpoint case). (See [28].) With the help of it, one may obtain the local +well-posedness for (1.1) with a subcritical nonlinearity in the energy space. We leave +it ifor interested readers. +3. One may also consider the tori analogue of (1.1) (replacing Rm × Tn by Td)) +and obtain some estimates. The reason we consider the waveguide case is that we +intend to study the scattering behavior, which is not expected for the tori case. +4. The results in the current paper is only about the estimates and the PDE-level +of many body Schr¨odinger equations. One may consider the many body Schr¨odinger +equations in the tori setting or waveguide setting from the perspectives of mathemat- +ical physics. (See [8, 12, 13] for examples.) +References +1. 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Anal. 53 (2021), no. 3, 3644–3660. +MR 4277925 +Zehua Zhao +Department of Mathematics and Statistics, Beijing Institute of Technology, Beijing, +China. +MIIT Key Laboratory of Mathematical Theory and Computation in Information Secu- +rity, Beijing, China. +Email address: zzh@bit.edu.cn + diff --git a/39FQT4oBgHgl3EQf3zah/content/tmp_files/load_file.txt b/39FQT4oBgHgl3EQf3zah/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1dd08ff3ceab1a94bdb2283119b65d470b65e46b --- /dev/null +++ b/39FQT4oBgHgl3EQf3zah/content/tmp_files/load_file.txt @@ -0,0 +1,657 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf,len=656 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='13429v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='AP] 31 Jan 2023 ON STRICHARTZ ESTIMATE FOR MANY BODY SCHR¨ODINGER EQUATION IN THE WAVEGUIDE SETTING ZEHUA ZHAO Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Preliminaries 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='6 8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Further remarks 9 References 9 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' In this short paper, we prove Strichartz estimates for N-body Schr¨odinger equations in the waveguide manifold setting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' on semiperiodic spaces Rm ×Tn where m ≥ 3), provided that interaction potentials are small enough (depending on the number of the particles and the universal constants, not on the initial data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The proof combines both the ideas of Tzvetkov-Visciglia [29] and Hong [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' As an immediate application, the scattering asymptotics for this model is also obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' This result extends Hong [17] to the waveguide case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Keywords: Strichartz estimate, many body Schr¨odinger equations, scattering, waveg- uide manifolds Mathematics Subject Classification (2020) Primary: 35Q55;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Secondary: 35R01, 37K06, 37L50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Background and Motivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Let d = m + n, m ≥ 3, n ≥ 1 and N ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We consider the many body Schr¨odinger equations in the waveguide setting as follows, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1) (i∂t + HN)u(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='xN) = 0, u(0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='xN) = u0(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='xN) ∈ L2 x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='xN, where HN = ∆x − VN = �N α=1 ∆xα − � 1≤α<β≤N V (xα − xβ), and α-th particle xα ∈ Rm × Tn for any α ∈ [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=', N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' From physical explanations, N ≥ 1 indicates the number of particles in a quantum system (which is often very large) and the interacting potentials of form V (xα − xβ) indicates the interactions of any two particles, which depends on their relative distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Moreover, the product spaces of form Rm × Tn is known as semi-periodic space or waveguide manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' d = m + n is the whole dimension while m is the dimension for the Euclidean component and n is the dimension for the tori component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' When N = 1, initial value problem(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1) is exactly the standard nonlinear Schr¨odinger equation (NLS) with a potential, which has been well studied (in the Euclidean case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' replacing Rm×Tn by Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' It is also known as ‘the one-body case’ and the research on the decay properties has a long history (see the Introduction of [17], the survey [25] and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' In this paper, we mainly concern the general case (N ≥ 1 can be arbitrarily large), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' the many body Schr¨odinger case, which will involve some new difficulties than the single body case such as the issue of interacting potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' 1 2 ZEHUA ZHAO The purpose of this paper is to investigate time decay properties of solutions to the N-body Schr¨odinger equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1) in the waveguide setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' In particular, we discuss the Strichartz-type estimate and the scattering behavior for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We note that the Euclidean case of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1) has been studied in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' (See also [7] for a recently result which deals with the two body case via the scheme of [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=') We intend to generalize [17] to the waveguide case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' That is one main motivation of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Another motivation is the recent developments for the topic: ‘Long time behavior for NLS on waveguides’ so the author is interested in combining both of ‘waveguides’ and ‘many body Schr¨odinger equations’ together, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' studying the estimates and the long time behavior for many body Schr¨odinger equations on waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We will briefly mention the background for ‘NLS on waveguides’ in the next paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Waveguide manifolds of form Rm ×Tn are of particular interest in nonlinear optics of telecommunications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Generally, well-posedness theory and long time behavior of NLS are hot topics in the area of dispersive equations and have been studied widely in recent decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Naturally, the Euclidean case is first treated and the theory, at least in the defocusing setting, has been well established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We refer to [9, 10, 21] for some typical Euclidean results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Moreover, we refer to [5, 4, 14, 15, 16, 18, 19, 22, 31, 32, 34, 35, 36] with regard to the torus and waveguide settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' (See also [24, 27, 33] for other dispersive equations on waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=') One may roughly think of the waveguide case as the “intermediate point” between the Euclidean case and the torus case since the waveguide manifold is a product of Euclidean spaces and the tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The techniques used in Euclidean and torus settings are frequently combined and applied to the waveguides problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' At last, we refer to [2, 11, 28] for some classical textbooks on the study of NLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Since the current paper concerns the estimates and the PDE-level of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1) rather than the mathematical physics level, we will not mention too much for the background of the many body problems/equations from physical perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We refer to the Introductions of [3, 6, 7, 8, 12, 26] and the references therein for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' To the authors’ best knowledge, the current paper is the first result towards un- derstanding long time dynamics for the many body Schr¨odinger equations within the context of waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' As last, we note that, as in [17], we need to assume some smallness for the potential V and this smallness does not depend on the initial data (only depends on the particle number N and the universal constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The statement of main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Now we are ready to state the two main results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We start with the Strichartz estimate as follows since the other one is an application of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 (Strichartz estimate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Let m ≥ 3, n ≥ 1 and 1 < p < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' There exists a small number ǫ such that if ∥V ∥ L d 2 ,∞ y L2z ≤ ǫ N 2 , then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='2) ∥1[0,+∞)e−itHN u0∥V p ∆x ≲ ∥u0∥L2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Here V p ∆x-norm (known as variation spaces) is introduced by Koch- Tataru [23] (see Section 2 for discussions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' See also [15, 16, 18, 19] for more information and some other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' In viewing of the properties of V p ∆x-type spaces, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 directly implies Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Let m ≥ 3 and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' There exists a small number ǫ such that if ∥V ∥ L d 2 ,∞ y L2z ≤ ǫ N 2 , then for any m-dimensional admissible pair (q, r) and 1 ≤ α ≤ N, we have (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3) ∥e−itHNu0∥Lq tLryαL2zαL2 ˆxα ≲ ∥u0∥L2x, MANY BODY SCHR ¨ODINGER EQUATION ON WAVEGUIDE MANIFOLDS 3 where ˆxα is the N − 1 spatial variables except the α-th variable xα, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=', (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='4) ˆxα = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='xα−1, xα+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=', xN) ∈ Rd(N−1), and xα is the α-th variable with Euclidean component yα and tori component zα, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=', (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='5) xα = (yα, zα) ∈ Rm × Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Moreover, for any mN-dimensional admissible pair (q, r), we have (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='6) ∥e−itHNu0∥Lq tLryL2z ≲ ∥u0∥L2x, where y is for the whole Euclidean component (mN-dimensional) and z is for the whole tori component nN-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' See Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='2 in [17] for the Euclidean case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We will give the proof for Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3 after the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 in the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' As shown above, the formulation of the Strichartz estimates for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1) combines both the ideas of [29] and [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' As in [29] (see also [30]), we fix the tori component by using L2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' (In other words, one decomposes the function along the tori direction and derive the Strichartz estimate using the dispersion from the Eu- clidean direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=') As in [17], we fix other particles by only considering the dispersion of one certain particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Thus, we consider the dispersion of the Euclidean component of one particle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' fixing other particles and the tori component of this particle by using L2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' As a direct application of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1, we show the scattering behavior for an N-body Schr¨odinger operator with rough small interactions in the following sense, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='6 (Scattering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Let m ≥ 3, n ≥ 1 and 1 < p < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' let ǫ be a small constant given in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' If ∥V ∥ L m 2 ,∞ y L2 z ≤ ǫ N 2 , then for each u0 ∈ L2 x, there exist scattering states u± such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='7) lim t→±∞ ��e−itHN u0 − eit∆xu± �� L2x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We note that for the tori case of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1), the scattering behavior is not expected due to the lack of dispersion, though a Strichartz estimate can still be possibly obtained with suitable modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We leave it for interested readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' For the above results, the dimension of the tori component n ≥ 1 does not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' (When n = 0, it is exactly the Euclidean case [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' However, if one considers the long time dynamics of a nonlinear problem on waveguide manifolds, the dimension of the tori component often matters a lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' In general, the difficulty of the critical NLS problem on waveguide manifolds increases if the whole dimension is increased or if the Euclidean component is decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' See the Introductions in [18, 19] for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' To be more general, the tori component Tn in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1) can be generalized to a compact Riemannian manifold M such that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='6 still hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Next, we briefly introduce the main strategy of the proofs for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1, Corol- lary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' In fact, the proofs for Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='6 are standard and less complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3 follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 (see Section 3) according to the transfer principle of the function space V p ∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='6 also follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 (see Section 4), together with some other basic estimates like in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Thus we will focus on the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 (Strichartz estimate) is based on the properties of func- tion space V p ∆x and a perturbation method (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The main idea is: one 4 ZEHUA ZHAO establishes nonlinear estimate for one arbitrary interacting potential (treating it as a perturbation) and then sum them up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The key estimate is Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 which deals with one arbitrary interacting potential by regarding it as a forcing term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' With the help of it, one can handle all of the interacting potentials by treating them as perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Eventually, according to the smallness assumption, one can use per- turbation method to show the Strichartz estimate as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Compared with the single potential case (N = 1), the interacting potentials (involves rotations) cause dif- ficulties thus the ‘rotation flexible’ function space V p ∆x is needed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' compared with the Euclidean analogue ([17]), the new difference is the appearance of the tori component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Structure of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The rest of the article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' In Section 2, we discuss function spaces and some estimates for this model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' in Section 3, we give the proof for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 (Strichartz estimate);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' in Section 4, we give the proof for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='6 (scattering asymptotics);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' in Section 5, we give a few further remarks on this research line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We write A ≲ B to say that there is a constant C such that A ≤ CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We use A ≃ B when A ≲ B ≲ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Particularly, we write A ≲u B to express that A ≤ C(u)B for some constant C(u) depending on u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We use C for universal constants and N for the number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We say that the pair (p, q) is d-(Strichartz) admissible if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='8) 2 p + d q = d 2, 2 ≤ p, q ≤ ∞ (p, q, d) ̸= (2, ∞, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Throughout this paper, we regularly refer to the spacetime norms (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='9) ∥u∥Lp t Lq z(It×Rm×Tn) = �� It �� Rm×Tn |u(t, z)|qdz � p q dt � 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Similarly we can define the composition of three Lp-type norms like Lp tLq xL2 y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' As in Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='6 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3, we use Lr,s for the Lorentz norm (see [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' One can define the composition of norms in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' As stated in the above Theorems, in general, we refer to x for the whole spatial variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' y for the whole Euclidean spatial variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' z for the whole tori spatial vari- able;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' xα for the α-th spatial variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' yα for the α-th Euclidean spatial variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' zα for the α-th tori spatial variable for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Similar to the Euclidean case, function spaces such as V p ∆ are also tightly involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' we will discuss them in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' (See also [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=') To deal with the interacting potentials, we define the rotation operator Rαβ by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='10) Rαβ(f(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='xα−1, xα − xβ √ 2 , xα+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='xβ−1, xα + xβ √ 2 , xβ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='xN)) = f(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='xN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The author was supported by the NSF grant of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' 12101046, 12271032), Chinese overseas high-level young talents program (2022) and the Beijing Institute of Technology Research Fund Program for Young Scholars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The author has learned many body Schr¨odinger model and related background during his postdoc career at University of Maryland (2019-2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Thus he highly appreciates Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Grillakis, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Machedon and their group (Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Chong and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Huang) for related discussions, especially the paper of Hong [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Preliminaries In this section, we discuss function spaces and some estimates for the model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' See Section 2 to Section 4 in [17] for the Euclidean analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' MANY BODY SCHR ¨ODINGER EQUATION ON WAVEGUIDE MANIFOLDS 5 First, similar to the Euclidean case, one can easily show: if the potential V is small enough, then the Strichartz estimate for operator eit(∆x−V ) also holds for the waveguide case as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Let m ≥ 3, n ≥ 1, and let c0 be the implicit constant given in Proposi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' If ∥V ∥ L d 2 ,∞ y L2z < 1 c0 , then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1) ∥eit(∆x−V )u0∥Lq tLr yL2 z(R×Rm×Tn) ≤ c0 1 − c0∥V ∥ L d 2 ,∞ y L2z ∥u0∥L2y,z(Rm×Tn), for all m-admissible pair (q, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' As in the Euclidean case, to finish the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1, recall the Strichartz estimates in the waveguide setting as follows (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 in [29], in fact, this result is more general since it concerns the compact Riemannian manifold case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' For every n ≥ 1 and for every compact Riemannian manifold M k y , one considers functions f(x, y), F(x, y) on Rn ×M k y , then the following estimate holds: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='2) ∥eit∆x,yf∥Lp t Lq xL2y+ �� � t 0 ei(t−s)∆x,yF(s, x, y)ds �� Lp t Lq xL2 y ≲ ∥f∥L2x,y+∥F∥L ˜ p t L˜ q xL2y, where (p, q) and (˜p, ˜q) are Strichartz admissible pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Since the potential is small, the proof is purely perturbative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' One can just use waveguide Strichartz estimate Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='2 to treat the potential as a perturbation term provided the potential is small (the Duhamel’s formula and the H¨older inequality are also used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Thus we omit the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' See Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 in [17] for the Euclidean analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' □ Next, we discuss Strichartz estimates with frozen spatial variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' (See Propo- sition 1 in [17] and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 in [3] for the Euclidean analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=') The difference is that: now we fix both of the tori component of a certain particle and the other particles by using L2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' In other words, the ‘frozen spatial variables’ are the tori component and the other particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Standard dispersive estimate and an important lemma in [20] which ‘lifts’ dispersive estimates to Strichartz estimates are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Let m ≥ 3 and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Then for any m-dimensional admissible pair (q, r), (˜q, ˜r) and 1 ≤ α ≤ N, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3) ∥eit∆xu0∥Lq tLr,2 yα L2zαL2 ˆxα ≲ ∥u0∥L2x, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='4) ∥ � R e−is∆xF(s)ds∥L2x ≲ ∥F∥ L˜ q′ t L˜r′ ,2 yα L2zαL2 ˆxα , and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='5) ∥ � t 0 e−i(t−s)∆xF(s)ds∥Lq t Lr,2 yα L2zαL2 ˆxα ≲ ∥F∥ L˜ q′ t L˜r′ ,2 yα L2zαL2 ˆxα , where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='6) ˆxα = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='xα−1, xα+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=', xN) ∈ Rd(N−1), and xα is the α-th variable with Euclidean component yα and tori component zα, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='7) xα = (yα, zα) ∈ Rm × Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We consider a complex-valued function f(x) : RdN x → C in Lr,2 yα L2 zαL2 ˆxα with the function-valued function f(yα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' zα, ˆxα) in Lr,2 yα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We note that r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Using unitarity property, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='8) ∥eit∆xu0∥LryαL2zαL2 ˆxα = ∥eit∆yαu0∥LryαL2zαL2 ˆxα = ∥eit∆yαu0∥L2zαL2 ˆxαLryα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' 6 ZEHUA ZHAO Then, by the standard dispersive estimate (for yα-direction which is m-dimensional) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='9) ∥eit∆yαu0∥Lryα ≲ 1 |t|m( 1 2 − 1 r ) ∥f∥Lr′ yα, we obtain (the Minkowski allows one change the order of norms) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='10) ∥eit∆xu0∥LryαL2zαL2 ˆxα ≲ 1 |t|m( 1 2 − 1 r ) ∥u0∥L2 zαL2 ˆxαLr′ yα ≲ 1 |t|m( 1 2 − 1 r ) ∥u0∥Lr′ yαL2 zαL2 ˆxα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The proposition follows from Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' □ Now we briefly discuss the function spaces and corresponding estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' They will be essentially used in the following two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' As mentioned in the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3 in [17], Strichartz estimates with frozen spatial variables are still not sufficient to complete the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 (Strichartz estimate) because of the interacting potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' That is why a space-time norm that plays the role of the rotated space- time norm is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' This part is almost the same as Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 in [17] with natural modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We also refer to [15, 16, 23] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We note that the definitions and properties in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' of [17] are general enough which can be applied for our model in the waveguide setting naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' They construct function spaces with nice properties for a separable Hilbert space H and self-adjoint operator S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' In this paper, we can just choose H to be L2 x and S to be ∆x in the waveguide setting, where x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=', xN) and xα ∈ Rm × Tn for α ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=', N} as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Then the definitions and associated properties for our case will hold as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Thus we refer to Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' of [17] for the function spaces and corresponding estimates/properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' For instance, we will use the following property of V p ∆-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' (It follows from the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' See Proposition 2 in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=') (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='11) ∥1[0,∞)]eit∆xu0∥V p ∆x = ∥u0∥L2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Moreover, the duality, the inclusion properties and the transference principle of V p ∆- space are also often used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' (See Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' of [17]) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 In this section, we discuss the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 (Strichartz estimate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3 will also be obtained using the properties of function space V (xα−xβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Like in [17], we handle the potential terms by treating them as perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The key estimate is as follows, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Let m ≥ 3, n ≥ 1 and 1 < p < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Consider u in the waveguide setting as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Then, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1) ��1[0,+∞) � t 0 ei(t−s)∆x(V (xα − xβ)u(s))ds �� V p ∆ ≤ C∥V ∥ L m 2 ,∞ y L2z∥u∥V p ∆, where C is for the universal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 indicates that one can regard the potential terms as perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' As we can see from the proof below, it suffices to consider one arbitrary interacting potential V (xα − xβ) since the V p ∆-norm is rotation-flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' See Proposition 4 in [17] for the Euclidean analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The main new difference for the waveguide case is the appearance of the tori component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' For notational convenience, we denote (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='2) w = 1[0,∞) � t 0 ei(t−s)∆x(F(s))ds, where F = V (xα −xβ)u(s) is treated as the forcing term (or say a perturbative term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' MANY BODY SCHR ¨ODINGER EQUATION ON WAVEGUIDE MANIFOLDS 7 We will estimate w by the duality argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Since we only expect w ∈ V p −, not w ∈ V p, we consider ˜w(t) = w(−t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Similar to Proposition 4 of [17], using duality, it suffices to show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3) J � j=1 ⟨a(tj−1), ˜w(j) − ˜w(tj−1)⟩L2x ≲ ∥V ∥ L m 2 ,∞ y L2z∥u∥V p ∆ for any fine partition of unity t = {tj}J j=0 and any U p ′ atom a(t) = �K k=1 1(sk−1,sk)φk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' (We note that the U p ′ space is the dual of the V p ∆-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=') Doing some standard simplifications as in Proposition 4 of [17] (expanding atoms a in terms of φk), one can get a simpler sum (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='4) K � k=1 ⟨φk−1, ˜w(sk) − ˜w(sk−1)⟩L2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We further write it as K � k=1 ⟨φk−1, ˜w(sk) − ˜w(sk−1)⟩L2x (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='5) = − K � k=1 � −sk−1 −sk ⟨φk−1, e−is∆x(F(s))⟩L2xds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='6) = − K � k=1 � −sk−1 −sk ⟨eis∆xRφk−1, R(F(s))⟩L2xds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='7) = − K � k=1 � R ⟨eis∆xRφk−1, 1[−sk,−sk−1]R(F(s))⟩L2 xds, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='8) where R denotes any rotation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' (It is just Rαβ for interacting potential V (xα − xβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=') We want to control it by ∥V ∥ L m 2 ,∞ y L2 z∥u∥V p ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Then, applying the H¨older inequality and the Strichartz estimate Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3, we estimate it by K � k=1 ⟨φk−1, ˜w(sk) − ˜w(sk−1)⟩L2x (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='9) ≲ K � k=1 ∥eit∆Rφk−1∥ L2 tL 2m m−2 ,2 yα L2zαL2 ˆxα ∥1[−sk,−sk−1]R(F(s))∥ L2 tL 2m m+2 ,2 yα L2zαL2 ˆxα (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='10) ≲ K � k=1 ∥φk−1∥L2 x∥V ∥ L m 2 ,∞ y L2z∥1[−sk,−sk−1]R(u)∥ L2 tL 2m m−2 ,2 yα L2zαL2 ˆxα (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='11) ≲ K � k=1 ∥φk−1∥L2 x∥V ∥ L m 2 ,∞ y L2z∥1[−sk,−sk−1](u)∥V p ∆x (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='12) ≲ ∥V ∥ L m 2 ,∞ y L2z ��∥φk−1∥L2 x �� lp′ · ��∥1[−sk,−sk−1](u)∥V p ∆x �� lp (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='13) ≲ ∥V ∥ L m 2 ,∞ y L2z ��∥1[−sk,−sk−1](u)∥V p ∆x �� lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='14) We note that we have used the inclusion property of discrete Lp spaces (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' lp-spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' (1 < p < 2 implies p ′ > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=') To close the argument, now it remains to show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='15) ��∥1[−sk,−sk−1](u)∥V p ∆x �� lp = � K � k=1 ∥1[−sk,−sk−1)u∥p V p ∆x � 1 p ≤ ∥u∥V p ∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' 8 ZEHUA ZHAO This estimate follows exactly as the Euclidean case (using the definition of V p ∆x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' There is no difference in the waveguide setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Thus the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' □ With the help of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1, we give the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We can now treat the potential terms as perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 to the Duhamel formula for u = e−itHN u0, we have, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='16) ∥1[0,+∞)u(t)∥V p ∆ ≤ ∥u0∥L2x + N(N − 1) 2 C∥V ∥ L d 2 ,∞ y L2 z ∥u∥V p ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 now follows from the smallness assumption of potential V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' ( N(N−1) 2 is the number of interacting potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=') □ Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3 follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 in viewing of the following lemma: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='4 (Transference principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Let d ≥ 1, 1 < p < 2, q ≥ 2 and X be a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' If a function u : R → X satisfies the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='17) ∥e∆xu0∥Lq tX ≲ ∥u0∥L2x, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='18) ∥u∥Lq tX ≲ ∥u∥V p ∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We note that the Bourgain spaces Xs,b (also known as Fourier restriction space) enjoy the similar transfer principle (see [28] for more info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' As summarized in [17], the Strichartz estimates in the V p ∆x sharpen the bounds in Xs,b by 0+ in that Strichartz estimates in the Xs,b space do not cover the endpoint Strichartz estimates, while those in the V p ∆x-space do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' See Proposition 3 in [17] for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' As a direct consequence, it shows that the V p ∆x-norm dominates the two Strichartz-type space-time norms in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Thus, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3 follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='6 Now we are ready to discuss the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='6, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' the scattering for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Since we have established proper Strichartz-type estimate, the proof will follow similarly as in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' For the sake of completeness, we include it as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Without loss of generality, we only consider for the positive time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' It suffices to show that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1) u+ = lim t→+∞ e−it∆xe−itHN u0 exists in L2 x as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Indeed, by the Duhamel formula ∥e−it2∆e−it2HN u0 − e−it1∆e−it1HN u0∥L2x (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='2) ≤ � 1≤α<β≤N �� � t2 t1 e−is∆x((V (xα − xβ))e−isHN u0)ds �� L2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3) MANY BODY SCHR ¨ODINGER EQUATION ON WAVEGUIDE MANIFOLDS 9 It suffices to consider one single potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' According to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3, we have that �� � t2 t1 e−is∆x((V (xα − xβ))e−isHN u0)ds �� L2x (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='4) = ��Rαβ � t2 t1 e−is∆x((V (xα − xβ))e−isHN u0)ds �� L2x (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='5) = �� � t2 t1 e−is∆x(V ( √ 2xα)(Rαβe−isHN u0))ds �� L2x (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='6) ≤ c0 ��V ( √ 2xα)(Rαβe−isHN u0) �� L2 t∈[t1,t2]L 2d d+2 ,2 yα L2zαL2 ˆxα (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='7) ≤ c0 2 ∥V ∥ L d 2 ,∞ z L2z ��(Rαβe−isHN u0) �� L2 t∈[t1,t2]L 2d d−2 ,2 yα L2zαL2 ˆxα → 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='8) as t1, t2 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Then we can see that the limit exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Further remarks In this section, we make a few more remarks for many body model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1) and Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='6 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The main results in this paper and [17] are based on perturbative scheme which are tightly dependent on the smallness assumption of the potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' One may consider removing the smallness assumption to prove Strichartz estimates like Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1 or Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' It may be hard to consider the general case thus the two body case may be a good model to start with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' (See [7] for the Euclidean case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' It is also interesting to consider many body equation with a nonlinearity F(t, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=', xN) and study the long time behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' There are few general theories and results regarding this topic, especially the scattering-type results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Also, it may be hard to consider the general case thus the two body case may still be a good model to start with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' (The Λ-equation in the Hartree–Fock–Bogoliubov equations is an example for the two body case, though it is in a coupled system which makes it more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' See [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=') We also note that via the standard T -T ∗ argument and the Christ-Kiselev lemma, one can obtain the inhomogeneous Strichartz analogue of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='3 (excluding the double endpoint case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' (See [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=') With the help of it, one may obtain the local well-posedness for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1) with a subcritical nonlinearity in the energy space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' We leave it ifor interested readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' One may also consider the tori analogue of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='1) (replacing Rm × Tn by Td)) and obtain some estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The reason we consider the waveguide case is that we intend to study the scattering behavior, which is not expected for the tori case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' The results in the current paper is only about the estimates and the PDE-level of many body Schr¨odinger equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' One may consider the many body Schr¨odinger equations in the tori setting or waveguide setting from the perspectives of mathemat- ical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' (See [8, 12, 13] for examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=') References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' J¨oran Bergh and J¨orgen L¨ofstr¨om, Interpolation spaces: an introduction, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' 223, Springer Science & 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Zhao and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Zheng, Long time dynamics for defocusing cubic nonlinear Schr¨odinger equa- tions on three dimensional product space, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' 53 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' 3, 3644–3660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' MR 4277925 Zehua Zhao Department of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' MIIT Key Laboratory of Mathematical Theory and Computation in Information Secu- rity, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content=' Email address: zzh@bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39FQT4oBgHgl3EQf3zah/content/2301.13429v1.pdf'} +page_content='edu.' metadata={'source': 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+1,1563 @@ +3-space orthogonal to uµ +3-space orthogonal to uµ +Observer’s worldline, uµ = dxµ +dτ + + +x1 +x2 +x3 +Before the wave passes +After the wave passes + + +arXiv:2301.11772v1 [gr-qc] 27 Jan 2023 +Electromagnetic memory in arbitrary curved space-times +Susmita Jana1, ∗ and S. Shankaranarayanan1, † +1Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India +Abstract +The gravitational memory effect and its electromagnetic (EM) analog are potential probes in +the strong gravity regime. In the literature, this effect is derived for static observers at asymptotic +infinity. While this is a physically consistent approach, it restricts the space-time geometries for +which one can obtain the EM memory effect. To circumvent this, we evaluate the EM memory +effect for comoving observers (defined by the 4-velocity uµ) in arbitrary curved space-times. Using +the covariant approach, we split Maxwell’s equations into two parts — projected parallel to the 4- +velocity uµ and into the 3-space orthogonal to uµ. Further splitting the equations into 1+1+2-form, +we obtain master equation for the EM memory in an arbitrary curved space-time. We provide a +geometrical understanding of the contributions to the memory effect. We then obtain EM memory +for specific space-time geometries and discuss the salient features. +∗ susmitajana@iitb.ac.in +† shanki@iitb.ac.in +1 + +I. +INTRODUCTION +LIGO-VIRGO-KAGRA has detected close to 100 gravitational wave (GW) sources. GW +signals emanating from a black hole or neutron star binaries have opened many new research +avenues in astronomy, cosmology, and fundamental physics [1–4]. GWs provide a unique +way to test gravity’s most extreme, non-linear regime in novel ways. The planned third- +generation ground-based detector (Cosmic Explorer and the Einstein Telescope) will allow +us to peer far deeper, and LISA will open a new observational window at low frequencies. +With more sensitive detectors shortly, the focus has been to understand the physical effects +of GWs. Gravitational wave memory is one such effect [5–13]. +GW memory effects — physically observable phenomena that modify the state of +gravitational-wave detectors a little bit from their original undisturbed state — are one +of the key predictions of general relativity [6, 7, 9, 14]. GW memory effects can be divided +into two types [12, 13]: null memory that occurs when radiation or massless particles es- +cape from a system to null infinity, and ordinary memory that occurs when the detector +recoils relative to its initial center of mass frame. The GW memory is characterized as a +gravitational wave signal approaching a nonzero finite value. This aspect of the GW signal +is yet to be observed, although LISA is predicted to observe it [15]. +Recently, it has been realized that the memory effect can be thought of as a vacuum +transition between two different states related by an asymptotic transformation [16, 17]. +Since such asymptotic transformations also occur for other gauge theories, there has been +an intense activity to obtain analogous memory effects in other gauge theories [18–22]. Since +electromagnetic (EM) theory is the simplest of all gauge theories and can be a potential +probe, electromagnetic memory has received much attention [23–33]. Like in GW memory, +an EM wave generates a permanent change in the relative velocity of test-charged particles +attached to a detector in the 2-D surface perpendicular to the direction of propagation of the +wave while passing through the detector [cf. (Fig. 1)]. In other words, EM waves directly +displace test particles by giving them a momentum (kick), resulting in a relative velocity +change. This is different from GW memory as the GW does not displace test particles. +Instead, GW distorts the space-time geometry itself, which causes a change in separation +between two test particles. +Bieri and Garfinkle were the first to propose the memory effect due to electromagnetic +2 + +x1 +x2 +x3 +Before the wave passes +After the wave passes +FIG. 1. Electromagnetic memory effect that lies in the 2-D surface orthogonal to the direction of +the coming wave. +waves [18]. Like in GW memory, they showed that EM waves produce two types of momen- +tum kicks. In Ref. [19], Winicour showed the absence of memory effect generated by the +electromagnetic field coming from distant sources for a bound charge distribution and the +non-existence of memory effect due to the magnetic field. +In the case of GW memory, gravitational radiation must reach the detector. Likewise, EM +radiation also has to reach null infinity to generate null kick memory. Hence to calculate +EM memory, one needs to know the properties of the electric field and radiation at null +infinity [18]. More specifically, the original approach by Bieri and Garfinkle requires prior +knowledge about the behavior of the fields in asymptotic limits. It can be extended to +conformally flat space-times +[32, 34]. +Also, the analysis does not provide any physical +understanding of why the EM memory has such a form in flat and conformally flat space- +times. +This leads us to the following questions: Can we derive a master equation for EM memory +in a generic curved space-time? What role does curved geometry play in EM memory? Can +we have a physical understanding of the various contributions to EM memory? This work +addresses these three questions using 1 + 3 covariant formalism [35–40]. +3 + +There are two reasons why covariant formalism is better suited to studying EM memory. +First, as mentioned earlier, when the EM wave propagates in a given spatial direction, +the net momentum experienced by the particle lies in the 2-D surface orthogonal to the +direction of propagation of the EM wave (for a pictorial representation, see Fig. 1). In other +words, the EM memory affects the test particle lying on the 2-D surface. Hence, it is more +natural to have a formalism that identifies such a dynamical 2-D surface and evaluates EM +memory. Second, like in fluid mechanics, we can observe the flow of EM radiation in two +ways. First, as in Refs. [18, 19], an asymptotic stationary observer monitors changes in +Electric and Magnetic fields of the incoming EM radiation. Second, a comoving observer +monitors changes in Electric and Magnetic fields. In fluid mechanics, these are referred to +as the Lagrangian and Eulerian descriptions of flow, respectively. It is well-known that the +Eulerian description is better suited for fluids and in cosmology [37, 38, 40]. +In this work, we evaluate the memory effect using the 1+1+2 covariant formalism [37, 41– +44]. The 1 + 1 + 2 decomposition of space-time is a natural extension of the 1 + 3 formalism +in which the three-space is further decomposed to a given spatial direction. This approach +is also referred to as semi-tetrad formalism [45–49]. The principle advantage is that we can +evaluate the net momentum (kick) vector on the 2-D surface for arbitrary space-time. Since +this affects all the test particles on the 2-D surface, we refer to this as memory vector. This +can also be understood using the fact that the electric and magnetic fields are transverse +to the direction of propagation of the EM wave. Using the 1 + 1 + 2 covariant formalism, +we obtain the master equation for the EM memory in arbitrary space-time. We provide +a geometrical understanding of the various contributions to the memory effect. We then +obtain the EM memory for specific space-times. +The rest of this work is organized as follows: In Sec. II, we provide an overview of the +two — 1+3 and 1+1+2 — covariant formalisms and obtain the key geometrical quantities. +Then, in Sec. III, we rewrite Maxwell’s equation in 1+3 and 1+1+2 covariant formalisms in +arbitrary space-time. Next, in Sec. IV, we obtain the master equation for the EM memory +in arbitrary space-time and discuss the key features. In Sec. V, we then obtain EM memory +for specific space-times and compare them with the known results in the literature. Finally, +in Sec. VI, we summarise our results and discuss possible future directions. +In this work, we use (−, +, +, +) metric signature and set c = 1/(4πǫ0) = 1. A dot +4 + +denotes a derivative with respect to the proper time τ. A prime denote derivative w.r.t the +space-like vector nµ. For easy comparison, we follow the notations of Ref. [40]. +II. +OVERVIEW OF COVARIANT FORMALISM +A covariant theory like general relativity does not favor any particular coordinates. How- +ever, splitting tensors in time and space is typically required for its physical meaning. Thus, +the splitting achieves this by rewriting Einstein’s equations as a set of constraint and evo- +lution equations in a three-dimensional framework. This allows for an intuitive evaluation +of the relevant physical system. +A choice of coordinates defines a threading of space-time into lines and a slicing into +hypersurfaces [50]. Thus, the splitting procedure can be carried out in two distinct ways: +First, by employing the so-called (3 + 1)− formalism or slicing of space-time [51]. Second, +by employing (1 + 3)− formalism, or threading of space-time [37, 38, 40]. In the (3 + 1)− +decomposition, the time is a label of space-like slices Σt with space coordinates xi. +In +contrast, in the (1+3)− splitting, the time-like world lines have coordinate τ and are labeled +by xµ. In the (3 + 1)− formulation, the construction only requires space-like hypersurfaces +and does not demand causality of the time curves. However, in the (1+3)− approach, every +tensor is split into the parallel and orthogonal directions to a time-like vector (curves). +Furthermore, it does not provide any condition on the causality of the spatial distances. +Though the two approaches provide different points of view, it has been shown that they +are equivalent for space-times with symmetries [50]. We use the covariant 1 + 3 formalism +in this work to obtain EM memory. As mentioned in the introduction, covariant formalism +provides a physical understanding of the origin of EM memory in arbitrary space-time. +A. +Covariant 1+3 Formalism +Heckmann, Schucking, and Raychaudhuri developed the covariant approach to General +relativity in the 1950s [35, 36] and was later used in different gravitational and cosmological +models [37–40]. To decompose the 4-D space-time in (1 + 3)− formalism, we introduce a +family of observers with worldlines tangent to a timelike 4-velocity vector uµ satisfy the +5 + +3-space orthogonal to uµ +3-space orthogonal to uµ +Observer’s worldline, uµ = dxµ +dτ +FIG. 2. Visualisation of 1 + 3 formalism. +following: +uµ = dxµ +dτ ; +uµuµ = −1 , +(1) +where τ is the proper time measured along the fundamental world line. See Fig. 2. Using +the 4-velocity (uµ) we can define the following projection tensors [38, 40]: +Uµ +ν = −uµuν; +Uµ +ν Uν +γ = Uµ +γ; +Uµ +µ = 1 +(2a) +hµν = gµν + uµuν; +hµ +ν hν +γ = hµ +γ; +hµ +µ = 3; +hµν uν = 0 +(2b) +uµ, and hence Uµ ν, projects physical quantities parallel to the 4-velocity of the observer +and hµν projects quantities into the 3-space orthogonal to uµ. The tensor hµν provides the +metric properties of the instantaneous 3-space as well in the absence of rotation or vorticity. +In this formalism, the projection of the vector (V ν) orthogonal to uµ is defined as V<µ>. +Similarly, the trace-less part of a rank-2 tensor (Sαβ) projected into space orthogonal to uµ +6 + +is defined as S<µν>. Mathematically, these are given by: +V<µ> := hµν V ν; +S<µν> := +� +hµαhνβ − 1 +3hµνhαβ +� +Sαβ +(3) +The projection of the time derivative and orthogonal spatial derivative of any vector (V ν) +and tensor (Sαβ) are defined as: +˙V <µ> := hµ +αuν∇ν V α; +Dα Sβγ := hµ +α hβ +ν hγ +ρ ∇µ Sνρ +(4) +The covariant derivative of uµ can be split into two parts: 1) directional derivative along +the tangent to the world line, 2) spatial derivative in the 3-space orthogonal to uν. This +can further be split into trace, traceless symmetric and anti-symmetric tensor: +∇νuµ = Θ +3 hµν + σµν + ωµν − ˙uµuν . +(5) +In the above equation, σµν is the symmetric expansion tensor that describes the distortion +in the matter flow, Θ corresponds to the expansion rate of the matter w.r.t the observer, +ωµν is the anti-symmetric vorticity tensor describing the rotation of the matter w.r.t a non- +rotating frame. The last term refers to the relativistic acceleration vector (the directional +derivative) ˙uµ = uν∇ν which corresponds to the degree to which the matter moves under +forces other than gravity plus inertia. Further, using the vorticity tensor, we can define the +following quantity called the vorticity vector: +ων = −1 +2ǫµναβωαβ uµ +(6) +where, ǫµναβ = +1 +√−gηµνρσ is fully antisymmetric tensor, ηµνρσ is Levi-Civita symbol whose +values are ±1 and we set η0123 = 1 = −η0123 [52]. The Levi-Civita 3-tensor is defined as: +ǫµνα ≡ ǫµναβuβ , +(7) +and satisfies the following relations: ǫµνuν = 0 and ǫµναβ = 2 +� +u[µǫν]αβ − ǫµν[αuβ] � +. The +square bracket w.r.t the indices refers to antisymmetrization. +B. +1+1+2 covariant formalism +The 1 + 3-covariant formalism is well-suited for relativistic cosmology because, at the +largest observable scales, the universe is homogeneous and isotropic [38]. These symmetries +7 + +allow the slicing or threading of the 4-D space-time manifold into a one-parameter family of +spacelike hypersurfaces corresponding to cosmic time. Interestingly, it is easy to show that +in the Friedmann-Lemaitre-Robertson-Walker (FLRW) background, all physical quantities +except for the volume expansion Θ and the energy density vanish. +Using the Stewart- +Walker lemma, in this formalism, it was possible to construct gauge invariant quantities +up to second order in cosmological perturbations [53, 54]. However, the 1 + 3-formalism is +not suited if the space-time is inhomogeneous, like spherical symmetry or space-times with +local rotational symmetry (LRS) [41]. In such cases, splitting the 3-space orthogonal to +the time-like congruence into one spacelike direction and a 2-space is apt [37]. Thus, the +1 + 1 + 2 decomposition of space-time is a natural extension of the 1 + 3 formalism in which +the three-space is further decomposed to a given spatial direction. This approach is called +semi-tetrad formalism [45–49]. +As mentioned in the Introduction, our interest is to evaluate the net momentum experi- +enced by a test particle after the electromagnetic wave passes through the space-time point. +In the covariant 1 + 3 formalism, the test particle is the fundamental time-like observer. +As depicted in (Fig. 1), when the EM wave propagates in a given spatial direction, the net +momentum experienced by the particle lies in the 2-D surface orthogonal to the direction +of propagation of the EM wave. In other words, the net momentum (kick) vector lies in the +2-D surface. Thus, the net memory effect of the test particle will lie on the 2-D surface; +hence, we will refer to this as the memory vector. This can also be understood using the +fact that the electric and magnetic fields are transverse to the direction of propagation of +the EM wave. Thus, it is cogent to further split the 3-space to 1 + 2-space. +More specifically, choosing a generic space-like vector (nµ), we split the 3-space into 1 + +2-space [41–44]. The space-like vector (nµ) satisfies the following conditions: +nµnµ = 1, +nµuµ = 0 . +Like in the 1 + 3-formalism, we project the vectors and tensors defined in 3-space along +the space-like direction (nµ) and into the 2-space that is orthogonal to nµ. Here again, the +projection tensor (˜hµν) need to be defined: +˜hµν = hµν − nµnν; +˜hµ +ν ˜hν +γ = ˜hµ +γ; +˜hµ +µ = 2; +˜hµν uν = 0; +˜hµν nν = 0 . +(8) +All the vectors and tensors defined in the 3-space in the 1 + 3-formalism can be split into +8 + +1 + 2 form. For instance, an arbitrary space-like vector V µ (defined in the 3-space) can be +written as: +V µ = V nµ + V µ +(9) +where, V = V µnµ and V µ = ˜hµ νV ν. Similarly an arbitrary tensor vµν on the 3-space can +be split as: +vµν = V +� +nµnν − 1 +2 +˜hµν +� ++ 2V(µnν) + Vµν , +(10) +where V(µnν) = (Vµnν +nνVµ)/2. Similarly, the relative acceleration of the time-like observer +and other geometrical quantities defined in 3-space can be written in 1 + 2 space as: +˙uµ = A nµ + A µ +(11) +˙nµ = A uµ + αµ +(12) +ωµ = Ωnµ + Ωµ +(13) +σµν = Σ +� +nµnν − 1 +2 +˜hµν +� ++ 2Σ(µnν) + Σµν +(14) +where ˙nµ := uν∇ν nµ is the relative acceleration of the space-like vector along the time-like +observer. Here, A µ, αµ, Σµν, Ωµ are orthogonal to nµ as well as uµ. Also, A µ, Ωµ(Σµν) are +the vectors (tensor) projected on the 2-space. In this formalism, we define the alternating +Levi-Civita 2-tensor +ǫµν ≡ ǫµναnα +(15) +which is orthogonal to nµ and has components only in the 2-space. Given an arbitrary +vector V µ in the 2-space, we can construct another vector ǫµνV ν that is orthogonal to V µ +which is in the 2-space and has the same length. +The 1+2 splitting of the 3-space leads to a new directional derivative along the space-like +vector nµ: +v′ +µν ≡ nαDαvµν +(16) +˜Dαvµν ≡ ˜hα +β˜hµ +ρ˜hν +σDβvρσ . +(17) +The derivative in Eq. (16) physically correspond to the variation of the physical quantities +on the 2-space along the space-like vector nµ. The derivative ( ˜D) in Eq. (17) corresponds +9 + +to the variation of the physical quantities that lie in the 2-space. These will contribute to +the memory vector. +As we split the covariant derivative of uµ in Eq. (5), similarly we can split the covariant +derivative of nµ as: +Dνnµ = ˜Dνnµ + nµn′ +ν = ˜σµν + ˜ωµν + 1 +2 +˜Θ˜hµν + nµn′ +ν +(18) +where, ˜σµν ≡ ˜D<νnµ>, ˜ωµν ≡ ˜D(νnµ) and ˜Θ = ˜Dµnµ are shear, vorticity and the surface +expansion-contraction scalar respectively and n +′ +µ is the spatial derivative along nµ. Thus, +˜Dνnµ describes the kinematic properties or the relative motion of the space-like curves in +the 2-surface orthogonal to nµ. We can obtain the relation between the kinematic quantities +derived from the motion of time-like vector uµ and kinematic quantities in 2-space derived +from the space-like vector nµ. See, for instance, Ref. [44]. +III. +ELECTROMAGNETIC THEORY IN COVARIANT FORMALISM +The covariant formalism has been extensively employed in studying the evolution of +electromagnetic fields in curved space-time [43]. In the covariant formulation, the dynamics +and kinematics are constricted by the Bianchi and Ricci identities. The (1 + 3)− covariant +formulation permits the classification of cosmological models, a fluid description of the +matter field in FLRW universes. +However, as mentioned earlier, the 1 + 3-formalism is +not suited if the space-time is inhomogeneous, like spherical symmetry or space-times with +LRS [41]. In such cases, the 1 + 1 + 2-covariant or semi-triad formalism are better suited. +Since we aim to derive EM memory for arbitrary space-times, we use 1 + 1 + 2-covariant +formalism. We obtain a generic form of the EM memory effect by evaluating the change +in the velocity vector ∆uµ that lie in the 2-space. In order to do so, we fix the space- +like direction to be the direction of the propagation of the wave. In the case of spherically +symmetric space-time, this naturally translates to the radial direction. One key advantage is +that the electromagnetic theory in the 1+1+2 formalism helps to understand the evolution +and dynamics of the EM fields along the space-like direction and in the 2-space normal +to nµ and uµ. Our approach makes geometrical contributions to the memory effect more +transparent. +In the next subsection, we rewrite Maxwell’s equations in 1+3 formalism in an arbitrary +10 + +space-time. Later, we formulate the evolution equations of the EM fields in the 2-space and +two constraint equations of the same along uµ and nµ [44]. The key advantage is that we +can obtain the memory vector from the projected acceleration vector onto the 2-space. +A. +In 1+3 formalism +The fundamental objects are the Maxwell electromagnetic field tensor F µν. The (1 + +3) covariant formalism of Maxwell’s electromagnetic theory provides a way to study the +interaction of EM fields with different components of general space-time geometry [43]. +With the (1 + 3) decomposition, it is possible to split F µν into the electric and magnetic +fields. Note that the local coordinates are mathematical parameters that label the points +of the space-time manifold M; therefore, the electric and magnetic fields may not have a +direct physical meaning. In order to make measurements, an observer brings in an additional +structure on M by introducing the orthonormal coframe field. This gives rise to the split of +Maxwell’s tensor F into the physical electric and magnetic fields. +Specifically, formalism allows us to split the equations of motion of the fields and currents +into two parts: +1. projected parallel to the 4-velocity uµ of the fundamental observer +2. projected into the 3-space orthogonal to uµ. +To keep the calculations tractable, we perform all the calculations in source-free and lossless +regions. +However, the EM memory analysis can be straightforwardly extended to these +regions. In the source-free regions, Maxwell’s equations are: +∇νF µν = 0 +(19) +∇[γFµν] = 0; +or +∇νF ∗µν = 0 , +(20) +where F ∗µν is the dual to F µν and is defined as F ∗µν = (1/2)ǫµναβFαβ. +In the 1 + 3 formalism, by projecting F µν and F ∗µν along the time-like 4-velocity vector, +we can decompose them into electric and magnetic parts. The electric (Eµ) and magnetic +(Bµ) 4-vectors are defined as: +Eµ := F µνuν +(21) +Bµ := F ∗µνuν +(22) +11 + +From the above definitions, we infer: +Eµuµ = 0; +Bµuµ = 0 +(23) +which implies Eµ and Bµ have only spatial components. Given this, we can rewrite Fµν and +F ∗µν as: +Fµν = uµEν − uνEµ + ǫµναβBαuβ +(24) +˜F αβ = ǫαβµνuµEν + +� +uαBβ − uβBα � +. +(25) +From the above expressions, we see that the simultaneous transformations Eµ → −Bµ, +Bµ → Eµ leads to F ∗µν → F µν. This implies that we can obtain the second Maxwell’s +equation (20) from the first Maxwell’s equation (19) or vice-versa. More specifically, if we +obtain the time-like part and space-like part of Maxwell’s equations (20), we can write the +time-like part and space-like part of the other Maxwell’s equations (19) by substituting +Eµ → −Bµ, Bµ → Eµ. +In the rest of this subsection, we obtain Maxwell’s equations by projecting along uµ +(time-like part) and hµν (space-like part) [55]. We first obtain the time-like part of Eq. (20) +by multiplying it with uµ: +uα +� +∇β ˜F αβ � += 0 +(26) +Using the decomposition in Eq. (25), the above expression becomes: +∇βBβ − Bβ ˙uβ + (∇βuα) ǫαβµνuµEν = 0 +(27) +We simplify the above equation using the following steps: First, we combine the first two +terms in the LHS. From Eq. (26), we have Bβ ˙uβ = −uβ ˙Bβ = −uβuα∇αBβ. Substituting +in the second term of the above expression, we have δα +β ∇αBβ + uβuα∇αBβ = hα +β +� +∇αBβ� +. +Substituting ∇βuα from Eq. (5) and using the definition of vorticity vector in Eq. (6), the +third term in the LHS of the above expression simplifies to −2ωβEβ. Thus, the time-like +part of Eq. (20) reduces to: +DβBβ = 2ωβEβ . +(28) +The space-like part of Eq. (20) can be obtained by multiplying it with hµ ν, +hα +ρ � +∇β ˜F αβ � += 0 +(29) +12 + +Using a series of steps, the above expression can be rewritten as: +˙B<ρ> = +� +σρ +β + ωρ +β − 2Θ +3 hρ +β +� +Bβ − ǫρµν ˙uµEν − ǫρµν ∇µEν . +(30) +where, ǫµνα is defined in Eq. (7). The above equation provides the dynamical evolution of +the magnetic field, while Eq. (28) is the constraint equation. +As mentioned above, performing simultaneous transformation Eµ → −Bµ and Bµ → Eµ +in Eqs. (31) and (32), we obtain the time-like and space-like parts of the first Maxwell’s +equation (19): +DβEβ = −2ωνBν +(31) +˙E<ρ> = +� +σρ +β + ωρ +β − 2Θ +3 hρ +β +� +Eβ + ǫρµν ˙uµBν + ǫρµν DµBν . +(32) +Similarly, the above equation provides the dynamical evolution of the electric field, while +Eq. (31) is the constraint equation. +B. +In 1+1+2 formalism +We aim to calculate the memory effect of EM fields. As the memory vector resides in +the 2-surface orthogonal to the direction of propagation of the in-coming wave, we need to +decompose the 3-space to 1 + 2-space w.r.t a given spatial direction. In this subsection, +we rewrite Maxwell’s equations (19, 20) using the space-like vector nν and the projection +tensor (8) in 1 + 1 + 2 formalism. +To do this, we first express the EM fields and currents in 3-space into 1 + 2 form: +Eµ = E nµ + E µ +(33) +Bµ = Bnµ + Bµ . +(34) +where, E ≡ Eµnµ, E µ ≡ ˜hµ νEν, B ≡ Bµnµ, and Bµ ≡ ˜hµ νBν. Following the discussion in +Sec. (II B), it follows that ǫµνE ν is orthogonal to E µ and, similarly, ǫµνBν is orthogonal to +Bµ. If electric and magnetic fields are orthogonal to each other in 2 space, then we have +E ν = ǫµνBν +Bν = − ǫµνE ν . +(35) +These relations will play an important role in Sec. (IV) to derive the memory effect. +13 + +The second step is to split the evolution equations (30, 32) interms of E , E µ, B, Bµ. +To do that, we project Eq. (32) along spacelike direction nµ and multiply Eq. (32) with +projection tensor (8). After a long calculation, we obtain the following evolution equations +for E (along nµ) and E µ (in the orthogonal 2-space): +˙E + ΘE = αµEµ − 2˜ωB + ǫµρ ˜DµBρ +(36) +˙E¯µ + Θ +2 Eµ = − (αµ + 2ǫµρΩρ) E + (Σµρ + Ωǫµρ) E ρ + ǫµρ +� +A ρ − n′ρ + ˜Dρ� +B +− ǫµρ +� +A Bρ + B′ρ − +� +˜DρBν +� +nν� +, +(37) +where, ˜ω = ˜ωµν ǫµν, Θ is the expansion factor defined in Eq. (5), A µ is the relative accel- +eration vector in 2-space defined in Eq. (11), ˜ω is the vorticity defined in Eq. (18). Ωµ, Ω +is defined in Eq. (13) and Σµν is in Eq. (14). The 2-space component of ˙nµ is αµ which is +defined in Eq. (12), whereas A = nµ ˙uµ = −uµ ˙nµ mentioned in Eq. (11), (12). +We want to highlight the following points regarding the above expressions: First, the +above equations generalize Ampere’s law for arbitrary space-time. For example, in Eq. 36, +the first term in the LHS corresponds to the time derivative of the electric field along space- +like direction nµ and the last term in RHS is the curl of the magnetic field in 2-space. +Similarly, the LHS of Eq. (37) is the time derivative of the electric field in 2-space, and in +the last term in the RHS is the curl of Bρ. Second, in the flat space-time, the expansion +factor (Θ), the relative acceleration vector (αµ), and vorticity (˜ω) vanish, and the above +expression lead to Ampere’s law in flat space-time. Thus, background space-time introduces +new couplings between the electric and magnetic field components. Lastly, we showed that +the simultaneous transformation Eµ → −Bµ, Bµ → Eµ leads to F ∗µν → F µν. Substituting +E → B; E µ → Bµ and B → −E ; Bµ → −E µ in Eqs. (36, 37), we have: +˙ +B + ΘB =Bµαµ + 2˜ωE − ǫµρ ˜DµE ρ +(38) +˙ +B¯µ + 1 +2ΘBµ = − (αµ + 2ǫµρΩρ) B + (Σµρ + Ωǫµρ) Bρ − ǫµρ +� +A ρ + ˜Dρ − n′ρ� +E ++ ǫµρ +� +A E ρ + ǫµρE ′ρ − +� +˜DρEν +� +nν� +(39) +Note that we obtain the above equations by projecting Eq. (30) along spacelike direction +nµ and multiply Eq. (30) with projection tensor (8). Again, the above equations generalize +Faraday’s law for arbitrary space-time. +14 + +The last step is to split the constraint equations (31, 28) interms of E , E µ, B, Bµ. Sub- +stituting (33, 34) and the kinematic quantities (11-14), we get: +˜DµEµ + nµE ′ +µ + E ′ + ˜ΘE + 2 (ΩB + ΩµBµ) = 0 +(40) +˜DµBµ − n′µBµ + B′ + ˜ΘB − 2 (ΩE + ΩµEµ) = 0 +(41) +where ˜Θ is the expansion along the space-like vector defined in Eq. (18). The above equations +are generalizations of Gauss law. Here again, in the flat space-time, the expansion factor +(˜Θ), the relative acceleration vector (αµ), vorticity (Ω) vanish, and the above expressions +lead to Gauss law in flat space-time. +C. +Energy-momentum tensor of the electromagnetic field +As we will show in the next section, the electromagnetic stress tensor plays a crucial role +in understanding the memory effect. This subsection evaluates the electromagnetic stress +tensor in 1 + 1 + 2 formalism for an arbitrary space-time. The EM action in an arbitrary +background is: +S = −1 +4 +� +d4x √−g FµνFρσgµρgνσ . +(42) +Varying the above action w.r.t the metric (gµν) leads to the following energy-momentum +tensor: +Tµν = 1 +2gρσFµρFνσ − 1 +8gµνgρσgαβFραFσβ . +(43) +In 1 + 3-formalism, the stress-tensor of matter field (Tµν) can written as: +Tµν = ρ uµuν + 2 S(µ uν) + Wµν , +(44) +where, the energy-density ρ, the energy flux Sα and stress-tensor W αβ as measured in the +observer’s worldline are given by [56]: +ρ = Tµνuµuν, +Sα = −hα +µ T µνuν, +W αβ = hα +µ T µνhβ +ν +(45) +For the electromagnetic fields in 1 + 3-formalism, ρ, Sµ and Wµν are: +ρ ≡ 1 +2 (EµEµ + BµBµ) ; +Sµ ≡ ǫµνρEνBρ +(46) +Wµν ≡ 1 +2 (EµEµ + BµBµ) hµν − EµEν − BµBν +(47) +15 + +Rewriting ρ interms of the variables (E , E µ, B, Bµ) in 1 + 1 + 2 formalism, we have: +ρ = 1 +2 +� +E 2 + B2� ++ 1 +2 (E µEµ + BµBµ) = ρ(n) + ρ2−space +(48) +Thus, ρ(n) corresponds to the energy of the EM field along nµ and ρ2−space corresponds to +the energy of the EM field in the 2-space. The energy flux Sµ (a vector in 3-space) can be +rewritten in 1 + 2 space as: +Sµ = S nµ + Sµ +(49) +where S is the Poynting vector of the EM field along the space-like vector nµ and Sµ is +the energy flux in the 2-space. These are given by: +S = Sµnµ = ǫµνE µBν +(50) +Sµ = −ǫµν (E Bν − BE ν) = − (E E ν + BBν) +(51) +In deriving the last expression, we have used the orthogonality condition between the electric +and magnetic fields in the 2-space, i. e., Eν = ǫνµBµ. As we will see in the next section, the +memory vector depends on the part of the electromagnetic energy density ρ and Sµ. +IV. +MEMORY EFFECT IN ARBITRARY SPACE-TIME +Having written Maxwell’s equations in 1 + 1 + 2 formalism for an arbitrary space-time, +we now evaluate the memory effect. Usually, in the literature, one uses the Lorentz force +equation to derive EM memory. The equation of motion of a charged body (of mass m and +charge e) in both gravitational and electromagnetic fields are: +mduα +dτ − m +2 gβγ,αuβuγ = eFαβuβ +(52) +However, the above expression does not consider the new couplings between the electric +and magnetic field components in Eqs. (36) - (39). Hence, we use the complete Maxwell’s +equations (36) - (41) and explicitly obtain the change in velocity (∆uµ) of the time-like +observer. More specifically, using Eqs. (37, 39), we first calculate the acceleration vector +A µ in the 2-space. We can then integrate the expression for the acceleration vector (A µ +in the 2-space) with respect to time t or null time coordinate u ≡ (t − r) leading to the +memory vector. +16 + +In the rest of this section, we calculate A µ for observers whose tangents are congruent to +the space-like geodesics. This implies nσDσnρ = n′ρ = 0, i. e., nµ is tangent to a congruence +of space-like geodesics [44]. +Using this condition and substituting +˙E¯µ = ˜hµν ˙E ν, B′ ρ = +nνDνBρ in Eqs. (37, 39), we get: +˜hµν ˙E ν + ǫµρnνDνBρ = − 1 +2ΘEµ − (αµ + 2ǫµρΩρ) E + (Σµρ + Ωǫµρ) E ρ ++ +� +ǫµρA ρ + ǫµν ˜Dν� +B − ǫµν +� +˜Dνnρ� +Bρ − ǫµρA Bρ +(53) +� +˜hµν ˙ +Bν − ǫµρnνDνE ρ� += −1 +2ΘBµ − (αµ + 2ǫµρΩρ) B + (Σµρ + Ωǫµρ) Bρ +− +� +ǫµρA ρ + ǫµν ˜Dν� +E + ǫµν +� +˜Dνnρ� +Eρ + ǫµρA E ρ +(54) +Multiplying Eq. (53) with B, multiplying Eq. (54) with E and subtracting the resultant +equations leads to: +ǫµνA ν = − ǫµν +2 +Dν(E 2 + B2) +(E 2 + B2) ++ +� +Σµν + Ωǫµν − Θ +2 +˜hµν +� (E Bν − BE ν) +(E 2 + B2) ++ ǫµν +� +˜σρν + ˜ωρν + +˜Θ +2 +˜hρν +� +(BBρ + E Eρ) +(E 2 + B2) ++ ǫµρA (E E ρ + BBρ) +(E 2 + B2) ++ +B +(E 2 + B2) +� +˜hµν ˙E ν + ǫµρnνDνBρ� +− +E +(E 2 + B2) +� +˜hµν ˙ +Bν − ǫµρnνDνE ρ� +(55) +To have a transparent understanding, we substitute the definitions (48) - (51) in the expres- +sion above, resulting in: +ǫµνA ν = − ǫµν +2 +Dνρ(n) +ρ(n) +− ǫνα +2 +� +Σµν + Ωǫµν − Θ +2 +˜hµν +� Sα +ρ(n) +− ǫµν +2 +� +˜σρν + ˜ωρν + +˜Θ +2 +˜hρν +� +Sρ +ρ(n) +− ǫµρS ρ A +2ρ(n) ++ +B +2ρ(n) +� +˜hµν ˙E ν + ǫµρnνDνBρ� +− +E +2ρ(n) +� +˜hµν ˙ +Bν − ǫµρnνDνE ρ� +. (56) +This is the master equation for the EM memory in arbitrary space-time regarding which we +would like to discuss the following points: First, to our understanding, this is a first time the +EM memory has been obtained for an arbitrary space-time. In the previous calculations [18, +19], the authors have restricted to asymptotic flat space-times. Second, the last two terms +in the RHS of the above expression vanishes in the asymptotic limit. To see this, let us +consider a spherically symmetric space-time. Let t refer to the time coordinate and r to the +radial coordinate and the null coordinate is u ≡ t − r. In the asymptotic limit ∂u ∼ ∂t and +17 + +∂u ∼ −∂r. Setting uµ ≡ (1, 0, 0, 0) and nµ ≡ (0, 1, 0, 0), the penultimate term in the RHS +of the above equation simplifies to: +˜hµν ˙E ν + ǫµρnνDνBρ ≃ ˜hµνu0∇0E ν + ǫµρn1∇1Bρ ≃ ˜hµν∂uE ν − ǫµρ∂uBρ += f(u)∂u +�¯˜hµνE ν − ¯EµνBν� +(57) +where, ˜hµν = f(u)¯˜hµν and ¯ǫµν = f(u)ǫµν. The terms with bar represent their time indepen- +dent parts. The above expression vanishes if E ν and Bν are orthogonal to each other in +the 2-space. As we mentioned earlier (35), in 2-space, the electric and magnetic fields are +always orthogonal to each other. Similarly, the last term can also be shown to vanish in the +asymptotic limit. Thus, the above expression reduces to: +ǫµνA ν = − ǫµν +2 +Dνρ(n) +ρ(n) +− ǫνα +2 +� +Σµν + Ωǫµν − Θ +2 +˜hµν +� Sα +ρ(n) +− ǫµν +2 +� +˜σρν + ˜ωρν + +˜Θ +2 +˜hρν +� +Sρ +ρ(n) +− ǫµρ +2ρ(n) +S ρ A +(58) +Third, the above expression provides a nice geometrical understanding of the various contri- +butions to memory effect. The first term in the RHS corresponds to the change in the EM +field energy (ρ(n)) along nµ in the 2-space. This does not contain any contribution from the +kinematical properties of the space-time. In other words, this term will vanish if the EM +field energy does not change in the 2-space, like a 2-D flat sheet. However, as we show in the +next section, this is non-zero in flat space-time expressed in spherical coordinates. The next +two terms in the RHS are proportional to the energy flux (Sα) in the 2-space. However, +both these terms have different kinematical information of the space-time and vanish for flat +space-time. The second term in the RHS carries information about shear (Σµν), vorticity +scalar (Ω) related to nµ and expansion scalar (Θ) corresponding to time-like observer uµ. +The third term in the RHS carries information about shear (˜σµν), vorticity tensor (˜ωµν) and +expansion scalar (˜Θ) corresponding to the space-like vector nµ. +Fourth, as mentioned earlier, we have not included external currents or charges in our +analysis. +Hence, the acceleration vector does not have contribution from the external +sources. +Hence, the memory vector we obtain is equivalent to the null-kick derived in +Refs. [18, 19]. It is also important to note that these authors did not obtain the contribu- +tions due to the kinematical properties of the space-time. However, as we will see in the +next section, their contribution can be significant. +18 + +Lastly, to obtain the memory vector, we need to integrate the above expression w.r.t the +proper time of the observer — ∆uµ is the memory vector. It is interesting to note that +initially if the observer has non-zero velocity only along the time direction, at a later time, +due to the memory effect, there is a non-zero velocity in the 2-space. +V. +APPLICATION TO SPECIFIC SPACE-TIMES +In the previous section, we obtained a master equation for the EM vector for an arbitrary +4-D space-time using 1 + 1 + 2-formalism. As we discussed, the memory vector has three +distinct contributions. In order to illustrate this fact, we consider specific examples and +obtain the memory vector. In this section we obtain memory vector for flat, FLRW, pp- +wave and Kerr space-times. +A. +Minkowski space-time +In order to compare the master equation with the existing results [18], we first consider +Minkowski space-time in spherical coordinates: +ds2 = −dt2 + dr2 + r2 γAB +(59) +where, +γAB = + + 1 +0 +0 sin2 θ + + +(60) +is the metric describing unit 2-sphere. In Minkowski space-time, the 4-velocity of the time- +like congruence observer is uµ ≡ (1, 0, 0, 0) and the space-like vector is nµ ≡ (0, 1, 0, 0). +Since ∇µuν = 0 and ∇µnν = 0, the kinematics quantities, defined in Sec. (II A, II B) vanish +for the Minkowski space-time. Hence only the first term in Eq. (56) will be non-zero, i. e., +A ν +Flat = − 1 +2 +Dνρn +ρn +. +(61) +As mentioned earlier, the acceleration vector corresponds to acceleration in the 2-Sphere. +Hence, it is appropriate to switch to the 2-Sphere index: +A A = uµ∇µuA = u0∂0uA + 2u0ΓA +0 BuB . +19 + +Since the 4-velocity uµ is zero in the 2-Sphere, we have A A = u0∂0uA = ∂tuA. In null +coordinate, this becomes A A = ∂tuA. Substituting the above expression in Eq. (61) and +integrating in the null coordinate, we have: +∆uA ≡ +� +du A A = −1 +2 +� +du DAρn +ρn +. +(62) +The above expression is velocity kick w.r.t the Eulerian observers. To compare this with +the net momentum (kick) vector as seen by the asymptotic static observers (Lagrangian +observers), we need to do a coordinate transformation. Specifically, we need to transform +from coordinate basis +� +⃗e θ,⃗e φ� +to orthogonal coordinate basis +� +ˆθ, ˆφ +� +. In terms of +� +ˆθ, ˆφ +� +, +we have ∆⃗u ≡ ∆uµ⃗eµ, where, ⃗e θ = ˆθ/r , ⃗e φ = ˆφ/(r sin θ). Thus, the velocity kick w.r.t the +asymptotic static observers is given by: +∆⃗uFlat = 1 +r +� +∆uθˆθ + ∆uφ +sin θ +ˆφ +� +(63) +Interestingly, the EM memory vector in Minkowski space-time is inversely proportional to r +and matches with Ref. [18]. This passes the first test that the master equation (56) indeed +describes the EM memory vector. In the rest of this section, we obtain the memory vector +for non-flat geometries and show the robustness of our approach. +B. +FLRW space-time +The conformally flat FLRW metric in spherical coordinates is: +ds2 = a(η)2 � +−dη2 + dr2 + r2 γAB +� +(64) +where, the conformal time (η) is related to the cosmic time (t) by dt = a(η) dη. In 1 + 3 +formalism, the fundamental observer with time-like 4−velocity in FLRW metric is uµ = +dxµ/dt = dxµ/(a(η)) dη = ( 1, 0, 0, 0 ) /a(η). +For this choice of observer, the 3−space +projection tensor (hµν) orthogonal to uµ is: +hµν = + + a2(η) +0 +0 +a2(η) r2 γAB + + . +(65) +Since the FLRW line-element is homogeneous and isotropic, only the expansion scalar (Θ) +is non-zero: +Θ = 3H (η) +a(η) +where +H = a′(η) +a(η) +20 + +where ′ refers to derivative w.r.t. η. Other kinematic quantities vanish, i. e., σµν = ωµν = 0. +We now spilt the 3-space into 1 + 2 by choosing the following space-like vector nµ = +(0, 1, 0, 0)/a(η). This satisfies the conditions: nµnµ = 1 and uµnµ = 0. Repeating the +steps discussed in Sec. (II B) for the line-element (64), we get: +˜Θ = +2 +a(η) +1 +r , ˜σµν = ˜ωµν = 0. +It is important to note that while Θ is a function of η only, ˜Θ depends on both η and r. +Also, Θ depends on the Hubble parameter H , while ˜Θ is inversely proportional of r. Hence, +at large distances, ˜Θ decays faster compared to Θ. +Substituting the above expressions in Eq. (58), we have: +A ν +FLRW = − 1 +2 +Dνρn +ρn ++ +1 +4 ρn +S ν(Θ − ˜Θ) . +(66) +Like Minkowski space-time, A ν will have components only in the 2-Sphere. Using the +fact that the fundamental observers have zero velocity in the 2-Sphere and repeating the +earlier analysis, we have +A A = u0∂0uA = +1 +a(η) +∂uA +∂η . +In terms of the null coordinate u(≡ η − r), we have: +A A = +1 +a(u) +∂uA +∂u . +Substituting the above expression in Eq. (66), we have: +∂uA +∂u = − a(u) +2 +DAρn +ρn ++ a(u) +4 ρn +S A(Θ − ˜Θ) . +(67) +Integrating the above expression w.r.t u, leads to the following memory vector: +∆uA +FLRW = −1 +2 +� +du a(u) +ρn +DAρn + 1 +4 +� +du a(u) +ρn +S A(Θ − ˜Θ) +(68) +This is the expression for the memory vector in FLRW space-time regarding which we want +to highlight the following points: First, unlike Minkowski space-time, here the fundamental +observers are Eulerian, and hence, we do not have to transform the above expression to +Lagrangian observers. Second, our results differ from the results of Ref. [34]. In Ref. [34], +the authors show that the EM memory effect in FLRW differs from the Minkowski only by +the conformal factor a(η) or a(u). In other words, their analysis did not account for the geo- +metric contribution to the memory effect. As mentioned earlier, the geometric contribution +21 + +leads to a non-zero energy flux (S A) contribution. Also note that the ordinary memory +derived in Ref. [34] is not present in Eq. (68) as we have assumed any external charge or +current to be zero. Third, we find that ρ(n) and the energy flux (S A) contribute oppositely. +It will be interesting to see whether the two contributions nullify the EM memory. +C. +pp-wave space-times +In this subsection, we derive the EM memory for a special kind of plane-fronted wave +with parallel rays (pp-waves) called plane-wave metric [57]: +ds2 = −2dudv − F(u, x, y) du2 + dx2 + dy2 +(69) +where, F(u, x, y) = A(u)(x2 − y2) + 2B(u)xy describes the plane wave and A(u), B(u) are +arbitrary functions such that F > 0. Note that u, v are not light-cone coordinates. u is +time-like coordinate and v is a null coordinate. +We split the above 4-D space-time into 1 + 3 form and later into 1 + 1 + 2-form by +considering the following time-like velocity vector (uµ) and space-like vector (nµ): +uµ ≡ +� +F(u, x, y)(−1/2), 0, 0, 0 +� +, +nµ ≡ +� +F(u, x, y)(−1/2), − F(u, x, y)(1/2), 0, 0 +� +. +For the above choice of time-like vector, the 3-space projection tensor (hµν) is: +hµν = + + +0 +0 +0 0 +0 +1 +F(u, x, y) 0 0 +0 +0 +1 0 +0 +0 +0 1 + + +(70) +Substituting these in the definitions in Sec. (II), only non-zero quantity is the expansion +scalar (Θ): +Θ = − +(x2 − y2) A′(u) + 2xy B′(u) +2 (2B(u) xy + A(u) (x2 − y2) )3/2 . +(71) +The non-zero projection tensor ˜hµν components in the 2-space are ˜hxx = 1, ˜hyy = 1. Thus, +the memory vector for the special kind of pp-wave space-times is: +A ν +PP = − 1 +2 +Dνρn +ρn ++ Θ +4 ρn +S ν . +(72) +22 + +Here, the acceleration of the time-like observer is confined to the x − y plane, i. e., +A A +PP = − 1 +2 +DAρn +ρn ++ Θ +4 ρn +S A , +(73) +where, the index A, B corresponds to (x, y). Evaluating the acceleration vector along x +and y, we have: +A (PP) +x(y) = − 1 +2 ρn +∂x(y) (ρn) + Θ +4 ρn +Sx(y) . +(74) +Integrating the above equation w.r.t u, we have: +∆uPP +x(y) = −1 +2 +� +du ∂x(y) (ρn) +ρn ++ Θ +4 +� +duSx(y) +ρn +. +(75) +The above expression for the velocity kick is for a generic plane-wave metric. To gain +some physical intuition, we consider two specific forms — Penrose limit of the Schwarzschild +and FLRW space-times [57]. For Schwarzschild space-time, we have +A(u) = +6 +25u2; +B(u) = 0 +Substituting these in Eq. (71), we have: +ΘPP,Sch = +5 +� +6(x2 − y2) +. +It is interesting to note that although the space-time metric does not differentiate between +the two spatial coordinates (x, y), in order for Θ to be real, the above expression demands +that x > y. Thus, velocity kick due to EM wave in PP-wave limit of Schwarzschild space- +time can only occur if x > y and is given by: +∆uPP Sch +x(y) += −1 +2 +� +du ∂x(y) (ρn) +ρn ++ +5 +4 +� +6(x2 − y2) +� +duSx(y) +ρn +. +(76) +In the case of Penrose limit of FLRW space-time with power-law scale factor a(t) ∼ th, we +have: +A(u) = − +h +(1 + h)u2, +B(u) = 0 . +Substituting these in Eq. (71), we have: +ΘPP,FLRW = +� +(1 + h) +h(y2 − x2); +∆uPP FLRW +x(y) += −1 +2 +� +du ∂x(y) (ρn) +ρn ++ +� +(1 + h) +4 +� +h(y2 − x2) +� +duSx(y) +ρn +. +(77) +23 + +Here again, we see that in-order for Θ to be real, the above expression demands that y > x. +Thus, velocity kick due to EM wave in PP-wave limit of FLRW space-time occurs in a +different region of the 2-space compared to Schwarzschild. Thus, EM memory has a distinct +signature for different space-times and can potentially be used as a probe. +D. +Kerr space-time +In this section, we derive the memory effect in Kerr space-time. +In Boyer-Lindquist +coordinates (t, r, χ, φ), the Kerr space-time is: +ds2 = +� +2mr +r2 + a2χ2 − 1 +� +dt2 + +� +r2 + a2χ2 +r2 − 2mr + a2 +� +dr2 + +�r2 + a2χ2 +1 − χ2 +� +dχ2 +− +�4mar (1 − χ2) +r2 + a2χ2 +� +dtdϕ + +� +1 − χ2� � +r2 + a2 + 2ma2r (1 − χ2) +r2 + a2χ2 +� +dϕ2 . +(78) +where χ ≡ cos θ. In this case, the time-like observer 4-velocity (uµ) and the space-like vector +(nµ) are [58]: +uµ = +�� +r2 − 2mr + a2 +r2 + a2χ2 +, 0, 0, 0 +� +, nµ = +� +0, +� +r2 − 2mr + a2 +r2 + a2χ2 +, 0, 0 +� +. +We give below the kinematical quantities (discussed in Sec. (II B)) for Kerr space-time in +1 + 1 + 2 formalism obtained in Ref. [58]: +Θ = 0; +Σµν = 0 ; +(79) +Ω = −2marχ +√ +L +J +√ +K 3 +; +˜Θ = +W +J +√ +K 3L +; +(80) +˜ωµν = ˜ωǫµν = 0; +A = −mD +√ +L +J +√ +K 3 ; +(81) +˜σµν = + + +0 0 +0 +0 +0 0 +0 +0 +0 0 −1 +2 +a2(m−r) +√ +K +J +√ +L +0 +0 0 +0 +1 +2 +a2(m−r)M 2√ +L K +J 2 + + +(82) +where, +M = χ2 − 1; +D = −r2 + a2χ2; +L = r2 − 2mr + a2 +(83) +J = r2 − 2mr + a2χ2; +K = r2 + a2χ2 +(84) +24 + +W = 2r3(r − 2m)2 + a4χ2 � +m + r − mχ2 + rχ2� ++ a2r2 � +−3m + r + χ2(3r − 5m) +� +(85) +Substituting these expressions in Eq. (58), and noting that the memory vector lies in +the 2-D surface, we get: +A A = − 1 +2 +DAρ(n) +ρ(n) +− Ω +2 +ǫAB SB +ρ(n) +− 1 +2 +� +˜σAB + +˜Θ +2 +˜hAB +� +SB +ρ(n) +− +A +2ρ(n) +S A +(86) +This is the EM memory vector for an Eulerian observer in Kerr space-time. Note that this +is a generic result for any value of angular momentum. For a better physical insight, we +consider a → 0 limit. Substituting a → 0 in Eqs. (79 - 85), we have +M0 = χ2 − 1; +D0 = −r2; +L0 = r2 − 2mr +(87) +J0 = r2 − 2mr; +K0 = r2; +W0 = 2r3(r − 2m)2 +(88) +Ω0 = ˜σ0 +µν = 0; +˜Θ0 = 2 +� +(r − 2m) +r3 +; +A = +m +� +r3(r − 2m) +(89) +Substituting the above quantities in Eq. (86), we have: +A A = − 1 +2 +DAρ(n) +ρ(n) +− 1 +2 +� +r − 2m +r3 +S A +ρ(n) +− +1 +2ρ(n) +m +� +r3(r − 2m) +S A . +(90) +This is the EM memory vector for an Eulerian observer in Schwarzschild space-time, regard- +ing which we want to mention the following points: First, in the limit, r → ∞, reduces to +Minkowski space-time expression (61). Second, in the limit r → ∞, the subleading term is +proportional to r−1. Third, to derive the memory vector ∆uA, we have to switch to the null +time coordinate u = t − r and integrate Eq. (90) with respect to u at the asymptotic limit. +Lastly, to evaluate the memory effect experienced by the static asymptotic (Lagrangian) +observer, we need to do the transformation from +� +⃗e θ,⃗e φ� +to the orthogonal coordinate basis +� +ˆθ, ˆφ +� +like in Sec. (V A). +VI. +CONCLUSIONS +In this work, we have derived a master equation for electromagnetic memory in an arbi- +trary space-time. We used the covariant formalism to obtain the same. More specifically, +we used the 1 + 1 + 2 covariant formalism. The 1 + 1 + 2 decomposition of space-time is +a natural extension of the 1 + 3 formalism in which the three-space is further decomposed +25 + +using a given spatial direction. This choice of covariant formalism is because the net mo- +mentum (kick) vector lies on the 2-D surface for arbitrary space-time. Also, the electric and +magnetic fields are transverse to the direction of propagation of the passing EM wave. +The EM memory (58) has three distinct contributions: First contribution is due to the +change in the EM field energy (ρ(n)) along nµ in the 2-space. This is non-zero for Minkowski +space-time. The second contribution is proportional to the energy flux (Sα) in the 2-space. +This has kinematical information of the space-time and vanishes for the flat space-time. +The third contribution is proportional to the acceleration A along the time-like vector uµ. +To our understanding, the earlier approaches could not isolate the different contributions +to the EM memory as done in this work. +We then obtained the EM memory for different space-times. In the case of FLRW space- +time, we showed that the earlier analysis did not account for the geometric contribution +to the memory effect [34]. Specifically, their analysis did not account for the geometric +contribution leading to a non-zero energy flux (S A) contribution. We have also obtained +the EM memory for Kerr space-time. We also showed that the EM memory has a distinct +signature for different pp wave space-times and can potentially be used as a probe. +It would be interesting to extend our analysis for black holes with multiple horizons +and those that are not asymptotically flat. These may be particularly relevant for using +EM memory as a probe to PBH. Finally, our analysis points to the possibility of using +1 + 1 + 2 covariant formalism to understand gravitational memory. These are currently +under investigation. +ACKNOWLEDGMENTS +The authors thank S. Mahesh Chandran and A. Kushwaha for comments on the earlier +version of the manuscript. The work is supported by the SERB-Core Research grant. +[1] B. S. Sathyaprakash and B. F. Schutz, “Physics, Astrophysics and Cosmology with +Gravitational Waves,” Living Rev. Rel. 12 (2009) 2, arXiv:0903.0338 [gr-qc]. +[2] K. G. Arun and A. Pai, “Tests of General Relativity and Alternative theories of gravity using +26 + +Gravitational Wave observations,” Int. J. 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Maharaj, “A semi-tetrad decomposition of the Kerr +spacetime,” arXiv:2109.04162 [gr-qc]. +31 + diff --git a/4dFKT4oBgHgl3EQfRy0L/content/tmp_files/load_file.txt b/4dFKT4oBgHgl3EQfRy0L/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b3d6f7e3e59798b42d812c5634ec225edacb4166 --- /dev/null +++ b/4dFKT4oBgHgl3EQfRy0L/content/tmp_files/load_file.txt @@ -0,0 +1,946 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf,len=945 +page_content='3-space orthogonal to uµ 3-space orthogonal to uµ Observer’s worldline, uµ = dxµ dτ x1 x2 x3 Before the wave passes After the wave passes arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='11772v1 [gr-qc] 27 Jan 2023 Electromagnetic memory in arbitrary curved space-times Susmita Jana1, ∗ and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Shankaranarayanan1, † 1Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India Abstract The gravitational memory effect and its electromagnetic (EM) analog are potential probes in the strong gravity regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In the literature, this effect is derived for static observers at asymptotic infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' While this is a physically consistent approach, it restricts the space-time geometries for which one can obtain the EM memory effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' To circumvent this, we evaluate the EM memory effect for comoving observers (defined by the 4-velocity uµ) in arbitrary curved space-times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Using the covariant approach, we split Maxwell’s equations into two parts — projected parallel to the 4- velocity uµ and into the 3-space orthogonal to uµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Further splitting the equations into 1+1+2-form, we obtain master equation for the EM memory in an arbitrary curved space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' We provide a geometrical understanding of the contributions to the memory effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' We then obtain EM memory for specific space-time geometries and discuss the salient features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' ∗ susmitajana@iitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='in † shanki@iitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='in 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' INTRODUCTION LIGO-VIRGO-KAGRA has detected close to 100 gravitational wave (GW) sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' GW signals emanating from a black hole or neutron star binaries have opened many new research avenues in astronomy, cosmology, and fundamental physics [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' GWs provide a unique way to test gravity’s most extreme, non-linear regime in novel ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The planned third- generation ground-based detector (Cosmic Explorer and the Einstein Telescope) will allow us to peer far deeper, and LISA will open a new observational window at low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' With more sensitive detectors shortly, the focus has been to understand the physical effects of GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Gravitational wave memory is one such effect [5–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' GW memory effects — physically observable phenomena that modify the state of gravitational-wave detectors a little bit from their original undisturbed state — are one of the key predictions of general relativity [6, 7, 9, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' GW memory effects can be divided into two types [12, 13]: null memory that occurs when radiation or massless particles es- cape from a system to null infinity, and ordinary memory that occurs when the detector recoils relative to its initial center of mass frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The GW memory is characterized as a gravitational wave signal approaching a nonzero finite value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This aspect of the GW signal is yet to be observed, although LISA is predicted to observe it [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Recently, it has been realized that the memory effect can be thought of as a vacuum transition between two different states related by an asymptotic transformation [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Since such asymptotic transformations also occur for other gauge theories, there has been an intense activity to obtain analogous memory effects in other gauge theories [18–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Since electromagnetic (EM) theory is the simplest of all gauge theories and can be a potential probe, electromagnetic memory has received much attention [23–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Like in GW memory, an EM wave generates a permanent change in the relative velocity of test-charged particles attached to a detector in the 2-D surface perpendicular to the direction of propagation of the wave while passing through the detector [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' 1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In other words, EM waves directly displace test particles by giving them a momentum (kick), resulting in a relative velocity change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This is different from GW memory as the GW does not displace test particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Instead, GW distorts the space-time geometry itself, which causes a change in separation between two test particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Bieri and Garfinkle were the first to propose the memory effect due to electromagnetic 2 x1 x2 x3 Before the wave passes After the wave passes FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Electromagnetic memory effect that lies in the 2-D surface orthogonal to the direction of the coming wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' waves [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Like in GW memory, they showed that EM waves produce two types of momen- tum kicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' [19], Winicour showed the absence of memory effect generated by the electromagnetic field coming from distant sources for a bound charge distribution and the non-existence of memory effect due to the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In the case of GW memory, gravitational radiation must reach the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Likewise, EM radiation also has to reach null infinity to generate null kick memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Hence to calculate EM memory, one needs to know the properties of the electric field and radiation at null infinity [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' More specifically, the original approach by Bieri and Garfinkle requires prior knowledge about the behavior of the fields in asymptotic limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' It can be extended to conformally flat space-times [32, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Also, the analysis does not provide any physical understanding of why the EM memory has such a form in flat and conformally flat space- times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This leads us to the following questions: Can we derive a master equation for EM memory in a generic curved space-time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' What role does curved geometry play in EM memory?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Can we have a physical understanding of the various contributions to EM memory?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This work addresses these three questions using 1 + 3 covariant formalism [35–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' 3 There are two reasons why covariant formalism is better suited to studying EM memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' First, as mentioned earlier, when the EM wave propagates in a given spatial direction, the net momentum experienced by the particle lies in the 2-D surface orthogonal to the direction of propagation of the EM wave (for a pictorial representation, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In other words, the EM memory affects the test particle lying on the 2-D surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Hence, it is more natural to have a formalism that identifies such a dynamical 2-D surface and evaluates EM memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Second, like in fluid mechanics, we can observe the flow of EM radiation in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' First, as in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' [18, 19], an asymptotic stationary observer monitors changes in Electric and Magnetic fields of the incoming EM radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Second, a comoving observer monitors changes in Electric and Magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In fluid mechanics, these are referred to as the Lagrangian and Eulerian descriptions of flow, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' It is well-known that the Eulerian description is better suited for fluids and in cosmology [37, 38, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In this work, we evaluate the memory effect using the 1+1+2 covariant formalism [37, 41– 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The 1 + 1 + 2 decomposition of space-time is a natural extension of the 1 + 3 formalism in which the three-space is further decomposed to a given spatial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This approach is also referred to as semi-tetrad formalism [45–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The principle advantage is that we can evaluate the net momentum (kick) vector on the 2-D surface for arbitrary space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Since this affects all the test particles on the 2-D surface, we refer to this as memory vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This can also be understood using the fact that the electric and magnetic fields are transverse to the direction of propagation of the EM wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Using the 1 + 1 + 2 covariant formalism, we obtain the master equation for the EM memory in arbitrary space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' We provide a geometrical understanding of the various contributions to the memory effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' We then obtain the EM memory for specific space-times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The rest of this work is organized as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' II, we provide an overview of the two — 1+3 and 1+1+2 — covariant formalisms and obtain the key geometrical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Then, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' III, we rewrite Maxwell’s equation in 1+3 and 1+1+2 covariant formalisms in arbitrary space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Next, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' IV, we obtain the master equation for the EM memory in arbitrary space-time and discuss the key features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' V, we then obtain EM memory for specific space-times and compare them with the known results in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' VI, we summarise our results and discuss possible future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In this work, we use (−, +, +, +) metric signature and set c = 1/(4πǫ0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' A dot 4 denotes a derivative with respect to the proper time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' A prime denote derivative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='t the space-like vector nµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' For easy comparison, we follow the notations of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' OVERVIEW OF COVARIANT FORMALISM A covariant theory like general relativity does not favor any particular coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' How- ever, splitting tensors in time and space is typically required for its physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Thus, the splitting achieves this by rewriting Einstein’s equations as a set of constraint and evo- lution equations in a three-dimensional framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This allows for an intuitive evaluation of the relevant physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' A choice of coordinates defines a threading of space-time into lines and a slicing into hypersurfaces [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Thus, the splitting procedure can be carried out in two distinct ways: First, by employing the so-called (3 + 1)− formalism or slicing of space-time [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Second, by employing (1 + 3)− formalism, or threading of space-time [37, 38, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In the (3 + 1)− decomposition, the time is a label of space-like slices Σt with space coordinates xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In contrast, in the (1+3)− splitting, the time-like world lines have coordinate τ and are labeled by xµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In the (3 + 1)− formulation, the construction only requires space-like hypersurfaces and does not demand causality of the time curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' However, in the (1+3)− approach, every tensor is split into the parallel and orthogonal directions to a time-like vector (curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Furthermore, it does not provide any condition on the causality of the spatial distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Though the two approaches provide different points of view, it has been shown that they are equivalent for space-times with symmetries [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' We use the covariant 1 + 3 formalism in this work to obtain EM memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' As mentioned in the introduction, covariant formalism provides a physical understanding of the origin of EM memory in arbitrary space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Covariant 1+3 Formalism Heckmann, Schucking, and Raychaudhuri developed the covariant approach to General relativity in the 1950s [35, 36] and was later used in different gravitational and cosmological models [37–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' To decompose the 4-D space-time in (1 + 3)− formalism, we introduce a family of observers with worldlines tangent to a timelike 4-velocity vector uµ satisfy the 5 3-space orthogonal to uµ 3-space orthogonal to uµ Observer’s worldline, uµ = dxµ dτ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Visualisation of 1 + 3 formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' following: uµ = dxµ dτ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' uµuµ = −1 , (1) where τ is the proper time measured along the fundamental world line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Using the 4-velocity (uµ) we can define the following projection tensors [38, 40]: Uµ ν = −uµuν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Uµ ν Uν γ = Uµ γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Uµ µ = 1 (2a) hµν = gµν + uµuν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' hµ ν hν γ = hµ γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' hµ µ = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' hµν uν = 0 (2b) uµ, and hence Uµ ν, projects physical quantities parallel to the 4-velocity of the observer and hµν projects quantities into the 3-space orthogonal to uµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The tensor hµν provides the metric properties of the instantaneous 3-space as well in the absence of rotation or vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In this formalism, the projection of the vector (V ν) orthogonal to uµ is defined as V<µ>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Similarly, the trace-less part of a rank-2 tensor (Sαβ) projected into space orthogonal to uµ 6 is defined as S<µν>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Mathematically, these are given by: V<µ> := hµν V ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' S<µν> := � hµαhνβ − 1 3hµνhαβ � Sαβ (3) The projection of the time derivative and orthogonal spatial derivative of any vector (V ν) and tensor (Sαβ) are defined as: ˙V <µ> := hµ αuν∇ν V α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Dα Sβγ := hµ α hβ ν hγ ρ ∇µ Sνρ (4) The covariant derivative of uµ can be split into two parts: 1) directional derivative along the tangent to the world line, 2) spatial derivative in the 3-space orthogonal to uν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This can further be split into trace, traceless symmetric and anti-symmetric tensor: ∇νuµ = Θ 3 hµν + σµν + ωµν − ˙uµuν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (5) In the above equation, σµν is the symmetric expansion tensor that describes the distortion in the matter flow, Θ corresponds to the expansion rate of the matter w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='t the observer, ωµν is the anti-symmetric vorticity tensor describing the rotation of the matter w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='t a non- rotating frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The last term refers to the relativistic acceleration vector (the directional derivative) ˙uµ = uν∇ν which corresponds to the degree to which the matter moves under forces other than gravity plus inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Further, using the vorticity tensor, we can define the following quantity called the vorticity vector: ων = −1 2ǫµναβωαβ uµ (6) where, ǫµναβ = 1 √−gηµνρσ is fully antisymmetric tensor, ηµνρσ is Levi-Civita symbol whose values are ±1 and we set η0123 = 1 = −η0123 [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The Levi-Civita 3-tensor is defined as: ǫµνα ≡ ǫµναβuβ , (7) and satisfies the following relations: ǫµνuν = 0 and ǫµναβ = 2 � u[µǫν]αβ − ǫµν[αuβ] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The square bracket w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='t the indices refers to antisymmetrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' 1+1+2 covariant formalism The 1 + 3-covariant formalism is well-suited for relativistic cosmology because, at the largest observable scales, the universe is homogeneous and isotropic [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' These symmetries 7 allow the slicing or threading of the 4-D space-time manifold into a one-parameter family of spacelike hypersurfaces corresponding to cosmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Interestingly, it is easy to show that in the Friedmann-Lemaitre-Robertson-Walker (FLRW) background, all physical quantities except for the volume expansion Θ and the energy density vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Using the Stewart- Walker lemma, in this formalism, it was possible to construct gauge invariant quantities up to second order in cosmological perturbations [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' However, the 1 + 3-formalism is not suited if the space-time is inhomogeneous, like spherical symmetry or space-times with local rotational symmetry (LRS) [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In such cases, splitting the 3-space orthogonal to the time-like congruence into one spacelike direction and a 2-space is apt [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Thus, the 1 + 1 + 2 decomposition of space-time is a natural extension of the 1 + 3 formalism in which the three-space is further decomposed to a given spatial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This approach is called semi-tetrad formalism [45–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' As mentioned in the Introduction, our interest is to evaluate the net momentum experi- enced by a test particle after the electromagnetic wave passes through the space-time point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In the covariant 1 + 3 formalism, the test particle is the fundamental time-like observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' As depicted in (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' 1), when the EM wave propagates in a given spatial direction, the net momentum experienced by the particle lies in the 2-D surface orthogonal to the direction of propagation of the EM wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In other words, the net momentum (kick) vector lies in the 2-D surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Thus, the net memory effect of the test particle will lie on the 2-D surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' hence, we will refer to this as the memory vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This can also be understood using the fact that the electric and magnetic fields are transverse to the direction of propagation of the EM wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Thus, it is cogent to further split the 3-space to 1 + 2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' More specifically, choosing a generic space-like vector (nµ), we split the 3-space into 1 + 2-space [41–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The space-like vector (nµ) satisfies the following conditions: nµnµ = 1, nµuµ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Like in the 1 + 3-formalism, we project the vectors and tensors defined in 3-space along the space-like direction (nµ) and into the 2-space that is orthogonal to nµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Here again, the projection tensor (˜hµν) need to be defined: ˜hµν = hµν − nµnν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' ˜hµ ν ˜hν γ = ˜hµ γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' ˜hµ µ = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' ˜hµν uν = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' ˜hµν nν = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (8) All the vectors and tensors defined in the 3-space in the 1 + 3-formalism can be split into 8 1 + 2 form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' For instance, an arbitrary space-like vector V µ (defined in the 3-space) can be written as: V µ = V nµ + V µ (9) where, V = V µnµ and V µ = ˜hµ νV ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Similarly an arbitrary tensor vµν on the 3-space can be split as: vµν = V � nµnν − 1 2 ˜hµν � + 2V(µnν) + Vµν , (10) where V(µnν) = (Vµnν +nνVµ)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Similarly, the relative acceleration of the time-like observer and other geometrical quantities defined in 3-space can be written in 1 + 2 space as: ˙uµ = A nµ + A µ (11) ˙nµ = A uµ + αµ (12) ωµ = Ωnµ + Ωµ (13) σµν = Σ � nµnν − 1 2 ˜hµν � + 2Σ(µnν) + Σµν (14) where ˙nµ := uν∇ν nµ is the relative acceleration of the space-like vector along the time-like observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Here, A µ, αµ, Σµν, Ωµ are orthogonal to nµ as well as uµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Also, A µ, Ωµ(Σµν) are the vectors (tensor) projected on the 2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In this formalism, we define the alternating Levi-Civita 2-tensor ǫµν ≡ ǫµναnα (15) which is orthogonal to nµ and has components only in the 2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Given an arbitrary vector V µ in the 2-space, we can construct another vector ǫµνV ν that is orthogonal to V µ which is in the 2-space and has the same length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The 1+2 splitting of the 3-space leads to a new directional derivative along the space-like vector nµ: v′ µν ≡ nαDαvµν (16) ˜Dαvµν ≡ ˜hα β˜hµ ρ˜hν σDβvρσ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (17) The derivative in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (16) physically correspond to the variation of the physical quantities on the 2-space along the space-like vector nµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The derivative ( ˜D) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (17) corresponds 9 to the variation of the physical quantities that lie in the 2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' These will contribute to the memory vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' As we split the covariant derivative of uµ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (5), similarly we can split the covariant derivative of nµ as: Dνnµ = ˜Dνnµ + nµn′ ν = ˜σµν + ˜ωµν + 1 2 ˜Θ˜hµν + nµn′ ν (18) where, ˜σµν ≡ ˜D<νnµ>, ˜ωµν ≡ ˜D(νnµ) and ˜Θ = ˜Dµnµ are shear, vorticity and the surface expansion-contraction scalar respectively and n ′ µ is the spatial derivative along nµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Thus, ˜Dνnµ describes the kinematic properties or the relative motion of the space-like curves in the 2-surface orthogonal to nµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' We can obtain the relation between the kinematic quantities derived from the motion of time-like vector uµ and kinematic quantities in 2-space derived from the space-like vector nµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' See, for instance, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' ELECTROMAGNETIC THEORY IN COVARIANT FORMALISM The covariant formalism has been extensively employed in studying the evolution of electromagnetic fields in curved space-time [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In the covariant formulation, the dynamics and kinematics are constricted by the Bianchi and Ricci identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The (1 + 3)− covariant formulation permits the classification of cosmological models, a fluid description of the matter field in FLRW universes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' However, as mentioned earlier, the 1 + 3-formalism is not suited if the space-time is inhomogeneous, like spherical symmetry or space-times with LRS [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In such cases, the 1 + 1 + 2-covariant or semi-triad formalism are better suited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Since we aim to derive EM memory for arbitrary space-times, we use 1 + 1 + 2-covariant formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' We obtain a generic form of the EM memory effect by evaluating the change in the velocity vector ∆uµ that lie in the 2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In order to do so, we fix the space- like direction to be the direction of the propagation of the wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In the case of spherically symmetric space-time, this naturally translates to the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' One key advantage is that the electromagnetic theory in the 1+1+2 formalism helps to understand the evolution and dynamics of the EM fields along the space-like direction and in the 2-space normal to nµ and uµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Our approach makes geometrical contributions to the memory effect more transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In the next subsection, we rewrite Maxwell’s equations in 1+3 formalism in an arbitrary 10 space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Later, we formulate the evolution equations of the EM fields in the 2-space and two constraint equations of the same along uµ and nµ [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The key advantage is that we can obtain the memory vector from the projected acceleration vector onto the 2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In 1+3 formalism The fundamental objects are the Maxwell electromagnetic field tensor F µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The (1 + 3) covariant formalism of Maxwell’s electromagnetic theory provides a way to study the interaction of EM fields with different components of general space-time geometry [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' With the (1 + 3) decomposition, it is possible to split F µν into the electric and magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Note that the local coordinates are mathematical parameters that label the points of the space-time manifold M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' therefore, the electric and magnetic fields may not have a direct physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In order to make measurements, an observer brings in an additional structure on M by introducing the orthonormal coframe field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This gives rise to the split of Maxwell’s tensor F into the physical electric and magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Specifically, formalism allows us to split the equations of motion of the fields and currents into two parts: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' projected parallel to the 4-velocity uµ of the fundamental observer 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' projected into the 3-space orthogonal to uµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' To keep the calculations tractable, we perform all the calculations in source-free and lossless regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' However, the EM memory analysis can be straightforwardly extended to these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In the source-free regions, Maxwell’s equations are: ∇νF µν = 0 (19) ∇[γFµν] = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' or ∇νF ∗µν = 0 , (20) where F ∗µν is the dual to F µν and is defined as F ∗µν = (1/2)ǫµναβFαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In the 1 + 3 formalism, by projecting F µν and F ∗µν along the time-like 4-velocity vector, we can decompose them into electric and magnetic parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The electric (Eµ) and magnetic (Bµ) 4-vectors are defined as: Eµ := F µνuν (21) Bµ := F ∗µνuν (22) 11 From the above definitions, we infer: Eµuµ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Bµuµ = 0 (23) which implies Eµ and Bµ have only spatial components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Given this, we can rewrite Fµν and F ∗µν as: Fµν = uµEν − uνEµ + ǫµναβBαuβ (24) ˜F αβ = ǫαβµνuµEν + � uαBβ − uβBα � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (25) From the above expressions, we see that the simultaneous transformations Eµ → −Bµ, Bµ → Eµ leads to F ∗µν → F µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This implies that we can obtain the second Maxwell’s equation (20) from the first Maxwell’s equation (19) or vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' More specifically, if we obtain the time-like part and space-like part of Maxwell’s equations (20), we can write the time-like part and space-like part of the other Maxwell’s equations (19) by substituting Eµ → −Bµ, Bµ → Eµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In the rest of this subsection, we obtain Maxwell’s equations by projecting along uµ (time-like part) and hµν (space-like part) [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' We first obtain the time-like part of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (20) by multiplying it with uµ: uα � ∇β ˜F αβ � = 0 (26) Using the decomposition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (25), the above expression becomes: ∇βBβ − Bβ ˙uβ + (∇βuα) ǫαβµνuµEν = 0 (27) We simplify the above equation using the following steps: First, we combine the first two terms in the LHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (26), we have Bβ ˙uβ = −uβ ˙Bβ = −uβuα∇αBβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Substituting in the second term of the above expression, we have δα β ∇αBβ + uβuα∇αBβ = hα β � ∇αBβ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Substituting ∇βuα from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (5) and using the definition of vorticity vector in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (6), the third term in the LHS of the above expression simplifies to −2ωβEβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Thus, the time-like part of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (20) reduces to: DβBβ = 2ωβEβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (28) The space-like part of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (20) can be obtained by multiplying it with hµ ν, hα ρ � ∇β ˜F αβ � = 0 (29) 12 Using a series of steps, the above expression can be rewritten as: ˙B<ρ> = � σρ β + ωρ β − 2Θ 3 hρ β � Bβ − ǫρµν ˙uµEν − ǫρµν ∇µEν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (30) where, ǫµνα is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The above equation provides the dynamical evolution of the magnetic field, while Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (28) is the constraint equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' As mentioned above, performing simultaneous transformation Eµ → −Bµ and Bµ → Eµ in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (31) and (32), we obtain the time-like and space-like parts of the first Maxwell’s equation (19): DβEβ = −2ωνBν (31) ˙E<ρ> = � σρ β + ωρ β − 2Θ 3 hρ β � Eβ + ǫρµν ˙uµBν + ǫρµν DµBν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (32) Similarly, the above equation provides the dynamical evolution of the electric field, while Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (31) is the constraint equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In 1+1+2 formalism We aim to calculate the memory effect of EM fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' As the memory vector resides in the 2-surface orthogonal to the direction of propagation of the in-coming wave, we need to decompose the 3-space to 1 + 2-space w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='t a given spatial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In this subsection, we rewrite Maxwell’s equations (19, 20) using the space-like vector nν and the projection tensor (8) in 1 + 1 + 2 formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' To do this, we first express the EM fields and currents in 3-space into 1 + 2 form: Eµ = E nµ + E µ (33) Bµ = Bnµ + Bµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (34) where, E ≡ Eµnµ, E µ ≡ ˜hµ νEν, B ≡ Bµnµ, and Bµ ≡ ˜hµ νBν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Following the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (II B), it follows that ǫµνE ν is orthogonal to E µ and, similarly, ǫµνBν is orthogonal to Bµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' If electric and magnetic fields are orthogonal to each other in 2 space, then we have E ν = ǫµνBν Bν = − ǫµνE ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (35) These relations will play an important role in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (IV) to derive the memory effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' 13 The second step is to split the evolution equations (30, 32) interms of E , E µ, B, Bµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' To do that, we project Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (32) along spacelike direction nµ and multiply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (32) with projection tensor (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' After a long calculation, we obtain the following evolution equations for E (along nµ) and E µ (in the orthogonal 2-space): ˙E + ΘE = αµEµ − 2˜ωB + ǫµρ ˜DµBρ (36) ˙E¯µ + Θ 2 Eµ = − (αµ + 2ǫµρΩρ) E + (Σµρ + Ωǫµρ) E ρ + ǫµρ � A ρ − n′ρ + ˜Dρ� B − ǫµρ � A Bρ + B′ρ − � ˜DρBν � nν� , (37) where, ˜ω = ˜ωµν ǫµν, Θ is the expansion factor defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (5), A µ is the relative accel- eration vector in 2-space defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (11), ˜ω is the vorticity defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Ωµ, Ω is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (13) and Σµν is in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The 2-space component of ˙nµ is αµ which is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (12), whereas A = nµ ˙uµ = −uµ ˙nµ mentioned in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (11), (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' We want to highlight the following points regarding the above expressions: First, the above equations generalize Ampere’s law for arbitrary space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' For example, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' 36, the first term in the LHS corresponds to the time derivative of the electric field along space- like direction nµ and the last term in RHS is the curl of the magnetic field in 2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Similarly, the LHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (37) is the time derivative of the electric field in 2-space, and in the last term in the RHS is the curl of Bρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Second, in the flat space-time, the expansion factor (Θ), the relative acceleration vector (αµ), and vorticity (˜ω) vanish, and the above expression lead to Ampere’s law in flat space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Thus, background space-time introduces new couplings between the electric and magnetic field components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Lastly, we showed that the simultaneous transformation Eµ → −Bµ, Bµ → Eµ leads to F ∗µν → F µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Substituting E → B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' E µ → Bµ and B → −E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Bµ → −E µ in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (36, 37), we have: ˙ B + ΘB =Bµαµ + 2˜ωE − ǫµρ ˜DµE ρ (38) ˙ B¯µ + 1 2ΘBµ = − (αµ + 2ǫµρΩρ) B + (Σµρ + Ωǫµρ) Bρ − ǫµρ � A ρ + ˜Dρ − n′ρ� E + ǫµρ � A E ρ + ǫµρE ′ρ − � ˜DρEν � nν� (39) Note that we obtain the above equations by projecting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (30) along spacelike direction nµ and multiply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (30) with projection tensor (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Again, the above equations generalize Faraday’s law for arbitrary space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' 14 The last step is to split the constraint equations (31, 28) interms of E , E µ, B, Bµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Sub- stituting (33, 34) and the kinematic quantities (11-14), we get: ˜DµEµ + nµE ′ µ + E ′ + ˜ΘE + 2 (ΩB + ΩµBµ) = 0 (40) ˜DµBµ − n′µBµ + B′ + ˜ΘB − 2 (ΩE + ΩµEµ) = 0 (41) where ˜Θ is the expansion along the space-like vector defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The above equations are generalizations of Gauss law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Here again, in the flat space-time, the expansion factor (˜Θ), the relative acceleration vector (αµ), vorticity (Ω) vanish, and the above expressions lead to Gauss law in flat space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Energy-momentum tensor of the electromagnetic field As we will show in the next section, the electromagnetic stress tensor plays a crucial role in understanding the memory effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This subsection evaluates the electromagnetic stress tensor in 1 + 1 + 2 formalism for an arbitrary space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The EM action in an arbitrary background is: S = −1 4 � d4x √−g FµνFρσgµρgνσ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (42) Varying the above action w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='t the metric (gµν) leads to the following energy-momentum tensor: Tµν = 1 2gρσFµρFνσ − 1 8gµνgρσgαβFραFσβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (43) In 1 + 3-formalism, the stress-tensor of matter field (Tµν) can written as: Tµν = ρ uµuν + 2 S(µ uν) + Wµν , (44) where, the energy-density ρ, the energy flux Sα and stress-tensor W αβ as measured in the observer’s worldline are given by [56]: ρ = Tµνuµuν, Sα = −hα µ T µνuν, W αβ = hα µ T µνhβ ν (45) For the electromagnetic fields in 1 + 3-formalism, ρ, Sµ and Wµν are: ρ ≡ 1 2 (EµEµ + BµBµ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Sµ ≡ ǫµνρEνBρ (46) Wµν ≡ 1 2 (EµEµ + BµBµ) hµν − EµEν − BµBν (47) 15 Rewriting ρ interms of the variables (E , E µ, B, Bµ) in 1 + 1 + 2 formalism, we have: ρ = 1 2 � E 2 + B2� + 1 2 (E µEµ + BµBµ) = ρ(n) + ρ2−space (48) Thus, ρ(n) corresponds to the energy of the EM field along nµ and ρ2−space corresponds to the energy of the EM field in the 2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The energy flux Sµ (a vector in 3-space) can be rewritten in 1 + 2 space as: Sµ = S nµ + Sµ (49) where S is the Poynting vector of the EM field along the space-like vector nµ and Sµ is the energy flux in the 2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' These are given by: S = Sµnµ = ǫµνE µBν (50) Sµ = −ǫµν (E Bν − BE ν) = − (E E ν + BBν) (51) In deriving the last expression, we have used the orthogonality condition between the electric and magnetic fields in the 2-space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=', Eν = ǫνµBµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' As we will see in the next section, the memory vector depends on the part of the electromagnetic energy density ρ and Sµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' MEMORY EFFECT IN ARBITRARY SPACE-TIME Having written Maxwell’s equations in 1 + 1 + 2 formalism for an arbitrary space-time, we now evaluate the memory effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Usually, in the literature, one uses the Lorentz force equation to derive EM memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The equation of motion of a charged body (of mass m and charge e) in both gravitational and electromagnetic fields are: mduα dτ − m 2 gβγ,αuβuγ = eFαβuβ (52) However, the above expression does not consider the new couplings between the electric and magnetic field components in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (36) - (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Hence, we use the complete Maxwell’s equations (36) - (41) and explicitly obtain the change in velocity (∆uµ) of the time-like observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' More specifically, using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (37, 39), we first calculate the acceleration vector A µ in the 2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' We can then integrate the expression for the acceleration vector (A µ in the 2-space) with respect to time t or null time coordinate u ≡ (t − r) leading to the memory vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' 16 In the rest of this section, we calculate A µ for observers whose tangents are congruent to the space-like geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This implies nσDσnρ = n′ρ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=', nµ is tangent to a congruence of space-like geodesics [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Using this condition and substituting ˙E¯µ = ˜hµν ˙E ν, B′ ρ = nνDνBρ in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (37, 39), we get: ˜hµν ˙E ν + ǫµρnνDνBρ = − 1 2ΘEµ − (αµ + 2ǫµρΩρ) E + (Σµρ + Ωǫµρ) E ρ + � ǫµρA ρ + ǫµν ˜Dν� B − ǫµν � ˜Dνnρ� Bρ − ǫµρA Bρ (53) � ˜hµν ˙ Bν − ǫµρnνDνE ρ� = −1 2ΘBµ − (αµ + 2ǫµρΩρ) B + (Σµρ + Ωǫµρ) Bρ − � ǫµρA ρ + ǫµν ˜Dν� E + ǫµν � ˜Dνnρ� Eρ + ǫµρA E ρ (54) Multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (53) with B, multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (54) with E and subtracting the resultant equations leads to: ǫµνA ν = − ǫµν 2 Dν(E 2 + B2) (E 2 + B2) + � Σµν + Ωǫµν − Θ 2 ˜hµν � (E Bν − BE ν) (E 2 + B2) + ǫµν � ˜σρν + ˜ωρν + ˜Θ 2 ˜hρν � (BBρ + E Eρ) (E 2 + B2) + ǫµρA (E E ρ + BBρ) (E 2 + B2) + B (E 2 + B2) � ˜hµν ˙E ν + ǫµρnνDνBρ� − E (E 2 + B2) � ˜hµν ˙ Bν − ǫµρnνDνE ρ� (55) To have a transparent understanding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' we substitute the definitions (48) - (51) in the expres- sion above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' resulting in: ǫµνA ν = − ǫµν 2 Dνρ(n) ρ(n) − ǫνα 2 � Σµν + Ωǫµν − Θ 2 ˜hµν � Sα ρ(n) − ǫµν 2 � ˜σρν + ˜ωρν + ˜Θ 2 ˜hρν � Sρ ρ(n) − ǫµρS ρ A 2ρ(n) + B 2ρ(n) � ˜hµν ˙E ν + ǫµρnνDνBρ� − E 2ρ(n) � ˜hµν ˙ Bν − ǫµρnνDνE ρ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (56) This is the master equation for the EM memory in arbitrary space-time regarding which we would like to discuss the following points: First, to our understanding, this is a first time the EM memory has been obtained for an arbitrary space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In the previous calculations [18, 19], the authors have restricted to asymptotic flat space-times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Second, the last two terms in the RHS of the above expression vanishes in the asymptotic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' To see this, let us consider a spherically symmetric space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Let t refer to the time coordinate and r to the radial coordinate and the null coordinate is u ≡ t − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In the asymptotic limit ∂u ∼ ∂t and 17 ∂u ∼ −∂r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Setting uµ ≡ (1, 0, 0, 0) and nµ ≡ (0, 1, 0, 0), the penultimate term in the RHS of the above equation simplifies to: ˜hµν ˙E ν + ǫµρnνDνBρ ≃ ˜hµνu0∇0E ν + ǫµρn1∇1Bρ ≃ ˜hµν∂uE ν − ǫµρ∂uBρ = f(u)∂u �¯˜hµνE ν − ¯EµνBν� (57) where, ˜hµν = f(u)¯˜hµν and ¯ǫµν = f(u)ǫµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The terms with bar represent their time indepen- dent parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The above expression vanishes if E ν and Bν are orthogonal to each other in the 2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' As we mentioned earlier (35), in 2-space, the electric and magnetic fields are always orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Similarly, the last term can also be shown to vanish in the asymptotic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Thus, the above expression reduces to: ǫµνA ν = − ǫµν 2 Dνρ(n) ρ(n) − ǫνα 2 � Σµν + Ωǫµν − Θ 2 ˜hµν � Sα ρ(n) − ǫµν 2 � ˜σρν + ˜ωρν + ˜Θ 2 ˜hρν � Sρ ρ(n) − ǫµρ 2ρ(n) S ρ A (58) Third, the above expression provides a nice geometrical understanding of the various contri- butions to memory effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The first term in the RHS corresponds to the change in the EM field energy (ρ(n)) along nµ in the 2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This does not contain any contribution from the kinematical properties of the space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In other words, this term will vanish if the EM field energy does not change in the 2-space, like a 2-D flat sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' However, as we show in the next section, this is non-zero in flat space-time expressed in spherical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The next two terms in the RHS are proportional to the energy flux (Sα) in the 2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' However, both these terms have different kinematical information of the space-time and vanish for flat space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The second term in the RHS carries information about shear (Σµν), vorticity scalar (Ω) related to nµ and expansion scalar (Θ) corresponding to time-like observer uµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The third term in the RHS carries information about shear (˜σµν), vorticity tensor (˜ωµν) and expansion scalar (˜Θ) corresponding to the space-like vector nµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Fourth, as mentioned earlier, we have not included external currents or charges in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Hence, the acceleration vector does not have contribution from the external sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Hence, the memory vector we obtain is equivalent to the null-kick derived in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' It is also important to note that these authors did not obtain the contribu- tions due to the kinematical properties of the space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' However, as we will see in the next section, their contribution can be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' 18 Lastly, to obtain the memory vector, we need to integrate the above expression w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='t the proper time of the observer — ∆uµ is the memory vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' It is interesting to note that initially if the observer has non-zero velocity only along the time direction, at a later time, due to the memory effect, there is a non-zero velocity in the 2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' APPLICATION TO SPECIFIC SPACE-TIMES In the previous section, we obtained a master equation for the EM vector for an arbitrary 4-D space-time using 1 + 1 + 2-formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' As we discussed, the memory vector has three distinct contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In order to illustrate this fact, we consider specific examples and obtain the memory vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In this section we obtain memory vector for flat, FLRW, pp- wave and Kerr space-times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Minkowski space-time In order to compare the master equation with the existing results [18], we first consider Minkowski space-time in spherical coordinates: ds2 = −dt2 + dr2 + r2 γAB (59) where, γAB = \uf8eb \uf8ed 1 0 0 sin2 θ \uf8f6 \uf8f8 (60) is the metric describing unit 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In Minkowski space-time, the 4-velocity of the time- like congruence observer is uµ ≡ (1, 0, 0, 0) and the space-like vector is nµ ≡ (0, 1, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Since ∇µuν = 0 and ∇µnν = 0, the kinematics quantities, defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (II A, II B) vanish for the Minkowski space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Hence only the first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (56) will be non-zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=', A ν Flat = − 1 2 Dνρn ρn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (61) As mentioned earlier, the acceleration vector corresponds to acceleration in the 2-Sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Hence, it is appropriate to switch to the 2-Sphere index: A A = uµ∇µuA = u0∂0uA + 2u0ΓA 0 BuB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' 19 Since the 4-velocity uµ is zero in the 2-Sphere, we have A A = u0∂0uA = ∂tuA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In null coordinate, this becomes A A = ∂tuA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Substituting the above expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (61) and integrating in the null coordinate, we have: ∆uA ≡ � du A A = −1 2 � du DAρn ρn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (62) The above expression is velocity kick w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='t the Eulerian observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' To compare this with the net momentum (kick) vector as seen by the asymptotic static observers (Lagrangian observers), we need to do a coordinate transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Specifically, we need to transform from coordinate basis � ⃗e θ,⃗e φ� to orthogonal coordinate basis � ˆθ, ˆφ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In terms of � ˆθ, ˆφ � , we have ∆⃗u ≡ ∆uµ⃗eµ, where, ⃗e θ = ˆθ/r , ⃗e φ = ˆφ/(r sin θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Thus, the velocity kick w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='t the asymptotic static observers is given by: ∆⃗uFlat = 1 r � ∆uθˆθ + ∆uφ sin θ ˆφ � (63) Interestingly, the EM memory vector in Minkowski space-time is inversely proportional to r and matches with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This passes the first test that the master equation (56) indeed describes the EM memory vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In the rest of this section, we obtain the memory vector for non-flat geometries and show the robustness of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' FLRW space-time The conformally flat FLRW metric in spherical coordinates is: ds2 = a(η)2 � −dη2 + dr2 + r2 γAB � (64) where, the conformal time (η) is related to the cosmic time (t) by dt = a(η) dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In 1 + 3 formalism, the fundamental observer with time-like 4−velocity in FLRW metric is uµ = dxµ/dt = dxµ/(a(η)) dη = ( 1, 0, 0, 0 ) /a(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' For this choice of observer, the 3−space projection tensor (hµν) orthogonal to uµ is: hµν = \uf8eb \uf8ed a2(η) 0 0 a2(η) r2 γAB \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (65) Since the FLRW line-element is homogeneous and isotropic, only the expansion scalar (Θ) is non-zero: Θ = 3H (η) a(η) where H = a′(η) a(η) 20 where ′ refers to derivative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Other kinematic quantities vanish, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=', σµν = ωµν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' We now spilt the 3-space into 1 + 2 by choosing the following space-like vector nµ = (0, 1, 0, 0)/a(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This satisfies the conditions: nµnµ = 1 and uµnµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Repeating the steps discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (II B) for the line-element (64), we get: ˜Θ = 2 a(η) 1 r , ˜σµν = ˜ωµν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' It is important to note that while Θ is a function of η only, ˜Θ depends on both η and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Also, Θ depends on the Hubble parameter H , while ˜Θ is inversely proportional of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Hence, at large distances, ˜Θ decays faster compared to Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Substituting the above expressions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (58), we have: A ν FLRW = − 1 2 Dνρn ρn + 1 4 ρn S ν(Θ − ˜Θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (66) Like Minkowski space-time, A ν will have components only in the 2-Sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Using the fact that the fundamental observers have zero velocity in the 2-Sphere and repeating the earlier analysis, we have A A = u0∂0uA = 1 a(η) ∂uA ∂η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In terms of the null coordinate u(≡ η − r), we have: A A = 1 a(u) ∂uA ∂u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Substituting the above expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (66), we have: ∂uA ∂u = − a(u) 2 DAρn ρn + a(u) 4 ρn S A(Θ − ˜Θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (67) Integrating the above expression w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='t u, leads to the following memory vector: ∆uA FLRW = −1 2 � du a(u) ρn DAρn + 1 4 � du a(u) ρn S A(Θ − ˜Θ) (68) This is the expression for the memory vector in FLRW space-time regarding which we want to highlight the following points: First, unlike Minkowski space-time, here the fundamental observers are Eulerian, and hence, we do not have to transform the above expression to Lagrangian observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Second, our results differ from the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' [34], the authors show that the EM memory effect in FLRW differs from the Minkowski only by the conformal factor a(η) or a(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In other words, their analysis did not account for the geo- metric contribution to the memory effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' As mentioned earlier, the geometric contribution 21 leads to a non-zero energy flux (S A) contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Also note that the ordinary memory derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' [34] is not present in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (68) as we have assumed any external charge or current to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Third, we find that ρ(n) and the energy flux (S A) contribute oppositely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' It will be interesting to see whether the two contributions nullify the EM memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' pp-wave space-times In this subsection, we derive the EM memory for a special kind of plane-fronted wave with parallel rays (pp-waves) called plane-wave metric [57]: ds2 = −2dudv − F(u, x, y) du2 + dx2 + dy2 (69) where, F(u, x, y) = A(u)(x2 − y2) + 2B(u)xy describes the plane wave and A(u), B(u) are arbitrary functions such that F > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Note that u, v are not light-cone coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' u is time-like coordinate and v is a null coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' We split the above 4-D space-time into 1 + 3 form and later into 1 + 1 + 2-form by considering the following time-like velocity vector (uµ) and space-like vector (nµ): uµ ≡ � F(u, x, y)(−1/2), 0, 0, 0 � , nµ ≡ � F(u, x, y)(−1/2), − F(u, x, y)(1/2), 0, 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' For the above choice of time-like vector, the 3-space projection tensor (hµν) is: hµν = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 0 0 1 F(u, x, y) 0 0 0 0 1 0 0 0 0 1 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb (70) Substituting these in the definitions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (II), only non-zero quantity is the expansion scalar (Θ): Θ = − (x2 − y2) A′(u) + 2xy B′(u) 2 (2B(u) xy + A(u) (x2 − y2) )3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (71) The non-zero projection tensor ˜hµν components in the 2-space are ˜hxx = 1, ˜hyy = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Thus, the memory vector for the special kind of pp-wave space-times is: A ν PP = − 1 2 Dνρn ρn + Θ 4 ρn S ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (72) 22 Here, the acceleration of the time-like observer is confined to the x − y plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=', A A PP = − 1 2 DAρn ρn + Θ 4 ρn S A , (73) where, the index A, B corresponds to (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Evaluating the acceleration vector along x and y, we have: A (PP) x(y) = − 1 2 ρn ∂x(y) (ρn) + Θ 4 ρn Sx(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (74) Integrating the above equation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content='t u, we have: ∆uPP x(y) = −1 2 � du ∂x(y) (ρn) ρn + Θ 4 � duSx(y) ρn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (75) The above expression for the velocity kick is for a generic plane-wave metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' To gain some physical intuition, we consider two specific forms — Penrose limit of the Schwarzschild and FLRW space-times [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' For Schwarzschild space-time, we have A(u) = 6 25u2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' B(u) = 0 Substituting these in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (71), we have: ΘPP,Sch = 5 � 6(x2 − y2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' It is interesting to note that although the space-time metric does not differentiate between the two spatial coordinates (x, y), in order for Θ to be real, the above expression demands that x > y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Thus, velocity kick due to EM wave in PP-wave limit of Schwarzschild space- time can only occur if x > y and is given by: ∆uPP Sch x(y) = −1 2 � du ∂x(y) (ρn) ρn + 5 4 � 6(x2 − y2) � duSx(y) ρn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (76) In the case of Penrose limit of FLRW space-time with power-law scale factor a(t) ∼ th, we have: A(u) = − h (1 + h)u2, B(u) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Substituting these in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (71), we have: ΘPP,FLRW = � (1 + h) h(y2 − x2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' ∆uPP FLRW x(y) = −1 2 � du ∂x(y) (ρn) ρn + � (1 + h) 4 � h(y2 − x2) � duSx(y) ρn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (77) 23 Here again, we see that in-order for Θ to be real, the above expression demands that y > x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Thus, velocity kick due to EM wave in PP-wave limit of FLRW space-time occurs in a different region of the 2-space compared to Schwarzschild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Thus, EM memory has a distinct signature for different space-times and can potentially be used as a probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Kerr space-time In this section, we derive the memory effect in Kerr space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In Boyer-Lindquist coordinates (t, r, χ, φ), the Kerr space-time is: ds2 = � 2mr r2 + a2χ2 − 1 � dt2 + � r2 + a2χ2 r2 − 2mr + a2 � dr2 + �r2 + a2χ2 1 − χ2 � dχ2 − �4mar (1 − χ2) r2 + a2χ2 � dtdϕ + � 1 − χ2� � r2 + a2 + 2ma2r (1 − χ2) r2 + a2χ2 � dϕ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (78) where χ ≡ cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In this case, the time-like observer 4-velocity (uµ) and the space-like vector (nµ) are [58]: uµ = �� r2 − 2mr + a2 r2 + a2χ2 , 0, 0, 0 � , nµ = � 0, � r2 − 2mr + a2 r2 + a2χ2 , 0, 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' We give below the kinematical quantities (discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (II B)) for Kerr space-time in 1 + 1 + 2 formalism obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' [58]: Θ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Σµν = 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (79) Ω = −2marχ √ L J √ K 3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' ˜Θ = W J √ K 3L ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (80) ˜ωµν = ˜ωǫµν = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' A = −mD √ L J √ K 3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (81) ˜σµν = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 0 0 0 0 0 0 0 −1 2 a2(m−r) √ K J √ L 0 0 0 0 1 2 a2(m−r)M 2√ L K J 2 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb (82) where, M = χ2 − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' D = −r2 + a2χ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' L = r2 − 2mr + a2 (83) J = r2 − 2mr + a2χ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' K = r2 + a2χ2 (84) 24 W = 2r3(r − 2m)2 + a4χ2 � m + r − mχ2 + rχ2� + a2r2 � −3m + r + χ2(3r − 5m) � (85) Substituting these expressions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (58), and noting that the memory vector lies in the 2-D surface, we get: A A = − 1 2 DAρ(n) ρ(n) − Ω 2 ǫAB SB ρ(n) − 1 2 � ˜σAB + ˜Θ 2 ˜hAB � SB ρ(n) − A 2ρ(n) S A (86) This is the EM memory vector for an Eulerian observer in Kerr space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Note that this is a generic result for any value of angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' For a better physical insight, we consider a → 0 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Substituting a → 0 in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (79 - 85), we have M0 = χ2 − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' D0 = −r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' L0 = r2 − 2mr (87) J0 = r2 − 2mr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' K0 = r2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' W0 = 2r3(r − 2m)2 (88) Ω0 = ˜σ0 µν = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' ˜Θ0 = 2 � (r − 2m) r3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' A = m � r3(r − 2m) (89) Substituting the above quantities in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (86), we have: A A = − 1 2 DAρ(n) ρ(n) − 1 2 � r − 2m r3 S A ρ(n) − 1 2ρ(n) m � r3(r − 2m) S A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (90) This is the EM memory vector for an Eulerian observer in Schwarzschild space-time, regard- ing which we want to mention the following points: First, in the limit, r → ∞, reduces to Minkowski space-time expression (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Second, in the limit r → ∞, the subleading term is proportional to r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Third, to derive the memory vector ∆uA, we have to switch to the null time coordinate u = t − r and integrate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (90) with respect to u at the asymptotic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Lastly, to evaluate the memory effect experienced by the static asymptotic (Lagrangian) observer, we need to do the transformation from � ⃗e θ,⃗e φ� to the orthogonal coordinate basis � ˆθ, ˆφ � like in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' (V A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' CONCLUSIONS In this work, we have derived a master equation for electromagnetic memory in an arbi- trary space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' We used the covariant formalism to obtain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' More specifically, we used the 1 + 1 + 2 covariant formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The 1 + 1 + 2 decomposition of space-time is a natural extension of the 1 + 3 formalism in which the three-space is further decomposed 25 using a given spatial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This choice of covariant formalism is because the net mo- mentum (kick) vector lies on the 2-D surface for arbitrary space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Also, the electric and magnetic fields are transverse to the direction of propagation of the passing EM wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The EM memory (58) has three distinct contributions: First contribution is due to the change in the EM field energy (ρ(n)) along nµ in the 2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This is non-zero for Minkowski space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The second contribution is proportional to the energy flux (Sα) in the 2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' This has kinematical information of the space-time and vanishes for the flat space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The third contribution is proportional to the acceleration A along the time-like vector uµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' To our understanding, the earlier approaches could not isolate the different contributions to the EM memory as done in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' We then obtained the EM memory for different space-times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' In the case of FLRW space- time, we showed that the earlier analysis did not account for the geometric contribution to the memory effect [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Specifically, their analysis did not account for the geometric contribution leading to a non-zero energy flux (S A) contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' We have also obtained the EM memory for Kerr space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' We also showed that the EM memory has a distinct signature for different pp wave space-times and can potentially be used as a probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' It would be interesting to extend our analysis for black holes with multiple horizons and those that are not asymptotically flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' These may be particularly relevant for using EM memory as a probe to PBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Finally, our analysis points to the possibility of using 1 + 1 + 2 covariant formalism to understand gravitational memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' These are currently under investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Mahesh Chandran and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' Kushwaha for comments on the earlier version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' The work is supported by the SERB-Core Research grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} +page_content=' 31' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQfRy0L/content/2301.11772v1.pdf'} diff --git a/4tAzT4oBgHgl3EQfuv3m/content/tmp_files/2301.01697v1.pdf.txt b/4tAzT4oBgHgl3EQfuv3m/content/tmp_files/2301.01697v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..540c4a663967328059af4e5d342e51e6377844ea --- /dev/null +++ b/4tAzT4oBgHgl3EQfuv3m/content/tmp_files/2301.01697v1.pdf.txt @@ -0,0 +1,4310 @@ +Spectral analysis and k-spine decomposition of inhomogeneous +branching Brownian motions. Genealogies in fully pushed fronts +Emmanuel Schertzer∗ and Julie Tourniaire† +January 5, 2023 +Abstract +We consider a system of particles performing a one-dimensional dyadic branching Brownian motion +with space-dependent branching rate, negative drift −µ and killed upon reaching 0. More precisely, the +particles branch at rate r(x) = (1 + f(x))/2, where f is a compactly supported and non-negative smooth +function and the drift µ is chosen in such a way that the system is critical in some sense. +This particle system can be seen as an analytically tractable model for fluctuating fronts, describing +the internal mechanisms driving the invasion of a habitat by a cooperating population. Recent studies +from Birzu, Hallatschek and Korolev suggest the existence of three classes of fluctuating fronts: pulled, +semi pushed and fully pushed fronts. Here, we focus on the fully pushed regime. We establish a Yaglom +law for this branching process and prove that the genealogy of the particles converges to a Brownian +Coalescent Point Process using a method of moments. +In practice, the genealogy of the BBM is seen as a random marked metric measure space and we use +spinal decomposition to prove its convergence in the Gromov-weak topology. We also carry the spectral +decomposition of a differential operator related to the BBM to determine the invariant measure of the +spine as well as its mixing time. +Contents +1 +Introduction +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +1 +2 +Outline of the proof +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +3 +The many-to-few theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +4 +Spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +5 +Convergence of moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +28 +6 +Survival probability (0th moment) +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +7 +Convergence of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +1 +Introduction +1.1 +The model and assumptions +We consider a dyadic branching Brownian motion (Xt)t>0 (BBM) with killing at 0, negative drift −µ +and position-dependent branching rate +r(x) = 1 +2f(x) + 1 +2, +(1) +for some function f : [0, +∞) → R. We assume that f satisfies the following assumptions: +∗Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria +†Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria. +1 +arXiv:2301.01697v1 [math.PR] 4 Jan 2023 + +(A1) the function f is non-negative, continuously differentiable and compactly supported. +(A2) the support of f is included in [0, 1]. +We denote by Nt the set of particles in the system at time t and for all v ∈ Nt, we denote by xv = xv(t) +the position of the particle v. Furthermore, we write Zt for the number of particles in the system at +time t. We write Px for the law of the process initiated from a point x ≥ 0 and Ex for the corresponding +expectation. +Critical regime. We aim at choosing µ in such a way that the number of particles in the system stays +roughly constant. +Fix L > 1 and consider the BBM (XL +t )t>0 with branching rate r(x), drift −µ and killed at 0 and L. +Denote by N L +t +the set of particles in this system at time t and define ZL +t = |N L +t |. By a slight abuse of +notations, we will also denote by xv the positions of the particles in the BBM XL. Let (t, x, y) �→ pt(x, y) +be the fundamental solution of the linear equation +� +∂tu(t, y) = 1 +2∂yyu(t, y) + µ∂yu(t, y) + r(y)u(t, y) +u(t, 0) = u(t, L) = 0. +(A) +We say that pt ≡ pL +t is the density of particles in XL in the sense that for any measurable set B ⊂ [0, L], +the expected number of particles in B at time t starting from a single particle at x is given by +� +B pt(x, y)dy +[Law18, p.188]. Let us now define +gt(x, y) := eµ(y−x)e +µ2−1 +2 +tpt(x, y). +(2) +A direct computation shows that gt is the fundamental solution of the self-adjoint PDE +� +∂tu(t, y) = 1 +2∂yyu(t, y) + 1 +2f(y)u(t, y) +u(t, 0) = u(t, L) = 0. +(B) +Let λ1 = λ1(L) be the maximal eigenvalue [Zet12, Chapter 4] of the Sturm–Liouville problem +1 +2v′′(x) + 1 +2f(x)v(x) = λ1v(x), +(SLP) +with boundary conditions +v(0) = v(L) = 0. +(BC) +It is known that λ1 is an increasing function of L [Pin95, Theorem 4.4.1] and that it converges to a +finite limit λ∞ +1 +∈ (−∞, +∞) as L → ∞ [Pin95, Theorem 4.3.2]. We now choose µ in such a way that +the expected number of particles is neither increasing nor decreasing exponentially. According to (2), we +expect that for large t +pt(x, y) ≈ eµ(y−x)e +µ2−1 +2 +teλ∞ +1 t v1(x)v1(y) +||v1||2 +, +where v1 denotes an eigenfunction associated to λ1 for the Sturm-Liouville problem (SLP). This motivates +the following definition. +Definition 1 (Critical regime). The BBM is in the critical regime iff +µ = +� +1 + 2λ∞ +1 . +(3) +Pushed and pulled waves. The next definitions are motivated by recent numerical simulations and +heuristics [BHK18, BHK21] for the noisy F-KPP equation with Allee effect +ut = 1 +2uxx + u(1 − u)(1 + Bu) + +� +u +N η, +where B > 0, N is a large demographic parameter and η is a space-time white noise. See [Tou21] for +more details and [EP22] for recent rigorous results on the bistable case. +2 + +Definition 2 (Pulled, semi pushed, fully pushed regimes). Consider the BBM (Xt) in the critical regime. +Define +β := +� +2λ∞ +1 , +and +α := µ + β +µ − β . +(4) +1. If λ∞ +1 = 0, or equivalently α = 1, the BBM is said to be pulled. +2. If λ∞ +1 ∈ (0, 1/16) or equivalently +α ∈ (1, 2) ⇐⇒ µ > 3β, +the BBM is said to be semi pushed. +3. If λ∞ +1 > 1/16 or equivalently +α > 2 ⇐⇒ µ < 3β, +(Hfp) +the BBM is said to be fully pushed. +We say that the BBM is pushed if it is either semi or fully pushed, that is when +λ∞ +1 > 0. +(Hp) +Example 1.1. Let ε > 0 and consider the BBM with inhomogeneous branching rate rε(x) = 1 +2 + εf(x). +By [Pin95, Theorem 4.6.4 and Theorem 4.4.3], for any function f satisfying (A1), there exists 0 < ε1 < ε2 +such that +1. The BBM is pulled for all ε ∈ (0, ε1). +2. The BBM is semi-pushed for all ε ∈ (ε1, ε2). +3. The BBM is fully-pushed for all ε > ε2. +It is conjectured that up to rescaling, the size and the genealogy at large time is undistinguishable +from the ones of a continuous-state branching process (CSBP). More precisely, +1. In the pulled regime, the population size should converge to a Neveu’s continuous-state branching +process and the genealogy of the BBM to the Bolthausen–Sznitman coalescent (see [BBS13] in the +case f ≡ 0). +2. In the semi pushed regime, the population size should converge to an α-stable CSBP. In the pre- +vious example, this has been proved only in the case where f = 1[0,1] [Tou21]. +Therein, it is +also conjectured that the genealogy should converge to a time-changed Beta(α, 2 − α) coalescent +[Pit99, Sag99, BBC+05]. +3. In the fully pushed regime, the rescaled population size should converge to a Feller diffusion and the +genealogy should be undistinguishable from the genealogy of a large critical Galton-Watson process +with finite second moment. This is the content of the present article. +1.2 +Comparaison with previous work +Branching Brownian motion with inhomogeneous branching rates have received quite a lot of attention in +the recent past [HKV20, HHK20, HHKW22, GHK22, FRS22, Tou21, RS21, LS21]. The general approach +always relies on a spinal decomposition of the BBM. Roughly speaking, the spine is constructed by +conditioning a typical particle to survive. This conditioning is achieved by a Doob-h transform. In our +setting, the harmonic function is approximated by h(x) ≈ eµxv1(x) and the resulting h-transform is given +by +dxt = v′ +1(xt) +v1(xt)dt + dBt +(5) +where Bt is a standard Brownian motion. +See Section 3 for more details. +A key assumption underlying [Pow19, HKV20, HHK20, HHKW22, GHK22] is that the harmonic +function h is bounded. +From a technical stand point, we emphasise that this assumption is the one +distinguishing our work from the previous ones. Indeed, we shall see that v1 decreases exponentially at +rate β so that the harmonic function h(x) ∝ e(µ−β)x blows up as x → ∞ since µ = +� +1 + β2. Due to the +explosion of the harmonic function, many of the previously developed technics break down in our setting. +3 + +At first sight, this assumption may only seem technical. However, it is the key assumption which +makes possible a transition from the semi to the fully pushed regime. Let us consider the spine dynamics +(5). In the pushed regime, the invariant distribution for the spine is given by +Π(x) = v2 +1(x) +||v1||2 , +so that h(x)Π(x) ∝ e(µ−3β)x +as x → ∞. +It then becomes clear from Definition 2 that, in the fully pushed regime (resp. semi pushed), the harmonic +function is integrable (resp. non-integrable) with respect to the invariant measure of the spine. As a +consequence, relaxing the assumption under which h is bounded is crucial for understanding the transition +between these two regimes. +This generalisation raises interesting technical challenges. A large fraction of the present work (Section +4.1) is dedicated to estimating the speed of convergence to the invariant measure of the spine in the +pushed regime. More precisely, we use a Sturm-Liouville approach in order to understand the spectral +decomposition of the differential operator (B). The difficulty arises from the fact that the negative part +of the spectrum of the Sturm-Liouville problem (SLP) becomes continuous as L → ∞ (see Figure 1). We +believe that this contribution is relevant for understanding not only the fully pushed case, but also the +semi pushed case which will be the subject of future work. This continuous spectrum already appeared +in the study of homogeneous BBM [BBS13], but the spectral analysis of (SLP) is quite straightforward +in this case. +Indeed, when f ≡ 0, the spectral decomposition of (SLP) is given by λi = − iπ2 +2L2 and +vi(x) = sin +� iπx +L +� +. When f is not trivial, the spectrum is not explicit and we use the Prüfer transformation +to derive the required estimates on the (vi) and the (λi). +Finally, one of the main contribution of the present work is the description of the genealogy spanned by +the population at a large time horizon. Beyond our Kolmogorov estimate and the Yaglom law reminiscent +of [Pow19, HKV20, HHK20, HHKW22, GHK22], we use k-spine decompositions [HR17] and moment +methods developed in [FRS22] to prove convergence of the genealogy in the Gromov weak topology to a +continuum random metric space known as the Brownian Coalescent Point Process [Pop04]. This approach +will be further explained in the next section. +Example 1.2. Consider f = 10.1[0,1]. It was calculated in [Tou21] that the negative part of the spectrum +of (SLP) with boundary conditions (BC) consists in the solutions of +tan( +√ +9 − 2λ) +√ +9 − 2λ += −tan( +√ +−2λ(L − 1)) +√ +−2λ +. +(6) +The solutions of this equation are plotted on Figure 1 +4 + +Figure 1: Negative spectrum of the Sturm–Liouville problem (SLP) with boundary conditions (BC) for f +defined as in Example 1.2 and different values of L. The blue line corresponds to RHS of (6) and the red +line to the LHS of (6). +1.3 +Main results +Proposition 1.3. Let v1 be the eigenfunction associated to the eigenvalue λ1 for the Sturm-Liouville +problem (SLP) with boundary conditions (BC), normalised in such a way that v1(1) = 1. Under (Hp), +v1 converges to a positive limiting function v∞ +1 +as L → ∞. Further, if in addition (Hfp) holds, then +� +R+ +eµx(v∞ +1 )3(x)dx < ∞. +To see why the latter proposition may hold true, recall that f ≡ 0 on [1, ∞). Hence, on this interval, +the problem reduces to +1 +2v′′ +1 (x) = λv1(x), +x ∈ [1, L], +v1(L) = 0. +If we impose the condition v1(1) = 1, a direct computation shows that v1(x) = sinh(√2λ1(L−x)) +sinh(√2λ1(L−1)) on [1, L] +so that, for all x ∈ [1, ∞), v1(x) → v∞ +1 (x) = e−β(x−1). The integrability condition then holds under the +extra assumption (Hfp). +In the following, we define +h∞(x) := ceµxv∞ +1 (x), +and +˜h∞(x) := ˜ce−µxv∞ +1 (x), +(7) +where +˜c := +�� ∞ +0 +e−µxv∞ +1 (x)dx +�−1 +and +c := (˜c∥v∞ +1 ∥2)−1. +The constants c and ˜c are thought as Perron-Frobenius renormalisation constants (see e.g. [AN72, p.185]), +in the sense that h∞ (resp. ˜h∞) is a left (resp. right) eigenfunction associated to the maximal eigenvalue +of the differential operator +Lu = 1 +2∂xxu + µ∂xu + r(x)u, +(8) +5 + +L=10 +4.0 +2.5 +2.0 +=50 +0.0 +L=100normalised in such a way that +� ∞ +0 +˜h∞(x)dx = 1 +and +� ∞ +0 +h∞(x)˜h∞(x)dx = 1. +From this perspective, the function ˜h∞ should correspond to the stable configuration of the system and +the function h∞ to the reproductive values of the individuals as a function of their positions. +Theorem 1 (Kolmogorov estimate). As N → ∞, for all x, t > 0, +NPx (ZtN > 0) → +2 +Σ2th∞(x), +where +Σ2 +2 +:= +� +R+ +r(z)(h∞(z))2˜h∞(z)dz. +This theorem is the continuous analogous of Kolmogorov estimate for multi-type Galton-Watson +process [AN72, p.187]: in our case, the variance in the offspring distribution is given by Σ2. +We now turn to the description of the genealogy and the Yaglom law. Intuitively, the next result states +that the genealogy is asymptotically identical to the one of a critical Galton Watson [Lam18, HJR20, +Joh19], whereas the marks are assigned independently according to ˜h∞. Let us now give a more precise +description of our result. +From now on, we condition on the event {ZtN > 0}. Let (v1, . . . , vk) be k individuals chosen uniformly +at random from NtN. Denote by dT (vi, vj) the time to the most recent common ancestor of vi and vj. +Let xvi be the position of the ith individual at time tN. +Let U be a uniform r.v. on [0, t] and θ > 0. Define U θ such that +∀s ≤ t, +P(U θ ≤ s) := (1 + θ)P(U ≤ s) +1 + θP(U ≤ s) . +(9) +Let (U θ +i ; i ∈ [k]) be k i.i.d. copies of U θ and set +∀1 ≤ i < j ≤ k, +U θ +i,j = U θ +j,i := max{U θ +k : k ∈ {i, · · · , j − 1}}. +Finally, define the random distance matrix (Hi,j) := (Hi,j; i ̸= j ∈ [k]) such that for every bounded and +continuous function ϕ : Rk2 → R, +E +� +ϕ +� +(Hi,j) +�� += k +� ∞ +0 +1 +(1 + θ)2 +� +θ +1 + θ +�k−1 +E +� +ϕ +� +(U θ +i,j) +�� +dθ. +(10) +Finally, (Wi) := (Wi; i ∈ [k]) will denote a sequence of i.i.d. copies of a random variable W with law ˜h∞. +Theorem 2 (Yaglom law and limiting genealogies). Let t > 0. +Start with a single particle at x > 0. +Conditional on ZtN > 0, as N → ∞, +(i) we have +ZtN +N +→ Σ2t +2 +E, +in distribution, +where E is a standard exponential distribution. +(ii) +�� dT (vi,vj) +N +� +, +� +xvi +�� +converges to the distribution of +� +(Hσi,σj), (Wσi) +� +where σ is a random uniform +permutation {1, . . . , k} and σ, (Hi,j) and (Wi) are independent. +Remark 1.4. The random distance matrix (Hi,j) is the one obtained from a critical Galton Watson with +finite second moment conditioned on surviving up to a large time. See [Lam18, HJR20, Joh19]. +1.4 +Notation +Given two sequences of positive real numbers (aN) and (bN), we write aN ≪ bn if aN/bN → 0 as N → ∞. +We write aN ≲ bN if aN/bN is bounded in absolute value by a positive constant and aN ≍ bN if aN ≲ bN +and bN ≲ aN. We write O(·) to refer to a quantity bounded in absolute value by a constant times what +the quantity inside the parentheses. Unless otherwise specified, these constants only depend on λ∞ +1 . +6 + +2 +Outline of the proof +Our approach relies on a method of moments devised in [FRS22]. To illustrate the approach, let us first +think about the Yaglom law of Theorem 2. To prove this result, one needs to show that the moments of +ZtN/N converge to the moments of an exponential. It turns out that this approach can be extended to +genealogies. +In Section 2.1, following the approach in [DGP11], we encode the genealogy at time tN as a random +marked measured metric space (mmm). In turn, the moments of a random mmm are obtained by biasing +the population by its kth moment and then picking k individuals uniformly at random (see Remark 2.1 +below). In section 2.2, we introduce a limiting random mmm called the marked Coalescent Point Process +(CPP) which corresponds to the limiting genealogy of a critical Galton-Watson process [Pop04]. The +remainder of the section is dedicated to the sketch of the proof for the convergence of the moments of our +BBM to the moments of the marked CPP using the spinal decomposition introduced in [FRS22]. +2.1 +Marked Metric Spaces +Let (E, dE) be a fixed complete separable metric space, referred to as the mark space. In our application, +E = (0, ∞) is endowed with the usual distance on the real line. A marked metric measure space (mmm- +space for short) is a triple [X, d, µ], where (X, d) is a complete separable metric space, and µ is a finite +measure on X × E. To define a topology on the set of mmm-spaces, for each k ≥ 1, we consider the map +Rk : +� +(X × E)k → Rk2 ++ × Ek +� +(xi, ui); i ≤ k +� +�→ +� +d(xi, xj), ui; i, j ≤ k +� +that maps k points in X×E to the matrix of pairwise distances and marks. We denote by νk,X = µ⊗k◦R−1 +k , +the marked distance matrix distribution of [X, d, µ], which is the pushforward of µ⊗k by the map Rk. Note +that µ is not necessarily a probability distribution. Let k ≥ 1 and consider a measurable bounded test +function ϕ: Rk2 ++ × Ek → R. One can define a functional +Φ +� +X, d, µ +� += ⟨νk,X, ϕ⟩ = +� +X×E +ϕ +� +d(vi, vj), xi; i ̸= j ∈ [k] +� +k +� +i=1 +µ(dvi ⊗ dxi). +(11) +Functionals of the previous form are called polynomials, and the set of all polynomials, obtained by varying +k and ϕ, is denoted by Π. Finally, the moment of [X, d, µ] associated to Φ is defined as E(Φ +� +X, d, µ +� +). +Let ϕ be of the form +ϕ +� +di,j, xi; i ̸= j ∈ [k] +� += +� +i,j +ψi,j(di,j) +� +i +ϕi(xi) +where ψi,j, ϕi are bounded measurable functions. We say that Φ(X, d, µ) is a product polynomial. We +denote by ˜Π the set of product polynomials. +Remark 2.1. The moments of a random mmm can be rewritten as +E +� +Φ +� +X, d, µ +�� += E(|X|k) × +1 +E(|X|k)E +� +|X|kϕ(d(vi, vj), Xvi, i ̸= j ∈ [k]) +� +, +where (vi, Xvi) are k points sampled uniformly at random with their marks and |X| = µ(X ×E) is thought +as the total size of the population. As a consequence, the moments of a random mmm are obtained by +biasing the population size by its kth moment and then picking k individuals uniformly at random. +Definition 3. The marked Gromov-weak (MGW) topology is the topology on mmm-spaces induced by +Π. A random mmm-space is a r.v. with values in M – the set of (equivalence classes of) mmm-spaces – +endowed with the Gromov-weak topology and the associated Borel σ-field. +Finally, the marked Gromov-weak (MGW) topology is identical to the topology induced by the product +polynomials ˜Π. +7 + +Many properties of the marked Gromov-weak topology are derived in [DGP11] under the further +assumption that µ is a probability measure. In particular, the following result shows that Π forms a +convergence determining class only when the limit satisfies a moment condition, which is a well-known +criterion for a real variable to be identified by its moments, see for instance [Dur19, Theorem 3.3.25]. +This result was already stated for metric measure spaces without marks in [DG19, Lemma 2.7] and was +proved in [FRS22]. +Proposition 2.2. Suppose that [X, d, µ] is a random mmm-space verifying +lim sup +p→∞ +E[µ(X × E)p]1/p +p +< ∞. +(12) +Then, for a sequence [Xn, dn, µn] of random mmm-spaces to converge in distribution for the marked +Gromov-weak topology to [X, d, µ] it is sufficient that +lim +n→∞ E +� +Φ +� +Xn, dn, µn +�� += E +� +Φ +� +X, d, µ +�� +for all Φ ∈ Π. +2.2 +Marked Brownian Coalescent Point Process (CPP) +Let T > 0 and m be a measure on R+. Assume that |m| := m(R+) > 0. Consider P a PPP +� dx +x2 ⊗ dt +� +. +Define +YT = inf {y : (y, t) ∈ P, t ≥ T} , +and +dT (x, y) = sup{t : (z, t) ∈ P and x ≤ z ≤ y}, +0 < x < y < YT . +The marked Brownian Coalescent Point Process (CPP) is defined as +MT := +� +[0, YT ], dT , dv ⊗ m(dx) +� +. +This object is a natural extension of the standard Brownian CPP [Pop04]. +Remark 2.3. A direct computation shows that YT m(R+) (which can be thought as the population size +at time T) is distributed as an exponential random variable with mean T|m|. If we think as the CPP +as the genealogy of critical branching processes, this consistent with Yaglom’s law for critical branching +processes. +Figure 2: Simulation of the unmarked Brownian CPP. On the RHS, a vertical line of height x at location s +represents an atom (x, s) of P. On the LHS, the tree corresponding to the right CPP; the distance dT is the +tree distance of the leaves. +8 + +Proposition 2.4. Let K ∈ N. Let (ϕi; i ∈ [K]) and (ψi,j; i, j ∈ [K]) be measurable bounded functions. +Consider an arbitrary product polynomial of the form +∀M = [X, d, µ], +Ψ(M) := +� � +i,j +ψi,j(dT (vi, vj)) +� +i +ϕi(xi)µ(dvi ⊗ dxi). +Then +E [Ψ(MT )] = K!T KE +�� +i,j +ψi,j(Uσi,σj) +� �� +i +� +m(dx)ϕi(x) +�K +, +where (Ui; i ∈ [K − 1]) is a vector of uniform i.i.d. random variables on [0, T] and for i < j +Ui,j = Uj,i = max{Uk : k = i, · · · , j − 1}. +Proof. The proof is identical to Proposition 4 in [FRS22]. +Proposition 2.5 (Sampling from the CPP). Let k ∈ N and sample k points (v1, · · · , vk) uniformly at +random from the CPP. Let xv1 · · · , xvk be the corresponding types. Then ((d(vi, vj))i,j , (xvi)i) is identical +in law to +� +(Hσi,σj), (Wσi) +� +where (Hi,j) is defined as in Theorem 2 (ii) and (Wi) are independent random +variables with law +m +|m|. +Proof. The proof is identical to the one in the case of the unmarked CPP. See [BFRS22, Proposition +4.3]. +2.3 +Convergence of mmm +Fix t > 0. Recall that Nt refers the set of particles alive at time t in the BBM X. Set +µt := +� +v∈Nt +δv,xv, +∀v, v′ ∈ Nt, +d(v, v′) = +� +t − |v ∧ v′| +� +, +where v∧v′ denotes the MRCA of v and v′, and |v| denotes the generation of vertex v. Let Mt = [Nt, d, µt] +be the resulting random mmm. Finally, set +¯µt := +1 +N +� +v∈NtN +δv,xv, +∀v, v′ ∈ NtN, +¯d(v, v′) = +� +t − 1 +N |v ∧ v′| +� +, +and define the rescaled metric space +¯ +Mt +:= +[NtN, ¯d, ¯µt]. The main idea underlying Theorem 2 is to +prove the convergence of ¯ +Mt to a limiting CPP whose size and sampling structure coincides with (i) and +(ii) in Theorem 2 — See Propositions 2.4 and 2.5. +Theorem 3. Conditional on the event {ZtN > 0}, ( ¯ +Mt; N ∈ N) converges in distribution for the Gromov +weak topology to a marked Brownian CPP with parameters (t, Σ2 +2 ˜h∞). +The proof of the theorem relies on a cut-off procedure. Namely, let +L := +1 +µ − β log(N). +(13) +Recall that XL refers to the BMM killed at 0 and L. Let µL +t (resp. ¯µL +t ) be the empirical measure obtained +by replacing N by N L in the definition of µt (resp. ¯µt). Let M L +t be the mmm obtained from XL, that +is M L +t += [N L +t , d, µL +t ]. +¯ +M L +t +is defined analogously to ¯ +Mt (i.e. accelerating time by N and rescaling the +empirical measure by 1/N). We will proceed in two steps. For our choice of L, we will show that +1. +¯ +M L +t converges to the limit described in Theorem 3. +2. +¯ +M L +t and ¯ +Mt converge to the same limit. +The choice for L will be motivated in Section 2.6. We start by motivating the fact that ¯ +M L +t converges to +the desired limit using a spinal decomposition introduced in [FRS22] in a discrete time setting. +9 + +2.4 +The K-spine +Definition 4. The 1-spine is the stochastic process on [0, L] with generator +1 +2∂xxu + v′ +1(x) +v1(x)∂xu, +u(0) = u(L) = 0. +In the following, qt(x, y) will denote the probability kernel of the 1-spine. +We can also directly determine the invariant distribution of the spine +Proposition 2.6. The 1-spine has a unique invariant measure given by Π(dx) = +� +v1(x) +||v1|| +�2 +dx. +Let (U1, ..., UK−1) be i.i.d. random variables uniformly distributed on [0, t]. Define +∀ 1 ≤ i < j ≤ K − 1, +d(i, j) = d(j, i) = max{Ui, ..., Uj−1}. +(14) +Let T be the unique planar ultrametric tree of depth t with K leaves labeled by {1, ..., K} such that the +tree distance between the leaves is d. See Fig 3. +We then assign marks on this tree such that, on each branch of the tree, marks evolve according to +the 1-spine (on [0, L]) and branch into independent particles at the branching points of T. See [FRS22] +for a more formal definition. +t +0 +U1 +U2 +V1 +V2 +V3 +t +0 +x +x0 +ζv1 +qt(x0, ·) +qt−U1(ζv1, ·) +Figure 3: K-spine with K = 3. Left panel: planar tree T generated from 2 i.i.d. uniform random variables +(U1, U2). Right panel: branching 1-spines running along the branches of the tree T. +The resulting planar marked ultrametric tree will be referred to as the K-spine. We will denote by +QK,t +x +the distribution of the K-spine rooted at x. The t superscript refers to the depth of the underlying +genealogy. Note that QK,t +x +has an implicit dependence on N by our choice of L – see (13). To ease the +notation, this dependence will be dropped in the notation. +In the following, B will denote the set consisting of the K − 1 branching points of the K-spine; L will +denote the set consisting of the K leaves. We will denote by ζv the mark (or the position) of the spine +at a node v ∈ B ∪ L. For v ∈ B, |v| will denote the time component of the branching point. Finally, +(Vi; i ∈ [K]) is the enumeration of the leaves from left to right in the K-spine (i.e., Vi is the leaf with +label i). +We will also need the accelerated version of the K-spine. +Definition 5 (Accelerated K-spine). Consider the 1-spine accelerated by N, i.e. the transition kernel +of the 1-spine is now given by qtN(x, y) ≡ qL +tN(x, y). We denote this kernel by ¯qt(x, y). Consider the +same planar structure as before, i.e., the depth is t and the distance between points at time t is given by +(14). We denote by ¯QK,t +x +the distribution of the K-spine obtained by running accelerated spines along the +branches. +For any vertex v in the K-spine, ¯ζv will denote the mark of the vertex v. +10 + +Proposition 2.7 (Rescaled many-to-few). Let t > 0. Let (ϕi; i ∈ [K]) and (ψi,j; i, j ∈ [K]) be measurable +bounded functions and define +∀M = [X, d, µ], +Ψ(M) = +� � +i,j +1(vi ̸= vj)ψi,j(d(vi, vj)) +� +i +ϕ(xi)µ(dvi ⊗ dxi). +Define h(t, x) := ce(λ∞ +1 −λ1)teµxv1(x) where c is the renormalisation constant in (7). Then +Ex +� +Ψ( ¯ +M L +t ) +� += +1 +N K! h(0, x) tK−1 ¯QK,t +x +� +¯∆ +� +i,j +ψi,j(Uσi,σj) +� +i +ϕi(¯ζVσi ) +� +, +where +¯∆ := +� +v∈B(U) +r(¯ζv)h(|v|N, ¯ζv) +� +v∈L(U) +1 +h(tN, ¯ζv). +The second crucial result is the following convergence theorem. +Theorem 4. Let ( ˜ϕi; i ∈ [K]) and (ψi,j; i, j ∈ [K]) be measurable bounded functions. Assume further +that the ˜ϕi’s are compactly supported in (0, ∞). As N → ∞, +¯QK,t +x +� +¯∆ +� +i,j +ψi,j(Uσi,σj) +� +i +˜ϕi(¯ζVσi )h∞(¯ζVσi ) +� +→ +�Σ2 +2 +�K−1 +E +�� +i,j +ψi,j(Uσi,σj) +� � +i +� +R+ +˜ϕi(x)Π∞(dx). +Let us now give a brief heuristics underlying the previous result. By definition +¯QK,t +x +� +¯∆ +� +i,j +ψi,j(Uσi,σj) +� +i +˜ϕi(¯ζVσi )h∞(¯ζVσi ) +� += +¯QK,t +x +�� +v∈B +r(¯ζv)h(|v|N, ¯ζv) +� +i,j +ψi,j(Uσi,σj) +� +i +˜ϕi(¯ζVσi ) h∞(¯ζVσi ) +h(tN, ¯ζVσi ) +� +. +The branching structure for the K-spine is binary a.s. whereas the spines running along the branches are +accelerated by N. Recall that the invariant measure for the 1-spine is Π ≈ Π∞ as L → ∞ (see 2.6). +Moreover, we will show later on that under (Hfp), h(tN, x) ≈ h∞(x) for N large. It is now reasonable to +believe that, provided enough mixing, the RHS can be approximated by +�� L +0 +h∞(x)r(x)Π∞(dx) +�K−1 +E +�� +i,j +ψi,j(Uσi,σj) +� � +i +� +˜ϕi(x)Π∞(dx), +assuming the values of the spine at the branching points and the leaves converge to a sequence of i.i.d. +random variables with law Π∞. This yields the content of Theorem 4. +Challenge 1. The previous argument relies on a k-mixing property of the 1-spine. This analysis will be +carried out in Section 4 using Sturm–Liouville theory. +2.5 +Limiting moments +Let us now demonstrate the importance of Theorem 4. Let +∀M = [X, d, µ], +˜Ψ(M) := +� +X +� +i,j +ψi,j(d(vi, vj)) +� +i +˜ϕi(xi)h∞(xi) µ(dvi ⊗ dxi). +11 + +for any bounded measurable functions ˜ϕi, ψi,j such that the ˜ϕi are compactly supported. +From the +many-to-few formulae, our result entails +Ex +� +˜Ψ( ¯ +M L +t ) +� += +Ex +� � � +i,j +ψi,j( ¯d(vi, vj)) +� +i +˜ϕi(xi)h∞(xi) ¯µL +t (dvi ⊗ dxi) +� +≈ +Ex +� � � +i,j +1(vi ̸= vj)ψi,j( ¯d(vi, vj)) +� +i +˜ϕi(xi)h∞(xi) ¯µL +t (dvi ⊗ dxi) +� +≈ +2h∞(x) +NΣ2t +× K! +� +tΣ2 +2 +�K +E +�� +i,j +ψi,j(Uσi,σj) +� � +i +� +R+ +˜ϕi(x)Π∞(dx). +Let us formally take ψi,j ≡ 1 and ˜ϕi ≡ 1/h∞ in the previous expression (note that this is problematic +since ˜ϕi is neither bounded nor compactly supported, see Challenge 4 below). Then for large N, +Ex +�� 1 +N ZL +tN +�K� +≈ +2h∞(x) +NΣ2t +× +K! +� +tΣ2 +2 +�K +� +�� +� +exponential moments +× +� ∞ +0 +˜h∞(x)dx +� +�� +� +=1 +, +where we used the fact that ˜h∞ = Π∞/h∞ under our Perron–Frobenius renormalisation (see (7)). Now, +the RHS coincides with the moments of a r.v. with law +� +1 − 2h∞(x) +NΣ2t +� +δ0(dx) + 2h∞(x) +NΣ2t exp +� +− 2x +Σ2t +� 2dx +Σ2t . +If we identify the Dirac measure at 0 with the extinction probability, this suggests the Kolmogorov +estimate and the Yaglom law exposed in Theorem 1 and Theorem 2. Further, if we replace ˜ϕi = ϕi/h∞ +in Theorem 4 (again a problematic step), the previous estimates entail +Ex +� +Ψ( ¯ +M L +t ) +��ZL +tN > 0 +� +≈ K!E +�� +i,j +ψi,j(Uσi,σj) +� � +i +� +R+ +ϕi(x) +� +tΣ2 +2 +� +˜h∞(x)dx. +where Ψ(M) is now an arbitrary polynomial of the form +Ψ(M) := +� � +i,j +ψi,j(d(vi, vj)) +� +i +ϕi(xi)µ(dvi ⊗ dxi). +According to Proposition 2.4, this coincides with the moments of the Brownian CPP described in Theorem +3. +Challenge 2. The previous computation only suggests that the probability for the population to be o( 1 +N ) +is given by the Kolmogorov estimate. Intuitively, the Dirac mass above corresponds to a population whose +size becomes invisible at the limit after rescaling the population by N. It thus remains to show that if the +population is small compared to N then it must be extinct. This will be carried out in Section 6. +Challenge 3. Going from Theorem 4 to the convergence of the M L +tN requires to use test functions ex- +ploding at the boundary. To overcome this technical difficultly, we will impose an extra thinning of the +population by killing all the particles close to the boundaries at time tN. This final technical step will be +carried in Section 7 using some general property of the Gromov-weak topology. +2.6 +Choosing the cutoff L +We now motivate our choice for L. According to the previous arguments, we want to choose L large +enough such that +(i) The particles do not reach L with high probability. This will imply that ¯ +M L +t and ¯ +Mt coincide with +high probability. +(ii) The 1-spine reaches equilibrium in a time o(N) regardless of its initial position on [0, L]. This is +needed to justify the calculations of Section 2.4. +12 + +(i) Hitting the right boundary. Let E be a compact set in the vicinity of the boundary L (say +[L − 2, L − 1]). Recall from the discussion after Proposition 1.3 that for x ≥ 1, +h∞(x) = c eµxv∞ +1 (x) ≈ c eβe(µ−β)x. +A direct application of the many-to-few lemma with K = 1 (many-to-one) and ϕ1(x) = 1x∈E implies that +Ex +� +� � +v∈N L +tN +1xv∈E +� +� += +� +E +h(0, x) +h(tN, y)qtN(x, y)dy ≈ +� +E +h∞(x) +h∞(y) Π∞(dy) ≈ h∞(x)O( +=N−α +� +�� +� +e− µ+β +µ−β log(N)), +where the last approximation holds under the assumption that E is a compact set close to L. Integrating +on [0, tN], this yields that the occupation time of the set E on the time interval [0, tN] is O(N 1−α). Recall +that the probability of survival is of order 1/N so that the occupation time of the conditioned process is +O(N 2−α). Recalling that α > 2 under (Hfp), this yields the desired estimate. +(ii) Mixing time. Recall from the discussion after Proposition 1.3 that v1(x) ≈ e−β(x−1) for x ≥ 1 +and L large enough. As a consequence, the 1-spine (see Definition 4) is well approximated by the diffusion +dzt = −βdt + dwt for zt ≥ 1. +A good proxy for the mixing time is the first returning time at 1 which is of the order log(N) = o(N) for +every x ∈ [1, L], as desired. A more refined analysis will be carried in Section 4. +3 +The many-to-few theorem +3.1 +The general case +In this section, we consider a general BBM killed at the boundary of a regular open domain Ω ⊂ Rd. +Unless otherwise specified, we used the same notation as in the previous sections. We will assume that +1. The generator of a single particle is given by a differential operator +Lf(x) = 1 +2 +� +i,j +aij(x)∂xixjf(x) + +� +i +bi(x)∂xif(x), +x ∈ Ω, +(15) +f(x) = 0, +x ∈ ∂Ω. +We assume that (aij) is uniformly elliptic, which means that there exists a constant θ > 0 such that +for all ξ ∈ Rd and a.e. x ∈ Ω, �d +i,j=1 aij(x)ξi, ξj ≥ θ∥ξ∥2 (see [Eva10, §6.1]). In addition, we assume +that ai,j ∈ C1(Ω) and supx∈Ω |bi(x)| < ∞ for all 1 ≤ i, j ≤ d. +2. A particle at location x branches into two particles at rate r(x) (we only consider binary branching). +We denote by Nt the set of particles alive at time t. For any pair of particles v, w ∈ Nt, d(u, w) will +denote the time to their most recent common ancestor. Finally, we define the random mmm space +Mt = [Nt, d, µt], +where µt = � +v∈Nt δv,xv. +We say that a function h(x) is harmonic if and only h satisfies the Dirichlet problem +Bh(x) := Lh(x) + r(x)h(x) = 0, +for every x ∈ Ω, +h(x) = 0, +for x ∈ ∂Ω. +Definition 6. The 1-spine whose generator is given by the Doob h-transform of the differential operator +B +B(hf) +h +(x) = 1 +2 +� +i,j +aij(x) +� +∂xixjf(x) + ∂xih +h +∂xjf(x) + ∂xjh +h +∂xif(x) +� ++ +� +i +bi(x)∂xif(x), +x ∈ Ω, +f(x) = 0, +x ∈ ∂Ω, +where the first equality is a direct consequence of the fact that h is harmonic. We will denote by qt(x, y) +the transition probability at time t. The K-spine distribution QK,t +x +is defined analogously to Section 2.4. +13 + +Our many-to-few formulae rely on a uniform planarisation of the BBM that we now describe. At +every time t > 0, every particle is now endowed with two marks (xv, pv) where pv ∈ ∪n∈N{0, 1}n. As +before, xv denotes the position of the particle. The planarisation marks pv are assigned recursively as +follows. We mark the root with ∅ and +1. At every branching point v, we distribute the marks (pv, 0) and (pv, 1) uniformly among the two +children. (pv, 0) (resp. (pv, 1)) is said to the left (resp. right) child of v. +2. The mark pv does not vary between two branching points, i.e., pv1 = pv2 if the trajectory connecting +v1 and v2 does not encounter any branching points. +Let N pl +t +be the the set particles at time t in the planar BBM. For every K-uplet v1, v2 · · · , vK in N pl +t , let +U(⃗v) = [X(⃗v), d(⃗v), µ(⃗v)] +be the planar (ultra-)mmm space induced by this set of vertices. The space consists of the set of vertices +ancestral to some vertex in ⃗v. In particular, U(⃗v) is binary and made of K leaves. Finally, the measure +is given by the counting measure on the leaves +µ(⃗v) = +� +⃗v +δv,(xv,pv). +Let us now introduce some definition. Consider a planar ultra-mmm U made of finitely many leaves. We +define σ(U) to be the diameter U. We say that U has no simultaneous branching iff for every distinct +pair of leaves (v1, v2), (v3, v4), we have d(v1, v2) ̸= d(v3, v4). For such a tree, there exists a unique element +v ∈ U such that +v = Argmax d(w, ¯w) +where the maximum is taken over the pair of leaves in U. The individual v is said to be the MRCA of +U. Moreover, we define T0(U) (resp. T1(U)) to be the left (resp. right) subtree attached to the MRCA. +Finally, B(U) will refer to the set of branching points and L(U) will refer to the set of leaves. Finally, +xMRCA(U) will be the spatial position of the MRCA. +Lemma 3.1 (Many-to-one). For every bounded continuous function f +Ex +� � +v∈Nt +f(xv) +� += +� +Ω +f(y)qt(x, y)h(x) +h(y) dy. +Proof. One can readily check that pt(x, y) := qt(x, y) h(x) +h(y) are the fundamental solutions of the same PDE +� +∂tu(t, y) = L∗u(t, y), +y ∈ Ω, +u(t, y) = 0, +y ∈ ∂Ω, +where L∗ is the adjoint of the differential operator (15). +Lemma 3.2. Let K ∈ N and t > 0. Define the measure RK,t +x +on the set of planar mmm so that for every +bounded measurable function F, +RK,t +x +(F) := Ex +� +� +� +� +v1̸=···̸=vK, vi∈N pl +t +F(U(⃗v)) +� +� +� . +Then +RK,t +x +(F) = K! h(x)tK−1 QK,t +x +(∆F) with ∆ = � +v∈B(U) r(ζv)h(ζv) � +v∈L(U) +1 +h(ζv). +Proof. We will show the result by an induction of K. The case K = 1 is the many-to-one lemma. Let us +now consider F of the following product form +F(U) = f(σ(U))ψ0(T0(U))ψ1(T1(U)). +14 + +Then +RK,t +x +(F) += +K!Ex +� +� +� +v1<··· 0. Let (ϕi; i ∈ [K]) and (ψi,j; i, j ∈ [K]) be measurable bounded +functions. Define the product polynomial +M = [X, d, µ], +Ψ(M) := +� � +i,j +1vi̸=vjψi,j(d(vi, vj)) +� +i +ϕi(xi)µ(dvi ⊗ dxi). +Then +Ex (Ψ(Mt)) = K! h(x) tK−1QK,t +x +� +∆ +� +i,j +ψi,j(Uσi,σj) +� +i +ϕi(ζVσi ) +� +, +where σ is an independent random permutation of [K]. +Proof. Let U be a planar mmm with K leaves. Let xVi be the ith leaf of U (where the ordering is the +one induced by the planarisation). Let σ be a permutation of [K]. Then +Fσ(U) = +� +i,j +ψi,j(d(xVσ(i), xVσ(j))) +� +i +ϕi(xVσi ). +From the previous result, we get +� +RK,t(Fσ(U))λ(dσ) = h(x)K!tK−1 +� +QK,t(Fσ(U))λ(dσ), +where λ is the uniform measure on permutation of K. The result follows from there. +3.2 +The subcritical BBM +We now apply the previous setting to the problem at hands. The idea is to enrich the mark space by +adding time as an extra variable. More precisely, for every v ∈ N L +t , we equip v with a two dimensional +mark (t, xv) where xv is the position of the particle on [0, L]. +Lemma 3.3. Let µ, λ1, λ∞ +1 , r(x) be defined as in Section 1. Define +Lf(t, x) = ∂tf(t, x) + 1 +2∂xxf(t, x) − µ∂xf(t, x), +f(0) = f(L) = 0. +Then h(t, x) = ce(λ∞ +1 −λ1)teµxv1(x) is harmonic. In addition, +B(hf) +h +(t, x) = ∂tf(t, x) + 1 +2∂xxf(t, x) + v′ +1(x) +v1(x)∂xf(t, x). +Proof. This is a straightforward consequence of Definition 6. +A direct application of Theorem 5 entails the following result. +Corollary 3.4 (Many-to-few). Let t > 0. Let (ϕi; i ∈ [K]) and (ψi,j; i, j ∈ [K]) be measurable bounded +functions. Define the product polynomial +M = [X, d, µ], +Ψ(M) := +� � +i,j +1vi̸=vjψi,j(d(vi, vj)) +� +i +ϕi(xi)µ(dvi ⊗ dxi). +Then +Ex +� +Ψ(M L +t ) +� += K! h(x) tK−1QK,t +x +� +�∆ +� +i,j +ψi,j(Uσi,σj) +� +Vi∈L +ϕi(ζVσi ) +� +� , +where σ is an independent permutation of [K] and +∆ := +� +v∈B(U) +r(ζv)h(|v|, ζv) +� +v∈L(U) +1 +h(t, ζv). +Proof of Proposition 2.7. This is a direct consequence Corollary 3.4 after rescaling the measure by N and +time by N. +16 + +4 +Spectral theory +In this section, we examine the density of the particles in XL. In Section 4.1 and Section 4.2, we give +precise estimates on the heat kernel pt associated to (A) and compute the relaxation time of the system. +All the lemmas in these sections hold under (Hp) and do not require the additional assumption (Hfp). +Sections 4.3 and 4.4 are aimed at quantifying the fluctuations in XL and are specific to the fully pushed +regime. +4.1 +Preliminaries +Consider the Sturm–Liouville problem (SLP) together with boundary conditions (BC). Let us first recall +some well-known facts about Sturm–Liouville theory following [Zet12, Section 4.6]: +(1) A solution of (SLP) is defined as a function v : [0, L] → R such that v and v′ are absolutely +continuous on [0, L] and satisfies (SLP) a.e. on (0, L). In particular, any solution v is continuously +differentiable on [0, L]. Since f is continuous on [0, L], the solutions are also twice differentiable on +[0, L] and (SLP) holds for all x ∈ (0, L). +(2) A complex number λ is an eigenvalue of the Sturm–Liouville problem (SLP) with boundary condi- +tions (BC) if Equation (SLP) has a solution v which is not identically zero on [0, L] and that satisfies +(BC). This set of eigenvalues will be referred to as the spectrum. +(3) It is known that this set of eigenvalues is infinite, countable and it has no finite accumulation point. +Besides, it is upper bounded and all the eigenvalues are simple and real so that they can be numbered +λ1 > λ2 > ... > λn > ... +where +λn → −∞ +as +n → +∞. +We will denote by K the largest integer such that +λK > 0. +(4) As a consequence, the eigenvector vi associated to λi is unique up to constant multiplies. Further- +more, the sequence of eigenfunctions can be normalised to be an orthonormal sequence of L2([0, L]). +This orthonormal sequence is complete in L2([0, L]) so that the fundamental solution of PDE (B) +can be written as +gt(x, y) = +∞ +� +k=1 +eλkt vk(x)vk(y) +∥vk∥2 +. +(17) +(5) v1 does not change sign in (0, L). More generally, the eigenfunction vk has exactly k − 1 zeros on +(0, L). +(6) The eigenvalues and the eigenvectors of (SLP) with boundary conditions (BC) can be characterised +through the Prüfer transformation (see [Zet12, Section 4.5]). For all λ ∈ R, consider the Cauchy +problems +˙θλ(x) = cos2(θλ(x)) + (f(x) − 2λ) sin2(θλ(x)), +θλ(0) = 0, +(18) +and +˙ρλ(x) = +�1 − f(x) +2 ++ λ +� +sin(2θλ(x))ρλ(x), +ρλ(0) = 1. +(19) +Note that (18) and (19) have a unique solution defined on [0, +∞) for each λ ∈ R. The eigenvalue +λk is characterised as the unique solution of θλ(L) = kπ. +Note that for all λ∗ < λ < 0, 0 ≤ θλ(x) ≤ θλ∗(x) for all x ∈ [0, L]. If uk is the eigenvector associated +to λk such that u′ +k(0) = 1, then +uk(x) = ρλk(x) sin(θλk(x)), +u′ +k(x) = ρλk(x) cos(θλk(x)), +∀x ∈ [0, L]. +(20) +17 + +(7) Denote by ¯λ1 > ¯λ2 > ... the eigenvalues of the Sturm-Liouville problem +1 +2v′′(x) + 1 +2∥f∥∞1[0,1](x)v(x) = λv(x), +v(0) = v(L) = 0. +and by λ1 > λ2 > ... the eigenvalues of the Laplacian with homogeneous Dirichlet boundary condi- +tions (at 0 and L). Then, for all k ∈ N, +λk ⩽ λk ⩽ ¯λk. +(21) +See [Zet12, Theorem 4.9.1] for a proof of this comparison principle. Recall that +λk = −k2π2 +2L2 , +(22) +and that the eigenvalues (¯λk) have been fully characterised in [Tou21]. See (6) for a characterisation +of the negative spectrum for a special case of f. In particular, we know that there exists ¯K ≥ K +which is fixed after some L and such that ¯λk > 0 for all k ≥ ¯K and ¯λk < 0 for all k < ¯K. +(8) For fixed i ∈ N, the eigenfunction λi is an increasing function of L (see e.g. [Zet12, Theorem 4.4.4]). +Since λ1 converges, the i-th eigenvalue λi also converges. Further, by (21), this implies that the +number of positive eigenvalues K is fixed after L large enough. For i > K, this limit, that we denote +by λ∞ +i +is positive and we have +λ∞ +1 ≥ λ∞ +2 ≥ λ∞ +3 ≥ ... ≥ λ∞ +K . +We will prove below that these inequalities are strict inequalities. +Throughout the article, the eigenvector v1 will be chosen such a that v1(1) = 1 and u1 in such a way that +u′ +1(0) = 1. Note that for x ∈ [1, L], we have +v1(x) = sinh +�√ +2λ1(L − x) +� +/ sinh +�√ +2λ1(L − 1) +� +. +(23) +On [0, 1], the eigenvector u1 is the unique solution of the Cauchy problem +u′′(x) = (2λ1 − f(x)) u(x), +u(0) = 0, u′(0) = 1. +(24) +Lemma 4.1 ([Zet12], Theorem 4.5.1). Let a, b, c ∈ C([0, 1]). Let θ and ρ be the solutions of the Cauchy +problems +� ˙θ(x) = a(x) cos(θ(x))2 + b(x) sin(θ(x))2, +θ(0) = 0, +˙ρ(x) = c(x) sin(2θ(x))ρ(x), +ρ(0) = 1. +Let Φ be the application from (C([0, 1])3, || · ||∞) to (C(0, 1)2, || · ||∞) which maps (a, b, c) to the solution +of the Cauchy problem. Then Φ is continuous. +Lemma 4.2. Assume that (Hp) holds. Then, +λ∞ +1 − λ1 = O +� +e−2βL� += O +� +1 +N α−1 +� +. +(25) +As a consequence, for all 0 < t ≲ N, +h(t, x) = +� +1 + O +� +e(µ−3β)L�� +h(0, x), +x ∈ [0, L]. +(26) +In addition, +v1(x) ≍ (1 ∧ x ∧ (L − x))e−βx, +x ∈ [1, L], +(27) +and we have v11∈[0,L] → v∞ +1 +as L → ∞ pointwise, in L2 and in C1([0, 1]). Further, v∞ +1 (x) = e−β(x−1) for +x ≥ 1. +18 + +Proof. We first prove the convergence of u1. +Recall from (20) that u1(x) = ρλ1(x) sin(θλ1(x)) and +u′ +1(x) = ρλ1(x) cos(θλ1(x)) for all x ∈ [0, L]. By Lemma 4.1, u1 converges to u∞ +1 += ρλ∞ +1 sin(θλ∞ +1 ) in +C1([0, 1]) as L → ∞. The pointwise convergence on [0, ∞], follows from the fact that +u1(x) = u1(1)sinh(√2λ1(L − x)) +sinh(√2λ1(L − 1)) +∀x ∈ [1, L]. +(28) +Further, the convergence in L2 follows from the uniform convergence on [0, 1] and the explicit form of u1 +on [1, L]. +It follows from the integrated version of (24) that u′′ +1 converges to (u∞ +1 )′′ in C([0, 1]) and that u∞ +1 +is +the unique solution of the Cauchy problem +1 +2u′′(x) + 1 +2u(x) = λ∞ +1 u(x), +x ∈ (0, 1), +u1(0) = 0, u′ +1(0) = 1. +(29) +As a consequence, u∞ +1 (x) ̸= 0 for every x ∈ [0, 1]. Indeed, u∞ +1 ≥ 0 so that if u∞ +1 (x) = 0 then (u∞ +1 )′(x) = 0 +and u∞ +1 ≡ 0 on [0, L] which contradicts the fact that (u∞ +1 )′(0) = 1. It easily follows that u∞ +1 (x) ≍ x on +[0, 1]. +The convergence of v1 then follows from the relation v1(x) = u1(x)/u1(1) and the convergence of u1. +Finally, Equation (27) then follows from (28). +Let us now prove the first part of the lemma. According to [Zet12, Theorem 4.4.4], L �→ λ1(L) is +differentiable on (1, +∞) and we have +λ′ +1(L) = |u′ +1(L)|2 +∥u1∥2 . +Yet, +u′ +1(L) = u1(1)v′ +1(L) = −u1(1) +√ +2λ1 +� +sinh +√ +2λ1(L − 1) +�−1 += O(e−√2λ1L). +Thus, +λ′ +1(L) = O(e−2√2λ1L). +Define ˜L such that λ1 > +λ∞ +1 +2 +for all L > ˜L. Hence, for L > ˜L +λ∞ +1 − λ1 ⩽ C +� ∞ +L +e−2√ +λ∞ +1 L′dL′ ⩽ Ce−2√ +λ∞ +1 L. +This implies that λ∞ +1 − λ1 ≪ 1 +L so that +sinh( +√ +2λ1(L − 1)) ∼ 1 +2eβ(L−1), +L → ∞, +and +λ′ +1(L) = +2λ1 +∥u1∥2 u1(1) sinh( +√ +2λ1(L − 1))−2 = O +� +e−2βL� +, +L → ∞. +Therefore, for L large enough, we obtain +λ∞ +1 − λ1 ⩽ C +2 +� ∞ +L +e−2βL′dL′ = O(e−2βL). +This concludes the proof of the (25) and (26). +Corollary 4.3. Assume that (Hp) holds. Let k ∈ N. Let g : [0, ∞)k → R and f : [0, ∞)2k → R be +bounded and continuous test functions. Let t > 0. We have +� +g(x1, ..., xk) +k +� +i=1 +Π(dxi) → +� +g(x1, ..., xk) +k +� +i=1 +Π∞(dxi), +as +L → ∞, +and uniformly in (t1, ..., tk) ∈ [0, T]k, as N goes to ∞, we have +� +f(x1, ..., xk, h(t1N, x1), ..., h(tkN, xk)) +k +� +i=1 +Π(dxi) → +� +f(x1, ..., xk, h∞(x1), ..., h∞(xk)) +k +� +i=1 +Π∞(dxi). +19 + +Proof. We restrict ourself to the case k = 1. The case k > 1 can be proved along the same lines. We +start with the first limit. Since ||v1|| → ||v∞ +1 ||, we need to prove +� +g(x1, ..., xk) +k +� +i=1 +v2 +1(xi)dxi → +� +g(x1, ..., xk) +k +� +i=1 +(v∞ +1 )2(xi)dxi, +as +L → ∞, +We first note that +v1(x) ≤ v∞ +1 (x) +∀x ∈ [1, L]. +The result then follows from the dominated convergence theorem and the fact that that v1 converges to +v∞ +1 +uniformly on [0, 1]. The second limit follows from (26) combined with similar arguments. +Lemma 4.4. Assume that (Hp) holds. For all k ∈ N, we have +|uk(x)| ⩽ e3¯ce|λk|(1 + 2|λk|)x, +x ∈ [0, 1], +with ¯c = 1 + ∥f∥∞. +Proof. First, note that for all λ ∈ R, the solution ρλ of the Cauchy problem (19) can be expressed as +ρλ(x) = ρλ(0) exp +�� x +y=0 +�1 − f(y) +2 ++ λ +� +sin(2θλ(y))dy +� +, +x ∈ [0, 1]. +Therefore, +0 ⩽ ρλ(x) ⩽ exp(¯c + |λ|), +x ∈ [0, 1]. +(30) +On the other hand, note that for all λ ∈ R, the unique solution θλ of (18) is such that +˙θλ(x) − ˙θ0(x) ⩽ | ˙θλ(x) − ˙θ0(x)| ⩽ 2¯c|θλ(x) − θ0(x)| + 2|λ|, +x ∈ [0, 1], +where we use that cos2 and sin2 are 2-Lipschitz. Moreover, remark that for all λ < 0, we have θλ ≥ θ0 +on [0, L]. Hence, for all λ < 0, we have +θλ(x) − θ0(x) ⩽ −2λx + +� x +y=0 +2¯c(θλ(y) − θ0(y))dy, +x ∈ [0, 1], +and Grönwall’s lemma yields +θλ(x) ⩽ θ0(x) − 2λx exp(2¯cx), +x ∈ [0, 1]. +(31) +In addition, we see from (18) that 0 ≤ θ0(x) ≤ ¯cx for all x ∈ [0, L], so that +θλ(x) ⩽ (¯c + 2|λ|e2¯c)x, +x ∈ [0, 1], +(32) +for all λ < 0. For λ > 0, we simply note that 0 ≤ θλ(x) ≤ θ0(x) for all x ∈ [0, L] so that the last +inequality still holds for positive λ. We finally get the result by combining (20), (30), (32) and the facts +that | sin(x)| ⩽ x for all x ≥ 0 and that ¯c ⩽ e2¯c. +Lemma 4.5. There exists a constant c4.5 > 0 such that for sufficiently large L, we have √−λK+1L > c4.5. +Proof. First, note that θ0(x) ∈ ((K − 1 +2)π, (K + 1 +2)π) for all x ∈ [1, L] since (K − 1 +2)π and (K + 1 +2)π are +fixed points of the ODE satisfied by θ0 on [1, L] and θ0(L) ∈ [Kπ, (K + 1)π]. +W.l.o.g. we assume that √−λK+1 → 0 as L → ∞. By arguing as in Lemma 4.1, uK+1 converges to +u∞ +K+1 = ρ0 sin(θ0) in C1([0, 1]). Recall that uK+1 has K zeroes in (0, L), and according to the characterisa- +tion of λK+1 by the Prüfer transformation, one zero is attained at x0 such that θλK+1(x0) = (K+ 1 +2)π. On +the other hand, θλK+1 converges to θ0 in C([0, 1]). Since θ0(1) < (K + 1 +2)π and thus θλK+1(1) < (K + 1 +2)π +for L large enough, uK+1 has a zero on [1, L]. +We now argue by contradiction and assume that limL→∞ +√−2λK+1L = 0. For all L > 1, there exists +cK+1 = cK+1(L) ̸= 0 such that +uK+1(x) = cK+1 sin( +� +−2λK+1(L − x)) on [1, L]. +(33) +This prevents uK+1 on vanishing on [1, L), which yields a contradiction. +20 + +Lemma 4.6. Assume that (Hp) holds. There exists c4.6 > 0 such that for all k ∈ N, +∥uk∥2 ⩾ c4.6 +� +1 ∧ +1 +|λk| +� +. +Proof. Define g1 : [−1, λ∞ +1 ] → [0, +∞), λ �→ +� 1 +x=0 ρλ(x)2 sin2(θλ(x))dx. The function g1 is continuous by +Lemma 4.1. Thus it attains a positive minimum at some λ0. +Let us now consider λ < −1 and remark that +� 1 +x=0 +ρλ(x)2 sin(θλ(x))2dx ⩾ +� +1 +|λ| +x=0 +ρλ(x)2 sin(θλ(x))2dx = 1 +|λ| +� 1 +x=0 +ρλ(x/|λ|)2 sin(θλ(x/|λ|))2dx. +For all x ∈ [0, 1], define ˜ρλ(x) = ρλ(x/|λ|) and ˜θλ(x) = θλ(x/|λ|). These functions correspond to the +unique solutions of the Cauchy problems +˙˜θλ(x) = 1 +|λ| +� +cos2(˜θλ(x)) + (f(x/|λ|) − 2λ) sin2(˜θλ(x)) +� +, +˜θλ(0) = 0, +and +˙˜ρλ(x) = 1 +|λ| +��1 − f(x/|λ|) +2 ++ λ +� +sin(2˜θλ(x))˜ρλ(x) +� +, +˜ρλ(0) = 1, +on [0, 1]. As before, g2 : [−1, 0] → [0, +∞), λ �→ +� 1 +x=0 ˜ρ1/λ(x)2 sin2(˜θ1/λ(x))dx attains its minimum at +some positive value. This completes the proof of the lemma. +Proposition 4.7. Assume that (Hp) holds. There exists a constant c4.7 such that for L large enough, +for all x ∈ [0, L], +∀k ≤ K, +1 +∥uk∥ +|uk(x)| +u1(x) +⩽ c4.7 eβx, +∀k > K, +1 +∥uk∥ +|uk(x)| +u1(x) +⩽ c4.7eβx e|λk|(1 + 2|λk|)2. +Proof. Let us consider the case k < K. By the same argument as (27), uk converges to a limiting function +u∞ +k in L2 and +|uk(x)| ≍ c1(x ∧ 1 ∧ (L − x))e−√ +2λ∞ +k , +on [0, L]. +This shows the result for k ≤ K. +We now turn to the case k > K. First, we use Lemma 4.4 along with Lemma 4.6 to bound uk/∥uk∥ +on [0, 1]: there exists c2 > 0 such that +|uk(x)| +∥uk∥ +⩽ c2e|λk|(1 + 2|λk|)(1 ∨ |λk|)1/2x ⩽ c2e|λk|(1 + 2|λk|)2x, +∀x ∈ [0, 1], k > K. +(34) +On [1, L], we have uk(x) = dk sin(√−2λk(L − x)) for some dk ∈ R. Hence, +∥uk∥2 ⩾ +� L +x=1 +uk(x)2dx = d2 +k(L − 1) +� +1 − sin(2√−2λk(L − 1)) +2√−2λk(L − 1) +� +. +Recall from Lemma 4.5 that, for L large enough, 2√−2λk(L − 1) > 2 +√ +2c4.5 for all k > K. Therefore, +there exists c3 > 0 such that +� +1 − sin(2√−2λk(L − 1)) +2√−2λk(L − 1) +� +> c3, +∀k > K. +Using that | sin(x)| ⩽ 1 ∧ x for all x ≥ 0, we get that for sufficiently large L, +|uk(x)| +∥uk∥ +⩽ +1 +� +c3(L − 1) +⩽ 1, +∀x ∈ [1, L − 1], +(35) +and +|uk(x)| +∥uk∥ +⩽ +1 +� +c3(L − 1) +� +−2λk(L − x) ⩽ +� +−2λk(L − x) ⩽ (1 + 2|λk|)2(L − x), +∀x ∈ [L − 1, L]. +(36) +Equations (34), (35) and (36) and Lemma 4.2 yield the result for k > K. +21 + +Proposition 4.8 (Spectral gap). Assume that (Hp) holds. Then, λ∞ +1 > λ∞ +2 . +Proof. Assume, for the sake of contradiction, that λ∞ +1 = λ∞ +2 . Then, it follows from Lemma 4.1 that u2 +converges to u∞ +1 +(see (29)) in C1([0, 1]). Recall that u∞ +1 (x) > 0 for all x ∈ (0, 1]. On the other hand, we +know that u2 has a single 0 located in (0, 1] (it can not be located in [1, L) since u2(x) is a multiple of +sinh(√2λ2(L−x)) in this interval). We denote by xL the position of this 0 and remark that u2(2xL) < 0. +Hence, xL → 0 as L → ∞ and u2(2xL)/(2xL) < 0 converges to (u∞ +1 )′(0) > 0 (e.g. using the mean value +theorem). This leads to a contradiction. +4.2 +Heat kernel estimate +Proposition 4.9. Assume that (Hp) holds. There exists a constant c4.9 > 0 such that for L large enough +and t > c4.9L, we have +����pt(x, y) − +1 +∥v1∥2 e(λ1−λ∞ +1 )teµ(x−y)v1(x)v1(y) +���� ⩽ e−βLe(λ1−λ∞ +1 )teµ(x−y) v1(x)v1(y) +∥v1∥2 +, +x, y ∈ [0, L]. +As a consequence, for L large enough and t > c4.9L, +|qt(x, y) − Π(y)| ⩽ e−βLΠ(y), +x, y ∈ [0, L]. +Proof. By Proposition 4.7, for L large enough and t > 2, +K +� +k=2 +eλkt +1 +∥uk∥2 +uk(x)uk(y) +v1(x)v1(y) ⩽ K(c4.7)2e2βL+λ∞ +2 t, +and +∞ +� +k=K+1 +eλkt +1 +∥uk∥2 +uk(x)uk(y) +v1(x)v1(y) ⩽ (c4.7)2e2βL +∞ +� +k=K+1 +(1 + 2|λk|)4eλk(t−2). +(37) +In order to evaluate the former sum, we rely on the comparaison principle (21). +First, assume that +∥f∥∞ ̸= (k − 1 +2)2π2, ∀k ∈ N. Define +Ai = 1 +2 +�� +¯K + 1 +2 + i +�2 +− ∥f∥∞ +� +, +Ni = +�(L − 1) +π +√ +2Ai + 1 +2 +� ++ i, +for all i ≥ 0 and A−1 = N−1 = 0. One can show by an explicit calculation that for all j ∈ N such that +Ni−1 < j ⩽ Ni, we have +λ ¯ +K+j ∈ (−Ai, −Ai−1). +(38) +See [Tou21] for further details. If ∥f∥∞ = (k − 1 +2)2π2 for some k ∈ N, one can replace ∥f∥∞ by ∥f∥∞ + ε +for some small ε > 0 and get similar estimates. We now use (21), (22) and (38) to bound the sum on the +RHS of (37). We get that for L large enough, +∞ +� +k=K+1 +(1 + 2|λk|)4eλk(t−2) ⩽ +¯ +K +� +k=K+1 +� +1 + k2π2 +L2 +�4 ++ +∞ +� +j=1 +� +1 + ( ¯K + j)2π2 +L2 +�4 +e +¯λ ¯ +K+j(t−2) +⩽ ¯K +� +1 + +¯K2π2 +L2 +�4 ++ +∞ +� +i=0 +Ni +� +j=Ni−1+1 +� +1 + ( ¯K + j)2π2 +L2 +�4 +e +¯λ ¯ +K+j(t−2) +⩽ 2 ¯K + +∞ +� +i=0 +(Ni − Ni−1) +� +1 + ( ¯K + Ni)2π2 +L2 +�4 +e−Ai−1(t−2). +One can easily show that, for i ∈ N, i2 +2 ≲ Ai, Ni ≲ iL and that Ni − Ni−1 ≲ iL (see [Tou21] for further +details). Therefore, there exists c0, c1 > 0 such that +∞ +� +k=K+1 +(1 + 2|λk|)4eλk(t−2) ⩽ c0 +� +1 + L +∞ +� +i=0 +i(1 + i2)4e−c1(i−1)2(t−2) +� +. +22 + +Note that �∞ +i=0 i(1 + i2)4e−c1(i−1)2(t−2) < �∞ +i=0 i(1 + i2)4e−c1(i−1)2 < ∞ for t large enough. Thus, there +exists c2 > 0 such that for all L large enough, +∞ +� +k=K+1 +(1 + 2|λk|)4eλk(t−2) ⩽ c2L, +say for all t > 3. Putting all of this together, we see that +∞ +� +k=2 +eλkt |vk(x)vk(y)| +∥vk∥2 +⩽ (c4.7)2v1(x)v1(y)e2βL � +Keλ∞ +2 t + c2L +� +. +Recalling that λ∞ +1 − λ∞ +2 > 0 under (Hp) (see Proposition 4.8), we obtain +����pt(x, y) − +1 +∥v1∥2 e(λ1−λ∞ +1 )teµ(x−y)v1(x)v1(y) +���� ⩽ (c4.7)2(K+c2L)e2βL−(λ∞ +1 −λ∞ +2 )te(λ1−λ∞ +1 )teµ(x−y)v1(x)v1(y), +for all t > 3 and L large enough. This concludes the proof of Proposition 4.9. +4.3 +Green function +The Green function can be expressed thanks to the fundamental solutions of the ODE +1 +2u′′ + 1 +2f(x)u = λu. +(39) +Let ϕλ (resp. ψλ) be a solution of (39) such that ϕλ(0) = 0 (resp. ψλ(L) = 0). Define the Wronskian as +ωλ = ψλ(1)ϕ′ +λ(1) − ψ′ +λ(1)ϕλ(1). +Note that ϕλ and ψλ are unique up to constant multiplies. Without loss of generality, we can set ψλ and +ϕλ so that +ψλ(x) = sinh( +√ +2λ(L − x) +sinh(√2λ1(L − 1), +ϕ′ +λ(0) = v′ +1(0). +(40) +Proposition 4.10. Let ξ > 0 and λ(ξ) ≡ λ := λ∞ +1 + ξ. Define +Gξ(x, y) = +� ∞ +0 +e−ξtpt(x, y)dt. +Then +Gξ(x, y) = +� +(ωλ)−1eµ(x−y)ψλ(x)ϕλ(y) +0 ≤ y ≤ x ≤ L, +(ωλ)−1eµ(x−y)ϕλ(x)ψλ(y) +0 ≤ x ≤ y ≤ L. +(41) +Proof. Recall the definition of g from (2) and define +Hλ(x, y) := +� ∞ +0 +e−λtgt(x, y) dt. +By [BS12, p.19], Hλ is given by +Hλ(x, y) = +� +(ωλ)−1ψλ(x)ϕλ(y) +0 ≤ y ≤ x ≤ L, +(ωλ)−1ϕλ(x)ψλ(y) +0 ≤ x ≤ y ≤ L. +On the other hand, we see from (2) that +Hλ(x, y) = +� ∞ +0 +e−λtgt(x, y) dt = eµ(y−x) +� ∞ +0 +e(λ∞ +1 −λ)tpt(x, y)dt. +This yields the result. +23 + +Lemma 4.11. Assume that (Hfp) holds and that +1 +N ≲ ξ ≪ 1 +L. Then, +ϕλ(x) = v1(x) + O(ξ)(1 ∧ x) +� +eβx + Le−βx� +, +(42) +and +ψλ(x) = v1(x) + O(ξL)(1 ∧ (L − x))e−βx. +(43) +In addition, we also have +ϕλ(x) = (1 ∧ x) +� +O(1)e−βx + O(ξ)eβx� +, +(44) +ψλ(x) = O(1)(1 ∧ (L − x))e−βx. +(45) +Proof. First, note that e−2βL ≪ N −1 ≲ ξ under (Hfp) so that λ > λ1 for sufficiently large L. +We have +ϕλ(x) − v1(x) = +� x +z=0 +� z +y=0 +� +ϕ′′ +λ(y) − v′′ +1 (y) +� +dy dz, +x ∈ [0, 1]. +(46) +Yet, we know that +ϕ′′ +λ(y) − v′′ +1 (y) = (2λ − f(y))(ϕλ(y) − v1(y)) + 2(λ − λ1)v1(y). +(47) +Besides, we see from Lemma 4.2 that 2(λ − λ1)v1 = O(ξ). Therefore, we obtain that for all x ∈ [0, 1] +|ϕλ(x) − v1(x)| ⩽ +� 1 +z=0 +� x +y=0 +|2λ − f(y)||ϕλ(y) − v1(y)|dy dz + Cξ +⩽ +� x +y=0 +(2λ∞ +1 + ∥f∥∞)|ϕλ(y) − v1(y)|dy + Cξ. +Grönwall’s lemma then yields +ϕλ(x) − v1(x) = O(ξ), +x ∈ [0, 1]. +Putting this together with (46) and (47) we get that +ϕλ(x) − v1(x) = O(ξ)x = O(ξ)(1 ∧ x)eβx, +x ∈ [0, 1]. +(48) +Using a similar argument, one can easily show that +ϕ′ +λ(1) − v′ +1(1) = O(ξ). +(49) +Then, we see from (39) that, on [1, L], ϕλ can be written as +ϕλ(x) = ϕλ(1)sinh( +√ +2λ(L − x)) +sinh( +√ +2λ(L − 1)) ++ Ae +√ +2λ(x−1) + Be− +√ +2λ(x−1), +for some A, B ∈ R. Applying this equality to x = 1, we get that A = −B. In addition, we know that +ϕλ(1) = v1(1) + O(ξ) = 1 + O(ξ) and one can show (by an explicit computation) that +sinh( +√ +2λ(L − x)) +sinh( +√ +2λ(L − 1)) += (1 + O(ξL))v1(x), +x ∈ [1, L]. +Besides, one can easily see that +ϕ′ +λ(1) = +√ +2λ +� +2A − ϕλ(1)cosh( +√ +2λ(L − 1)) +sinh( +√ +2λ(L − 1)) +� +. +Then, recalling from (49) that ϕ′ +λ(1) = v′ +1(1) + O(ξ) with v′ +1(1) = −√2λ1/ tanh(√2λ1(L − 1)) and +remarking that +√ +2λ = +� +2λ∞ +1 (1 + O(ξ)), √2λ1 = +� +2λ∞ +1 (1 + O(e−2βL)), tanh(√2λ1(L − 1))−1 = 1 + +O(e−2βL), tanh( +√ +2λ(L − 1))−1 = 1 + O(e−2βL) and that e−2βL = O(ξ) under (Hfp), we get that +A = O(ξ). +24 + +Therefore, +ϕλ(x) = (1+O(ξL))v1(x)+O(ξ) sinh( +√ +2λ(x−1)) = v1(x)+O(ξL)(1∧x)e−βx+O(ξ)(1∧x)eβx, +x ∈ [1, L]. +Putting this together with (48) yields (42). Equation (44) can then be deduced from the first part of the +above: one can easily show that sinh( +√ +2λx) = (1 + O(ξL))eβx for all x ∈ [1, L]. We then use (48) to +conclude. +We now move to the estimate on ψλ. We recall from (40) that +ψλ(x) = sinh( +√ +2λ(L − x)) +sinh(√2λ1(L − 1)), +x ∈ [1, L]. +Hence, the mean value theorem yields +|ψλ(x) − v1(x)| +v1(x) +⩽ cosh( +√ +2λ(L − x)) +sinh(√2λ1(L − x))( +√ +2λ − +√ +2λ1)(L − x), +x ∈ [1, L). +Note that the RHS is O(ξL). Thus, +ψλ(x) = (1 + O(ξL))v1(x), +x ∈ [1, L]. +(50) +Using a similar argument, we get that ψ′ +λ(1) = v′ +1(1) + O(ξL). On [0,1] we use that ψλ also satisfies (47). +Hence, we get that for all x ∈ [0, 1] +ψλ(x) − v1(x) = ψλ(1) − v1(1) + +� x +z=1 +ψ′ +λ(z) − v′ +1(z)dz += ψλ(1) − v1(1) +� +�� +� +=O(ξL) ++ +� x +z=1 +� +� +�ψ′ +λ(1) − v′ +1(1) +� +�� +� +=O(ξL) ++ +� z +y=1 +(ψ′′ +λ(y) − v′′ +1 (y)) dy +� +� +� dz += O(ξL) + +� x +z=1 +� z +y=1 +(2λ − f(y))(ψλ(y) − v1(y))dy dz. +Applying Grönwall’s inequality in the same way as for ϕλ we get that +ψλ(x) = v1(x) + O(ξL), +x ∈ [0, 1]. +This concludes the proof of the lemma. +Lemma 4.12. Assume that (Hfp) holds and that +1 +N ≲ ξ ≪ 1 +L. Then, +ωλ = ξ +�� L +y=0 +v1(y)2dy + O(ξL) +� +. +Proof. Recall that qt(x, ·) is a probability density function and that h(t, y) = e(λ∞ +1 −λ1)th(0, y). Thus, +� ∞ +0 +e−ξtqt(x, y)dt = h(0, y) +h(0, x)Gξ−(λ∞ +1 −λ1)(x, y), +(51) +and +1 +ξ = +� ∞ +0 +e−ξt +=1 +� +�� +� +� L +0 +qt(x, y)dy dt = +1 +h(0, x) +� L +0 +h(0, y)G˜ξ(x, y)dy, +with ˜ξ = ξ − (λ∞ +1 − λ1). Note that under (Hfp), λ∞ +1 − λ1 ≪ ξ. +Set λ = λ∞ +1 + ˜ξ. We see from (41) that +� L +y=0 +h(0, y)G˜ξ(x, y)dy = eµxω−1 +λ +� +ψλ(x) +� x +y=0 +ϕλ(y)v1(y)dy + ϕλ(x) +� L +y=x +ψλ(y)v1(y)dy +� +. +25 + +Using Lemma 4.2 and Lemma 4.11, and remarking that for 0 < x1 < x2 < L, we have +� x2 +x1 +v1(y)2dy = O(1)(1 ∧ (x2 − x1)), +we get that +ψλ(x) +� x +y=0 +ϕλ(y)v1(y)dy = ψλ(x) +� x +y=0 +� +v1(y)2 + O(˜ξ)(1 + Le−2βy) +� +dy += ψλ(x) +�� x +y=0 +v1(y)2dy + O(˜ξL)(1 ∧ y) +� += v1(x) +� x +y=0 +v1(y)2dy + O(˜ξL)(1 ∧ x ∧ (L − y))e−βx += v1(x) +�� x +y=0 +v1(y)2dy + O(˜ξL) +� +, +and +ϕλ(x) +� L +y=x +ψλ(y)v1(y)dy = ϕλ(x) +� L +y=x +� +v1(y)2 + O(˜ξL)e−2βy� +dy += ϕλ(x) +�� L +y=x +v1(y)2dy + O(˜ξL)(1 ∧ (L − x)e−2βx) +� += v1(x) +� L +y=x +v1(y)2dy + O(˜ξL)(x ∧ 1 ∧ (L − x))e−βx += v1(x) +�� L +y=x +v1(y)2dy + O(˜ξL) +� +. +Putting all of this together, we get that +ωλ = ξ +�� L +0 +v1(y)2dy + O(˜ξL) +� +. +We conclude the proof by remarking that under (Hfp), ˜ξ = O(ξ). +Remark 4.13. For all t > 0, +� t +0 +ps(x, y)dy ⩽ eG 1 +t (x, y). +This follows from the fact that 1s∈[0,t] ⩽ e +t−s +t +for all 0 < s < t. +4.4 +The number of particles escaping the bulk +In this section, γ denotes a real number in (0, 1]. We are interested in the number of particles reaching +the level γL during the time interval [0, tN]. The following lemma allows to prove that, for a suitable +choice of γ, this number is exponentially small (in L). +For t > 0, we denote by Rγ([0, tN]) the number of particles that reach (for the first time) the level +γL between times 0 and tN. We also define pγ +t (x, y), vγ +1 , λγ +1 and Gγ +· (x, y) (as well as ϕγ +λ, ψγ +λ and ωγ +λ) in +the same way as pt(x, y), v1, λ1 and G·(x, y) but with γL instead of L. +Lemma 4.14. Assume that (Hfp) holds. Let T > 0 and γ ∈ (0, 1]. For N large enough, for all x ∈ [0, γL] +and t ∈ [0, T], +Ex [Rγ([0, tN])] = O(1)(1 ∧ x) +� +e(µ−β)xe[(µ−β)−γ(µ+β)]L + e(µ+β)(x−γL)� +where the O(·) may depend on T but not on x nor on t. +26 + +Proof. It is known (see e.g. [MS20, Lemma 5.7]) that +Ex [Rγ([0, tN])] = −1 +2 +� tN +s=0 +∂ +∂y pγ +s(x, y)|y=γLds. +In words, this means [MS20] that the expected number of particles “killed” at γL between times 0 and +tN is equal to the integral of the heat flow out of the boundary γL. Given the boundary condition in +(A), the flow out of γL is exactly − 1 +2 +∂ +∂y pγ +s(x, y)|y=γL . +Hence, Ex [Rγ([0, tN])] is an increasing function of t (as expected) and an argument similar to Remark +4.13 yields that +Ex [Rγ([0, tN])] ⩽ Ex [Rγ([0, TN])] ⩽ −1 +2e ∂ +∂y Gγ +(T N)−1(x, y)|y=γL. +Let ξ = +1 +T N and λ = λ∞ +1 + ξ. We know from (41) that +∂ +∂y Gγ +(T N)−1(x, y)|y=γL = (ωγ +λ)−1eµ(x−γL)ϕγ +λ(x)(ψγ +λ)′(γL). +We then see from (40) and Lemma 4.2 that +(ψγ +λ)′(γL) = +√ +2λ sinh +�� +2λγ +1(γL − 1) +�−1 += O(e−βγL). +By Lemmas 4.11 and 4.12, we have +(ωγ +λ)−1ϕγ +λ(x) = (ωγ +λ)−1(1 ∧ x) +� +O(1)e−βx + O(ξ)eβx� += (1 ∧ x) +� +O(N)e−βx + O(1)eβx� +Putting all these estimates together, we see that +−1 +2 +∂ +∂y Gγ +(T N)−1(x, y)|y=γL = O(1)(1 ∧ x)e−(µ+β)γL � +e(µ−β)Le(µ−β)x + e(µ+β)x� +, +which concludes the proof of the lemma. +Corollary 4.15. Assume that (Hfp) holds. Let T > 0 and γ ∈ +� +0, µ−β +2β +� +. For N large enough, for all +x ∈ [0, γL] and t ∈ [0, T], +Ex [Rγ([0, tN])] = O(1)(1 ∧ x)e(µ−β)xe[(µ−β)−γ(µ+β)]L. +Proof. Note that for γ ≤ µ−β +2β +and x ≤ γL, we have +e(µ+β)x = e(µ−β)xe2βx = O(1)e2βγLe(µ−β)x = O(1)e(µ−β)Le(µ−β)x. +Corollary 4.16. Let 0 < δ < +� +µ−β +2β +� +∧ +� +1 +2β +(µ−β)2 +µ+β +� +and γ = µ−β +2β − δ. Then, +Px(Rγ([0, tN]) > 0) = O +� +N +− +� µ−β +2β −δ µ+β +µ−β +�� +(1 ∧ x)e(µ−β)x, +∀x ∈ [0, γL], t ∈ [0, T]. +Proof. This follows from Markov’s inequality and the fact that (µ−β)−γ(µ+β) = (µ−β) +� +− µ−β +2β + δ µ+β +µ−β +� +. +Corollary 4.17. Assume that (Hfp) holds and let x > 0. Then, +1 +h(0, x)NPx(R1([0, tN]) > 0) → 0, +N → ∞. +Proof. According to Lemma 4.14, Lemma 4.2 and Markov’s inequality, it is enough to prove that +Ne−2βL → 0, +and, +Ne2βxe−(µ+β)L → 0, +as N → ∞. The first assertion is a direct consequence of (Hfp). The second convergence is clear when x +is fixed. +27 + +5 +Convergence of moments +5.1 +K-Mixing +Proposition 5.1. Let (ψi; i ∈ [K]) be bounded measurable functions and (Xi) be a sequence i.i.d. random +variables with density Π∞. Under ¯QK,t +x +, define +¯RN := +� +v∈B +r +�¯ζv +� +h +� +|v|N, ¯ζv +� � +v∈L +ψi(¯ζv). +Then as N → ∞, +¯RN ⇒ +�K−1 +� +i=1 +r (Xi) h∞ (Xi) +� � K +� +i=1 +ψi(Xi+K−1) +� +(52) +is distribution. +Proof. Let ϕ : [0, ∞) → R be a continuous and bounded function. We first condition ¯RN on the tree +structure of the K-spine (see (14)). Combining Proposition 4.9 and Corollary 4.3, we see that +lim +N→+∞ +¯QK,t +x +� +ϕ( ¯RN)|U1, ...UK−1 +� += +� +ϕ +�K−1 +� +i=1 +r (xi) h∞ (xi) +K +� +i=1 +ψi(xi+K−1) +� �2K−1 +� +i=1 +Π∞(dxi) +� +, +where we use that, conditionally on the (Ui), for N large enough, +N|Ui − Uj| > c4.9L +and +N(Ui ∧ (1 − Ui)) > c4.9L +a.s. +for all 1 ≤ i ≤ j ≤ K − 1. In words, the accelerated spine reaches its stable distribution between the +branching points. The dominated convergence theorem then yields +lim +N→+∞ +¯QK,t +x +� +ϕ +� +¯RN�� += E +� +lim +N→+∞ +¯QK,t +x +� +ϕ +� +¯RN� ����U1, ..., UK−1 +�� +, +which concludes the proof of the proposition. +5.2 +Uniform integrability +Lemma 5.2. Let T > 0, n ∈ N and let +0 ≤ ε < +3β − µ +n(µ − β). +(53) +There exists a constant c5.2 = c5.2(T, ε, n) such that for sufficiently large N, +In−1,ε,t(x) := +� t +s=0 +� L +y=0 +h((t − s)N, y)1+εeε(n−1)(µ−β)y ¯qs(x, y) dy ds ⩽ c5.2eεn(µ−β)x, +∀x ∈ [0, L], ∀t ∈ [0, T]. +Proof. Recall from Lemma 4.2 that h((t − s)N, y) ≤ 2h(0, y) and h(sN, y) ≤ 2h(0, y) for N large enough +(that only depends on T). Using Fubini’s theorem along with Remark 4.13, we see that +In,ε,T (x) = +� t +s=0 +� L +y=0 +h((t − s)N, y)1+εeε(n−1)(µ−β)y +�h(sN, y) +h(0, x) psN(x, y) +� +dy ds +⩽ 22+ε +� T +s=0 +� L +y=0 +h(0, y)1+εeε(n−1)(µ−β)y +� h(0, y) +h(0, x)psN(x, y) +� +dy ds +⩽ 21+εe 1 +N +� L +y=0 +h(0, y)1+εeε(n−1)(µ−β)y +� h(0, y) +h(0, x)G(T N)−1(x, y) +� +dy +� +�� +� +=: Jn,ε,T (x) +. +28 + +Let ξ = +1 +T N and λ = λ∞ +1 + ξ. By definition of Gξ (see Section 4.3), +v1(x)Jn,ε,T (x) = (Nωλ)−1 +� +ψλ(x) +� x +y=0 +h(0, y)1+εeε(n−1)(µ−β)yϕλ(y)v1(y)dy ++ ϕλ(x) +� L +y=x +h(0, y)1+εeε(n−1)(µ−β)yψλ(y)v1(y)dy +� +=: (Nωλ)−1(ψλ(x)A(x) + ϕλ(x)B(x)) +Yet, we see from Equation (Hfp), Equation (53), Lemma 4.2 and Lemma 4.11 that +A(x) = +� x +y=0 +e(µ−2β)y � +O(1)e−βy + O(ξ)eβy� +eεn(µ−β)ydy += O(1) +� x +y=0 +e(µ−3β)yeεn(µ−β)ydy + O(ξ) +� x +y=0 +e(µ−β)yeεn(µ−β)ydy += O(1)(1 ∧ x) + O(ξ)(1 ∧ x)e(µ−β)xeεn(µ−β)x, +and remarking that ξe(µ−β)x = O(1) for x ∈ [0, L], we get that +ψλ(x)A(x) = +� +v1(x) + O(ξL)(1 ∧ (L − x))e−βx� +A(x) = O(1)v1(x)eεn(µ−β)x. +Similarly, +B(x) = O(1) +� L +y=x +e(µ−3β)yeεn(µ−β)ydy = O(1)(1 ∧ (L − x))e(µ−3β)xeεn(µ−β)x, +so that (using that ξe(µ−β)x = O(1) again) +ϕλ(x)B(x) = (1 ∧ x)(O(ξ)eβx + O(1)e−βx)(1 ∧ (L − x))e(µ−3β)xeεn(µ−β)x += O(1)v1(x)eεn(µ−β)x. +Applying Lemma 4.12 to ξ = (TN)−1, we get that ω−1 +λ += O(N) so that +Jn,ε,T (x) = O(1)eεn(µ−β)x, +which concludes the proof of the lemma. +Lemma 5.3. Assume that (Hfp) holds. Let K ≥ 2 and T > 0. For +0 ≤ ε < +1 +K − 1 +3β − µ +µ − β , +(54) +there exists a constant c5.3 = c5.3(ε, T, K) > 0 such that +¯QK,t +x +�� +v∈B +h(|v|N, ¯ζv)1+ε +� +⩽ c5.3 +1 +tK−1 eε(K−1)(µ−β)x, +∀x ∈ [0, L], ∀t ∈ (0, T]. +Proof. We prove by induction that for every n ≤ K there exists a constant cn = cn(ε, T) such that +¯Qn,t +x +�� +v∈B +h(|v|N, ¯ζv)1+ε +� +⩽ cn +1 +tn−1 eε(n−1)(µ−β)x, +∀x ∈ [0, L], +t ∈ (0, T]. +(55) +We start with n = 2. The 2-spine has a single branching point v. Hence, +¯Q2,t +x +� +h(|v|N, ¯ζv)1+ε� += 1 +t +� t +s=0 +� L +y=0 +h(sN, y)1+ε¯qt−s(x, y) dy ds. +29 + +Note that ε < (3β − µ)/(µ − β). Thus we get from Lemma 5.2 that for any t ≤ T +¯Q2,t +x +� +h(|v|N, ¯ζv)1+ε� += 1 +t I1,ε,t(x) ⩽ 1 +t c5.2(T, ε, 1)eε(µ−β)x. +Assume that the property (55) holds at rank n ≤ K − 1. We know from (16) that +¯Qn+1,t +x +�� +v∈B +h(|v|N, ¯ζv)1+ε +� +(56) += +� t +0 +� L +0 +r(y)¯qt−s(x, y)h(sN, y)1+ε nsn−1 +tn +n +� +m=1 +1 +n +¯Qm,s +y +�� +v∈B +h(|v|N, ¯ζv)1+ε +� +¯Qn+1−m,s +y +�� +v∈B +h(|v|N, ¯ζv)1+ε +� +. +It then follows by induction (see (55)) that +¯Qn+1,t +x +�� +v∈B +h(|v|N, ¯ζv)1+ε +� +⩽ n +� +max +i≤n ci +2 +� +1 +tn In−1,ε,t. +Finally, we see from Lemma 5.2 that +¯Qn+1,t +x +�� +v∈B +h(|v|N, ¯ζv)1+ε +� +⩽ n +� +max +i≤n ci +2 +� +c5.2(n, ε, T) 1 +tn eεn(µ−β)x. +This concludes the proof of the lemma. +Corollary 5.4. The sequence of r.v.’s +�� +v∈B +r(¯ζv)h(|v|N, ¯ζv) +� +i,j +ψi,j(Uσi,σj) +� +i +˜ϕi(¯ζVσi ) h∞(¯ζVσi ) +h(tN, ¯ζVσi ), N ∈ N +� +from Theorem 4 is uniformly integrable under ¯QK,t +x +. +Proof of Theorem 4. Recall that the functions ˜ϕi are compactly supported. Hence, there exists a constant +A > 0 such that ˜ϕi ≡ 0 on (A, ∞), ∀i ∈ [K]. Yet, we know from the proof of Lemma 4.2 that v1 converges +uniformly to v∞ +1 +on [0, A]. Putting this together with (26), we get that +���� +� +v∈B +r(¯ζv)h(|v|N, ¯ζv) +� +i,j +ψi,j(Uσi,σj) +� +i +˜ϕi(¯ζVσi ) h∞(¯ζVσi ) +h(tN, ¯ζVσi ) +���� ≤ C +� +v∈B +h(|v|N, ¯ζv). +The result then follows from Lemma 5.3. +6 +Survival probability (0th moment) +In this section, we assume that (Hfp) holds. Let L be as in (13) and ˜c as is (7). Define +u(t, x) := Px(ZL +t > 0), +and +a(t) := ˜c +� L +0 +e−µyv1(y)u(t, y)dy. +This section is aimed at proving Theorem 1. Essentially, we will show that for N large +u(tN, x) ≈ h(0, x)a(tN), +so that the problem boils down to estimating a(tN). Fix 0 < η < T. The idea is to prove that for N +large enough, a satisfies +˙a(tN) ≈ −Σ2 +2 a(tN)2, +∀t ∈ [η, T]. +(57) +30 + +As a consequence, +a(tN) ≈ +1 +Σ2 +2 (t − η)N + +1 +a(ηN) +. +Finally, the result will follow provided that +lim inf +η→0 lim inf +N→∞ +ηNa(ηN) > 0. +(58) +6.1 +Step 1: Rough bounds +In the remainder of this section, we will often make use of a union bound that we describe now. +We first remark from the branching property that for all t > 0, +u(tN, x) = Px +� +∪v∈N L +L2 {Z(v) +tN−L2 > 0} +� +, +(59) +where Z(v) +s +refers to the number of descendants of the particles v, whose ancestors stayed in (0, L) until +time |v| + s > 0. Using a union bound and the the many-to-one lemma (see Lemma 3.1), we see that +u(tN, x) ⩽ +� L +0 +pL2(x, y)u(tN − L2, y)dy. +Proposition 4.9 along with (26) then yields +u(tN, x) ≤ (1 + O(e−bL))h(0, x)a(tN − L2), +(60) +for some constant b = b(t) (note that this bound can be uniform in t if we consider t in a compact subset +of [0, ∞)). +Lemma 6.1 (Rough upper bounds). Let T > 0. There exist two positive constants γ and c6.1 = c6.1(T) +such that for all η > 0, there exists ˜ +N = ˜ +N(T, η) such that, for all N ≥ ˜ +N, we have +a(tN) ≤ c6.1 +N γ , +∀t ∈ [η, T], +and +u(tN, x) ⩽ c6.1 h(0, x) +N γ +, +x ∈ [0, L], +∀t ∈ [η, T]. +Proof. In this proof the quantities O(·) can depend on T but not on η and we will make use of the +notation defined in the beginning of Section 4.4. +Basically, we prove that, in order to survive a period of time of order N, the BBM has to reach the +level +µ−β +2β L < L. We established in Corollary 4.16 that the probability to reach this level is of order +(N −˜γ) for some ˜γ > 0. +Let 0 < δ < +� +µ−β +2β +� +∧ +� +1 +4β +(µ−β)2 +µ+β +� +and ˜γ = µ−β +2β − δ. For all x ∈ [0, ˜γL], we have +Px(ZL +tN > 0) = Px(Z ˜γL +tN > 0, R˜γ([0, tN]) = 0) + Px(ZL +tN > 0, R˜γ([0, tN]) > 0) +⩽ Px(Z ˜γL +tN > 0) + Px(R˜γ([0, tN]) > 0). +The first probability can be bounded by Proposition 4.9. Indeed, for all x ∈ [0, ˜γL], we have +Px(Z ˜γL +tN > 0) ⩽ Ex[Z ˜γL +tN ] = +� ˜γL +0 +p˜γ +tN(x, y)dy = (1 + O(e−βL))e(λ˜γ +1 −λ∞ +1 )tNeµx v˜γ +1 (x) +∥v˜γ +1 ∥2 +� ˜γL +0 +e−µyv˜γ +1 (y)dy. +Lemma 4.2 yields that +� ˜γL +0 +e−µyv˜γ +1 (y)dy < ∞, ∥v˜γ +1 ∥ → ∥v∞ +1 ∥ as N → ∞ and that +v˜γ +1 (x) ⩽ C(x ∧ 1 ∧ (˜γL − x))e−βx ⩽ Cv1(x), +∀x ∈ [0, ˜γL], +(61) +31 + +for L large enough so that L − ˜γL > 1. In addition, it implies that there exists a positive constant +C = C(λ∞ +1 ) > 0 such that +e(λ˜γ +1 −λ∞ +1 )tN ≤ exp(−Cte−2β˜γLe(µ−β)L) = exp(−Cte2βδL). +Hence, +Px(Z ˜γL +tN > 0) = O(exp(−Cηe2βδL))h(0, x), +x ∈ [0, ˜γL]. +(62) +It then follows from Corollary 4.16 and (61) that for all x ∈ (0, ˜γL) +Px(R˜γ([0, tN]) > 0) = O +� +N +− +� µ−β +2β −δ µ+β +µ−β +�� +h(0, x). +Note that the exponential factor in (62) is smaller than any power of N (for N large enough) so that +Px(ZL +tN > 0) = O +� +N +− +� µ−β +2β −δ µ+β +µ−β +�� +h(0, x), +∀x ∈ [0, ˜γL]. +(63) +Remark that this prove the second part of the lemma for all x ≤ ˜γL < L. We now establish an upper +bound on a(t) to get a control on u(t, x) for larger values of x. +We split the integral a(tN) into two parts: we see from (63) and Lemma 4.2 that +a(tN) = +� ˜γL +0 +˜ce−µxv1(x)u(tN, x)dx + +� L +˜γL +˜ce−µxv1(x)u(tN, x)dx += O +� +N +− +� µ−β +2β −δ µ+β +µ−β +�� � ˜γL +0 +v1(y)2dy + O(1) +� L +˜γL +e−(µ+β)xdx, +where we bound u(tN, x) by 1 in the second integral. Moreover, note that (µ − β)(µ + β) = 1, so that +� L +˜γL +e−(µ+β)xdx = O +� +e−(µ+β)˜γL� += O +� +e +− +� +1 +2β −δ(µ+β) +� +L +� += O +� +N +− +� µ+β +2β −δ µ+β +µ−β +�� +. +Finally, since we chose δ < 1 +2 +(µ−β)2 +2β(µ+β) in the beginning of the proof, we get that for sufficiently large N, +a(tN) = O +� +N − µ−β +4β +� +, +∀t ∈ [η, T]. +Note that the constant depends on T but not on η. +The upper bound on u follows from (60) (and the remark following the equation). Indeed, for N large +enough, L2 < 1 +2ηN and we see from the first part of the result that for N large enough, +a(tN − L2) ⩽ c6.1(λ∞ +1 , T) +N +µ−β +4β +. +Lemma 6.2 (Rough lower bounds). Let 0 < T1 < T2. There exist two positive constants C1 = C1(λ∞ +1 ), +C2 = C2(λ∞ +1 ) and an integer ˜ +N = ˜ +N(T1, T2) such that for all N ≥ ˜ +N, we have +u(tN, x) ⩾ C1 h(x) +1 + tN , +∀x ∈ [0, L], +∀t ∈ [T1, T2], +and +a(tN) ≥ +C2 +1 + tN , +∀t ∈ [T1, T2]. +Proof. The idea of the proof is adapted from [HHKW21, Lemma 7.2]. +Let ˜Pt +x be the probability measure +absolutely continuous w.r.t. to Px and with Radon-Nykodim derivative +d˜Pt +x +dPx += +1 +h(0, x) +� +v∈N L +tN +h(t, xv). +32 + +This change of measure combined with Jensen’s inequality yields +Px(ZL +tN > 0) = Ex +� +1ZL +tN >0 +� += ˜PtN +x +� +h(0, x) +� +v∈N L +tN h(tN, xv) +� +≥ +h(0, x) +˜PtN +x +�� +v∈N L +tN h(tN, xv) +�. +Yet, +˜PtN +x +� +� � +v∈N L +tN +h(tN, xv) +� +� = +Ex +��� +v∈N L +tN h(tN, xv) +�2� +h(0, x) +. +(64) +Corollary 3.4 then yields +Ex +� +� +� +� � +v∈N L +tN +h(tN, xv) +� +� +2� +� = Ex +� +� � +v∈N L +tN +h(tN, xv)2 +� +� + 2h(0, x)tNQ2,tN +x +[r(ζv)h(|v|N, ζv)], +where v refers to the unique branching point in the 2-spine of depth tN. The first term on the RHS of +the above can be calculated using the many-to-one lemma: we see again from Corollary 3.4 that +Ex +� +� � +v∈N L +tN +h(tN, xv)2 +� +� = h(0, x)Q1,tN +x +[h(tN, ζv)] +where v is the unique leaf of the 1-spine at time tN. It also follows from Proposition 5.1 that +lim +N→+∞ Q1,tN +x +[h(|v|N, ζv)] += +E[h∞(Y )], +lim +N→+∞ Q2,tN +x +[r(ζv)h(|v|N, ζv)] += +E[r(Y )h∞(Y )], +where Y is a r.v. distributed according to Π∞ and we used Lemma 5.2 with ε = 0 to get the desired +uniform integrability. Putting all of this together, we see that there exists a constant C1 > 0 such that +lim +N→∞ Ex +� +� +� +� � +v∈N L +tN +h(tN, xv) +� +� +2� +� ≤ C1(1 + tN)h(0, x), +and that C1 does not depend on T1 nor on T2. This equation combined with (64) yields the first part of +the lemma. The second part of the result follows from an integration. +6.2 +Step 2. Comparing a(t) and u(t, x) +Lemma 6.3. For all t > 0, we have +˙a(t) = −˜c +� L +0 +e−µxv1(x)r(x)u(t, x)2dx + O +� +1 +N α−1 +� +a(t). +Proof. By definition of a(t), we see that +˙a(t) = ˜c +� L +0 +e−µxv1(x)∂tu(t, x)dx. +Yet, u is solution of the FKPP equation +∂tu(t, x) = 1 +2∂xxu(t, x) − µ∂xu(t, x) + r(x)(u(t, x) − u(t, x)2), +u(t, 0) = u(t, L) = 0. +(65) +On the other hand, note that x �→ e−µxv1(x) is solution to the ODE 1 +2y′′ + µy′ + r(x)y = (λ1 − λ∞ +1 )y. +An integration by parts then entails +˙a(t) = (λ1 − λ∞ +1 )a(t) − ˜c +� L +0 +r(x)e−µxv1(x)u(t, x)2dx. +Lemma 4.2 finally yields the result. +33 + +Corollary 6.4. Let 0 < T1 < T2. For N large enough, +|˙a(tN)| = O +� 1 +N γ +� +a(tN), +t ∈ [T1, T2]. +Proof. Recall from (60) that +u(tN, x) = O(1)h(0, x)a(tN − L2) +for N large enough (that only depends on T1). In addition, we see from Lemma 6.1 that for any k ∈ N +and N large enough (that only depends on T1, T2 and k), we have +a(tN − kL2) ⩽ c6.1(T2) +N γ +. +(66) +Thus, the union bound (60), Lemma 6.3 and Proposition 1.3 yield +|˙a(tN)| = O +� +1 +N α−1 +� +a(tN) + O +� 1 +N γ +� +a(tN − L2), +t ∈ [T1, T2]. +Without loss of generality, one can assume that γ < α−1. Hence, using that for N ∈ N fixed, the function +t �→ a(t) is decreasing, we see that +|˙a(tN)| = O +� 1 +N γ +� +a(tN − L2), +t ∈ [T1, T2]. +(67) +Using the mean value theorem, we then obtain +|a(tN) − a(tN − L2)| = O +� 1 +N γ +� +a(tN − 2L2), +t ∈ [T1, T2], +so that (67) implies that +|˙a(tN)| = O +� 1 +N γ +� +a(tN) + O +� +1 +N 2γ +� +a(tN − 2L2), +t ∈ [T1, T2]. +Let k ∈ N be such that kγ > 1. Iterating the above estimates, we see that +|˙a(tN)| = O +� 1 +N γ +� +a(tN) + O +� +1 +N kγ +� +a(tN − kL2), +t ∈ [T1, T2]. +Note that it suffices to choose N large enough such that kL2/N < T1/2. Then, recall from Lemma 6.2 +that +1 +N ≲ a(tN). This, combined with (66) yields the result. +Corollary 6.5. Let ε > 0 and 0 < T1 < T2. There exists ˜ +N = ˜ +N(ε, T1, T2) large enough such that for +every N ≥ ˜ +N, +(1 − ε)h(0, x)a(tN) ≤ u(tN, x) ≤ (1 + ε)h(0, x)a(tN). +Proof. It follows from (59) along with Bonferroni inequalities that +u(tN, x) ≥ Ex +� +�� +� +v∈N L +L2 +Pxv +� +Z(v) +tN−L2 > 0 +� +� +�� − 1 +2Ex +� +�� +� +v̸=w∈N L +L2 +Pxv +� +Z(v) +tN−L2 > 0 +� +Pxw +� +Z(w) +tN−L2 > 0 +� +� +�� +≥ Ex +� +�� +� +v∈N L +L2 +Pxv +� +Z(v) +tN > 0 +� +� +�� − 1 +2Ex +� +�� +� +v̸=w∈N L +L2 +Pxv +� +Z(v) +tN−L2 > 0 +� +Pxw +� +Z(w) +tN−L2 > 0 +� +� +�� . (68) +By Proposition 4.9 +Ex +� +�� +� +v∈N L +L2 +Pxv +� +Z(v) +tN > 0 +� +� +�� = +� L +0 +pL2(x, y)u(tN, y)dy = +� +1 + O +� +e−bL�� +h(0, x)a(tN), +34 + +where b is as in (60). Hence, for N large enough (that only depends on ε), we have +Ex +� +�� +� +v∈N L +L2 +Pxv +� +Z(v) +tN > 0 +� +� +�� ⩾ (1 − ε)h(0, x)a(tN). +(69) +Corollary 3.4 then yields +Ex +� +�� +� +v̸=w∈N L +L2 +Pxv +� +Z(v) +tN−L2 > 0 +� +Pxw +� +Z(w) +tN−L2 > 0 +� +� +�� = 2h(0, x)L2Q2,L2 +x +� +r(ζv)h(|v|, ζv) +� +i=1,2 +u(tN − L2, ζvi) +h(L2, ζvi) +� +, +(70) +where v is the unique branching point of the 2-spine tree of depth L2 and v1, v2 are the two leaves. Using +a similar argument to (60), we get that for N large enough, +u(tN − L2, y) ⩽2h(0, y)a(tN − 2L2). +On the other hand, h(L2, y) ≥ +1 +2h(0, y) for N large enough (see Lemma 4.2). These upper and lower +bounds, combined with (70) and Proposition 1.3, yield +Ex +� +�� +� +v̸=w∈N L +L2 +Pxv +� +N (v) +tN−L2 > 0 +� +Pxw +� +N (w) +tN−L2 > 0 +� +� +�� ⩽ 16CL2h(0, x)a(tN −2L2)2Q2,L2 +x +(r(ζv)h(|v|, ζv)), +for some C = C(λ∞ +1 ) and N large enough. The term Q2,L2 +x +(r(ζv)h(|v|, ζv)) can be shown to be uniformly +bounded in N and x for ε = 0 by the same technic as Lemma 5.3. +We know from Corollary 6.4 and the mean value theorem that +a(tN) = +� +1 + O +� L2 +N γ +�� +a(tN − 2L2), +so that, for N large enough, +(1 − ε)a(tN) ≤ a(tN − 2L2) ≤ (1 + ε)a(tN) +(71) +Putting this together with (68), (69) and Lemma 6.1, we get that for sufficiently large N, +u(tN, x) ≥ (1 − 2ε)a(tN)h(0, x). +(72) +The upper-bound of the lemma can be obtained in a similar manner, using only the many-to-one lemma +and (71). +6.3 +Step 3. Kolmogorov estimates +Lemma 6.6. Let ε > 0 and 0 < T1 < T2. For N large enough, we have +−Σ2 +2 (1 + ε)a(tN)2 ⩽ ˙a(tN) ⩽ −Σ2 +2 (1 − ε)a(tN)2, +T1 < t < T2. +Proof. We see from (Hfp) and Lemma 6.2 that there exists a small δ > 0 such that +1 +N α−1 = O +� 1 +N δ +� +a(tN) +(73) +The result follows from a direct application of Lemma 6.3 and Corollary 6.5. +Proof of Theorem 1. Integrating the inequality in Lemma 6.6 allows to approximate a(tN) by the solu- +tion of (57) on [ηN, TN]. On the other hand, Lemma 6.2 yields that (58) is satisfied. As a result, we +obtain the following: +for all ε > 0 and x > 0, there exists ˜ +N = ˜ +N(ε, t) such that for all N ≥ ˜ +N, +(1 − ε)2h(0, x) +Σ2t +≤ Px(ZL +tN > 0) ≤ (1 + ε)2h(0, x) +Σ2t +. +It remains to prove that for N large enough {ZL +tN > 0} = {ZtN > 0} with high probability. Note that +{ZL +tN > 0} = {ZtN > 0} on the event {R1([0, TN]) = 0}. Corollary 4.17 then yields the result. +35 + +7 +Convergence of metric spaces +7.1 +Some useful lemmas on marked metric spaces +Let M = [X, d, µ] ∈ M. For any measurable set X′ ⊂ X we will write |X′| = µ(X′ × E) We define the +Marked Gromov Prokhorov distance between two elements of M as +∀[Xi, di, νi] ∈ M, +dMGP ([X1, d1, ν1], [X2, d2, ν2]) = +inf +Z,ϕ1,ϕ2 dP r(ϕ1 ⋆ µ1, ϕ2 ⋆ µ2), +where the infimum is taken over all complete metric spaces (Z, dZ) and over all isometric embeddings ϕi +from Xi to Z, i = 1, 2. Finally, dP r is the standard Prokhorov distance between measures. It is now a +standard result that the Gromov weak topology is metrisable by the metric dMGP [GPW09, Glö13, Gro99]. +Let M>0 be the set of mmm with strictly positive measure. We introduce the polar Marked Gromov- +Prokhorov distance on M>0 as +∀Mi = [Xi, di, µi] ∈ M>0, +dpMGP (M1, M2) := dMGP ( ˜ +M1, ˜ +M2) + +��|X1| − |X2| +�� +where ˜ +Mi := [Xi, di, +µi +|Xi|] is obtained from Mi by renormalising the sampling measure µi to make it a +probability measure. In the case of measured metric space (with no mark), the polar Gromov Prokhorov +distance induces the Gromov weak topology [GPW09, Glö13, Gro99]. It can readily be checked that the +same argument applies in the case of mmm. For the sake of conciseness, the details are left to the reader +Lemma 7.1. Let (X, d, µ) in M>0 and let X′ be a closed subset of X such that µ(X′ × E) > 0. +dpGMP +� +[X′, d, µ], [X, d, µ] +� +≤ |X \ X′| + +� +1 − |X′| +|X| +� +. +Proof. The proof is identical to Lemma 3.4 [BFRS22] in the case of measured metric spaces (with no +mark). +Corollary 7.2. Let (Mn = [Xn, dn, µn]; n ≥ 0) be a sequence of random mmm and for every n, let X′ +n +be a closed measurable subset of Xn. Assume further +1. |X′ +n| converges in distribution to a strictly positive r.v. +2. E(|Xn| − |X′ +n|) → 0 as n → ∞. +Assume that [X′ +n, dn, µn] converges to [X∞, d∞, µ∞] in distribution for the Gromov weak topology. Then +[Xn, dn, µn] converges in distribution to the same limit. +Proof. The proof goes along the same as in Corollary 3.5 [BFRS22]. We repeat the argument for the sake +of completeness. Define M ′ +n = [X′ +n, dn, µn]. The sequence X′ +n converges to a positive r.v. so that we can +assume w.l.o.g that M ′ +n, Mn ∈ M>0 by conditioning on the event {|M ′ +n| > 0}. The result will follow by +showing that the polar Gromov–Prokhorov distance between the two spaces converges to 0 in probability. +By Lemma 7.1, we have that +P +� +dpMGP +� +[Xn, dn, µn], [X′ +n, dn, µn] +� +≥ 2ϵ +� +≤ P(|Xn| − |X′ +n| > ϵ) + P(|Xn| − |X′ +n| > ϵ|Xn|). +The first term goes to 0 by our assumptions. For the second term, note that for any ϵ′ > 0, +P(|Xn| − |X′ +n| > ϵ|Xn|) ≤ P(|Xn| − |X′ +n| > ϵϵ′) + P(|X′ +n| ≤ ϵ′). +The first term goes to 0 by our assumption while the second term can be made arbitrarily small by using +the convergence of |X′ +n| to a positive limit. Since the distance between the two spaces converges to 0 in +probability, they have to converge to the same limit in distribution. +In the following, we restrict ourself to the case where the mark space is an open set E ⊆ R+. We also +consider an increasing sequence of closed finite sets (Eε, ε > 0) such that � +ε>0 Eε = E. For every finite +mmm M = [X, d, µ], we define +Xε := +� +(x, m) : µ({(x, m)}) > 0 +and m ∈ Eε +� +, and M ε := [Xε, d, µ]. +36 + +Corollary 7.3. Consider a sequence of finite random mmm (Mn = [Xn, dn, µn]; n ≥ 0). Assume that +1. lim supε→0 lim supn→∞ E (|Xn| − |Xε +n|) = 0 +2. For every ε > 0, (|Xε +n|; n ≥ 0) converges in distribution to a strictly positive random variable. +Then +lim sup +ε→0 +lim sup +n→∞ +dMGP (M ε +n, Mn) = 0 in probability. +Proof. The argument goes along the same lines as the proof of Corollary 7.2. +7.2 +Proof of Theorem 2 and 3 +For any ε ∈ (0, 1), set Eε := [ε, 1 +ε] and define M L,ε +t +out of M L +t by killing all the particles which do not +belong to the interval Eε at time t. In other words, we only keep the particle in Eε at t with no ancestor +outside of [0, L] on the time interval [0, t]. Finally, ¯ +M L,ε +t +is defined analogously to ¯ +M L +t by rescaling the +total mass of the space and by accelerating time by N. +Proposition 7.4. Fix ε > 0 and define +˜h∞ +ε (x) := 1(x ∈ Eε)˜h∞(x). +Conditional on {ZtN > 0}, +� +¯ +M L,ε +t +, N ≥ 0 +� +converges in distribution for the Gromov weak topology to a +marked Brownian CPP with parameters (t, Σ2 +2 ˜h∞ +ε (x)dx). +Proof. We follow the heuristics of Section 2.5. Let ( ˜ϕi, i ∈ [K]) and (ψi,j, i, j ∈ [K]) be bounded and +measurable functions. Assume that the ˜ϕi’s are compactly supported on (0, ∞). For a mmm [X, d, µ], +define the polynomial +∀M = [X, d, µ], +˜Ψ(M) = +� � +i,j +1(vi ̸= vj)ψi,j(d(vi, vj)) +� +i +˜ϕi(xi)h∞(xi)µ(dvi ⊗ dxi). +From Theorem 4, our Kolmogorov estimate and the many-to-few formula (see Theorem 2.7), we have +lim +N→∞ Ex +� +˜Ψ( ¯ +M L +t ) +��ZtN > 0 +� += K! +�Σ2 +2 t +�K +E +�� +i,j +ψi,j(Uσi,σj) +� � +i +� +Π∞(x) ˜ϕi(x)dx. +Let ϕi be a bounded and measurable function and consider +˜ϕi(x) = ϕi(x)1(x ∈ Eε) +h∞(x) +. +Note that ˜ϕi is now bounded and compactly supported. Define +∀M = [X, d, µ], +Ψ′(M) := +� � +i,j +1(vi ̸= vj)ψi,j(d(vi, vj)) +� +i +ϕi(xi)µ(dvi ⊗ dxi). +The previous limit translates into +lim +N→∞ Ex +� +Ψ′( ¯ +M L,ε +t +) +��ZtN > 0 +� += K!E +�� +i,j +ψi,j(Uσi,σj) +� � +i +� tΣ2 +2 +˜h∞ +ε (x)ϕi(x)dx. +Let us now define +∀M = [X, d, µ], +Ψ(M) := +� � +i,j +ψi,j(d(vi, vj)) +� +i +ϕi(xi)µ(dvi ⊗ dxi). +From Proposition 2.4, it remains to show that +lim +N→∞ +����Ex +� +Ψ( ¯ +M L,ε +t +) − Ψ′( ¯ +M L,ε +t +) +��ZtN > 0 +� ���� = 0. +37 + +In turns, it is is sufficient to show by induction on K that +lim +N→∞ Ex +� � +1 (∪1≤i 0 +� += 0. +On the one hand, +Ex +� � +1 (∪1≤i 0 +� +≤ +� +1≤i 0 +� +. +On the other hand, the RHS vanishes since by induction and the first part of the proof +lim +N→∞ Ex +�K−1 +� +k=1 +¯µL,ε +t +(dvk ⊗ dxk) +�� ZtN > 0 +� += +lim +N→∞ Ex +�� +1 (∪1≤i 0 +� += +K! +�Σ2t +2 +� +Eε +˜h∞(x)dx +�K +. +This completes the proof of the proposition. +Proposition 7.5. Conditional on the event {ZtN > 0}, +¯ +M L +t +converges in distribution for the Gromov +weak topology to the marked Brownian CPP with parameters (t, Σ2 +2 ˜h∞(x)dx). +Proof. By observing the moments of Brownian CPPs in Proposition 2.4, it is clear that a marked Brownian +CPP with parameters (t, Σ2 +2 ˜h∞ +ε (x)dx) converges to a marked CPP with parameters (T, Σ2 +2 ˜h∞(x)dx) as +ε → 0. Next, a simple triangular inequality shows that our proposition now boils down to proving that +Conditional on ZtN > 0, +lim +ε→0 lim +N→∞ dMGP ( ¯ +M L,ε +t +, ¯ +M L +t ) = 0 in probability. +From Proposition 7.4, +1 +N ZL,ε +tN +conditional on {ZL +tN > 0} converges to an exponential random variable +with mean tΣ2 +2 +� +Eε ˜h∞(x)dx and by Corollary 7.3, it remains to show that +lim sup +ε→0 +lim sup +N→∞ +1 +N Ex +� +ZL +tN − ZL,ε +tN +�� ZtN > 0 +� += 0. +Since +1 +N Ex +� +ZL +tN − ZL,ε +tN +�� ZtN > 0 +� += +1 +NPx(ZtN > 0) +� +x/∈Eε +ptN(x, y)dy, +the proof is complete after a direct application of Proposition 4.9 and Theorem 1. +Proof of Theorem 3. By Proposition 7.5, it is enough to prove that conditioned on {ZtN > 0}, MtN and +M L +tN are coupled in such a way that they coincide with a probability going to 1 as N → ∞. In light +of Corollary 7.2 and of our Kolmogorov estimate, it is sufficient to show that NEx[R1([0, tN])] → 0 as +N → ∞. Corollary 4.17 then yields the result. +Proof of Theorem 2. This is a simple corollary of Theorem 3. The proof goes along the exact same lines +as Theorem 2 in [BFRS22] where the convergence of the population size and the genealogy of a finite +sample is deduced from the convergence of the discrete metric space to the CPP. We recall the main steps +of the argument for completeness. +Both maps +[X, d, ν] �→ |X|, +[X, d, ν] �→ +� +X, d, +ν +|X| +� +are continuous w.r.t. the Gromov-weak topology. Since the r.v. ¯ +Mt conditioned on survival converges in +the MGW topology, (i) readily follows from the fact that the limiting CPP has a total mass exponentially +distributed with mean Σ2t +2 +(see Remark 2.3). +38 + +For (ii), let [X, d, m] be a general random mmm space. Sample k points (v1, · · · , vk) uniformly at ran- +dom with replacement. Let (xv1, · · · , xvk) be the types the sampled individuals. Then E +� +ψ +�� dT (vi,vj) +N +� +i,j, (xvi) +�� +is nothing but the moment of order k of [X, d, +ν +|X|]. Since ¯ +Mt conditioned on survival converges to a Brow- +nian CPP, (ii) follows from Proposition 2.5. +39 + +Acknowledgments +This project has received funding from the European Union’s Horizon 2020 research and innovation +programme under the Marie Skłodowska-Curie grant agreement No 101034413. +References +[AN72] +K. B. Athreya and P. E. Ney. Branching Processes. Grundlehren der mathematischen Wis- +senschaften. Springer-Verlag Berlin Heidelberg, 1972. +[BBC+05] +M. Birkner, J. Blath, M. Capaldo, A. Etheridge, M. Möhle, J. Schweinsberg, and A. Wakol- +binger. +Alpha-stable branching and beta-coalescents. +Electronic Journal of Probability, +10:303–325, 2005. +[BBS13] +J. Berestycki, N. Berestycki, and J. Schweinsberg. The genealogy of branching brownian +motion with absorption. The Annals of Probability, 41(2):527–618, 2013. +[BFRS22] +F. Boenkost, F. Foutel-Rodier, and E. 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American Mathematical Soc., 2012. +41 + diff --git a/4tAzT4oBgHgl3EQfuv3m/content/tmp_files/load_file.txt b/4tAzT4oBgHgl3EQfuv3m/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cfd80c4392103d9734f62603fcb7e45e11c43455 --- /dev/null +++ b/4tAzT4oBgHgl3EQfuv3m/content/tmp_files/load_file.txt @@ -0,0 +1,1786 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf,len=1785 +page_content='Spectral analysis and k-spine decomposition of inhomogeneous branching Brownian motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Genealogies in fully pushed fronts Emmanuel Schertzer∗ and Julie Tourniaire† January 5, 2023 Abstract We consider a system of particles performing a one-dimensional dyadic branching Brownian motion with space-dependent branching rate, negative drift −µ and killed upon reaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' More precisely, the particles branch at rate r(x) = (1 + f(x))/2, where f is a compactly supported and non-negative smooth function and the drift µ is chosen in such a way that the system is critical in some sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This particle system can be seen as an analytically tractable model for fluctuating fronts, describing the internal mechanisms driving the invasion of a habitat by a cooperating population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Recent studies from Birzu, Hallatschek and Korolev suggest the existence of three classes of fluctuating fronts: pulled, semi pushed and fully pushed fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Here, we focus on the fully pushed regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We establish a Yaglom law for this branching process and prove that the genealogy of the particles converges to a Brownian Coalescent Point Process using a method of moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In practice, the genealogy of the BBM is seen as a random marked metric measure space and we use spinal decomposition to prove its convergence in the Gromov-weak topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We also carry the spectral decomposition of a differential operator related to the BBM to determine the invariant measure of the spine as well as its mixing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Contents 1 Introduction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 1 2 Outline of the proof .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 28 6 Survival probability (0th moment) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 30 7 Convergence of metric spaces .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 36 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1 The model and assumptions We consider a dyadic branching Brownian motion (Xt)t>0 (BBM) with killing at 0, negative drift −µ and position-dependent branching rate r(x) = 1 2f(x) + 1 2, (1) for some function f : [0, +∞) → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We assume that f satisfies the following assumptions: ∗Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria †Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='01697v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='PR] 4 Jan 2023 (A1) the function f is non-negative, continuously differentiable and compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (A2) the support of f is included in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We denote by Nt the set of particles in the system at time t and for all v ∈ Nt, we denote by xv = xv(t) the position of the particle v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Furthermore, we write Zt for the number of particles in the system at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We write Px for the law of the process initiated from a point x ≥ 0 and Ex for the corresponding expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We aim at choosing µ in such a way that the number of particles in the system stays roughly constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Fix L > 1 and consider the BBM (XL t )t>0 with branching rate r(x), drift −µ and killed at 0 and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Denote by N L t the set of particles in this system at time t and define ZL t = |N L t |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' By a slight abuse of notations, we will also denote by xv the positions of the particles in the BBM XL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let (t, x, y) �→ pt(x, y) be the fundamental solution of the linear equation � ∂tu(t, y) = 1 2∂yyu(t, y) + µ∂yu(t, y) + r(y)u(t, y) u(t, 0) = u(t, L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (A) We say that pt ≡ pL t is the density of particles in XL in the sense that for any measurable set B ⊂ [0, L], the expected number of particles in B at time t starting from a single particle at x is given by � B pt(x, y)dy [Law18, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='188].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let us now define gt(x, y) := eµ(y−x)e µ2−1 2 tpt(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (2) A direct computation shows that gt is the fundamental solution of the self-adjoint PDE � ∂tu(t, y) = 1 2∂yyu(t, y) + 1 2f(y)u(t, y) u(t, 0) = u(t, L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (B) Let λ1 = λ1(L) be the maximal eigenvalue [Zet12, Chapter 4] of the Sturm–Liouville problem 1 2v′′(x) + 1 2f(x)v(x) = λ1v(x), (SLP) with boundary conditions v(0) = v(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (BC) It is known that λ1 is an increasing function of L [Pin95, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1] and that it converges to a finite limit λ∞ 1 ∈ (−∞, +∞) as L → ∞ [Pin95, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We now choose µ in such a way that the expected number of particles is neither increasing nor decreasing exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' According to (2), we expect that for large t pt(x, y) ≈ eµ(y−x)e µ2−1 2 teλ∞ 1 t v1(x)v1(y) ||v1||2 , where v1 denotes an eigenfunction associated to λ1 for the Sturm-Liouville problem (SLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This motivates the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Definition 1 (Critical regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The BBM is in the critical regime iff µ = � 1 + 2λ∞ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (3) Pushed and pulled waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The next definitions are motivated by recent numerical simulations and heuristics [BHK18, BHK21] for the noisy F-KPP equation with Allee effect ut = 1 2uxx + u(1 − u)(1 + Bu) + � u N η, where B > 0, N is a large demographic parameter and η is a space-time white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' See [Tou21] for more details and [EP22] for recent rigorous results on the bistable case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2 Definition 2 (Pulled, semi pushed, fully pushed regimes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Consider the BBM (Xt) in the critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Define β := � 2λ∞ 1 , and α := µ + β µ − β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (4) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' If λ∞ 1 = 0, or equivalently α = 1, the BBM is said to be pulled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' If λ∞ 1 ∈ (0, 1/16) or equivalently α ∈ (1, 2) ⇐⇒ µ > 3β, the BBM is said to be semi pushed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' If λ∞ 1 > 1/16 or equivalently α > 2 ⇐⇒ µ < 3β, (Hfp) the BBM is said to be fully pushed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We say that the BBM is pushed if it is either semi or fully pushed, that is when λ∞ 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (Hp) Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let ε > 0 and consider the BBM with inhomogeneous branching rate rε(x) = 1 2 + εf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' By [Pin95, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3], for any function f satisfying (A1), there exists 0 < ε1 < ε2 such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The BBM is pulled for all ε ∈ (0, ε1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The BBM is semi-pushed for all ε ∈ (ε1, ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The BBM is fully-pushed for all ε > ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' It is conjectured that up to rescaling, the size and the genealogy at large time is undistinguishable from the ones of a continuous-state branching process (CSBP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' More precisely, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In the pulled regime, the population size should converge to a Neveu’s continuous-state branching process and the genealogy of the BBM to the Bolthausen–Sznitman coalescent (see [BBS13] in the case f ≡ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In the semi pushed regime, the population size should converge to an α-stable CSBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In the pre- vious example, this has been proved only in the case where f = 1[0,1] [Tou21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Therein, it is also conjectured that the genealogy should converge to a time-changed Beta(α, 2 − α) coalescent [Pit99, Sag99, BBC+05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In the fully pushed regime, the rescaled population size should converge to a Feller diffusion and the genealogy should be undistinguishable from the genealogy of a large critical Galton-Watson process with finite second moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This is the content of the present article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='2 Comparaison with previous work Branching Brownian motion with inhomogeneous branching rates have received quite a lot of attention in the recent past [HKV20, HHK20, HHKW22, GHK22, FRS22, Tou21, RS21, LS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The general approach always relies on a spinal decomposition of the BBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Roughly speaking, the spine is constructed by conditioning a typical particle to survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This conditioning is achieved by a Doob-h transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In our setting, the harmonic function is approximated by h(x) ≈ eµxv1(x) and the resulting h-transform is given by dxt = v′ 1(xt) v1(xt)dt + dBt (5) where Bt is a standard Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' See Section 3 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A key assumption underlying [Pow19, HKV20, HHK20, HHKW22, GHK22] is that the harmonic function h is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' From a technical stand point, we emphasise that this assumption is the one distinguishing our work from the previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Indeed, we shall see that v1 decreases exponentially at rate β so that the harmonic function h(x) ∝ e(µ−β)x blows up as x → ∞ since µ = � 1 + β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Due to the explosion of the harmonic function, many of the previously developed technics break down in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 3 At first sight, this assumption may only seem technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' However, it is the key assumption which makes possible a transition from the semi to the fully pushed regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let us consider the spine dynamics (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In the pushed regime, the invariant distribution for the spine is given by Π(x) = v2 1(x) ||v1||2 , so that h(x)Π(x) ∝ e(µ−3β)x as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' It then becomes clear from Definition 2 that, in the fully pushed regime (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' semi pushed), the harmonic function is integrable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' non-integrable) with respect to the invariant measure of the spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' As a consequence, relaxing the assumption under which h is bounded is crucial for understanding the transition between these two regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This generalisation raises interesting technical challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A large fraction of the present work (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1) is dedicated to estimating the speed of convergence to the invariant measure of the spine in the pushed regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' More precisely, we use a Sturm-Liouville approach in order to understand the spectral decomposition of the differential operator (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The difficulty arises from the fact that the negative part of the spectrum of the Sturm-Liouville problem (SLP) becomes continuous as L → ∞ (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We believe that this contribution is relevant for understanding not only the fully pushed case, but also the semi pushed case which will be the subject of future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This continuous spectrum already appeared in the study of homogeneous BBM [BBS13], but the spectral analysis of (SLP) is quite straightforward in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Indeed, when f ≡ 0, the spectral decomposition of (SLP) is given by λi = − iπ2 2L2 and vi(x) = sin � iπx L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' When f is not trivial, the spectrum is not explicit and we use the Prüfer transformation to derive the required estimates on the (vi) and the (λi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, one of the main contribution of the present work is the description of the genealogy spanned by the population at a large time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Beyond our Kolmogorov estimate and the Yaglom law reminiscent of [Pow19, HKV20, HHK20, HHKW22, GHK22], we use k-spine decompositions [HR17] and moment methods developed in [FRS22] to prove convergence of the genealogy in the Gromov weak topology to a continuum random metric space known as the Brownian Coalescent Point Process [Pop04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This approach will be further explained in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Consider f = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1[0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' It was calculated in [Tou21] that the negative part of the spectrum of (SLP) with boundary conditions (BC) consists in the solutions of tan( √ 9 − 2λ) √ 9 − 2λ = −tan( √ −2λ(L − 1)) √ −2λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (6) The solutions of this equation are plotted on Figure 1 4 Figure 1: Negative spectrum of the Sturm–Liouville problem (SLP) with boundary conditions (BC) for f defined as in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='2 and different values of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The blue line corresponds to RHS of (6) and the red line to the LHS of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3 Main results Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let v1 be the eigenfunction associated to the eigenvalue λ1 for the Sturm-Liouville problem (SLP) with boundary conditions (BC), normalised in such a way that v1(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Under (Hp), v1 converges to a positive limiting function v∞ 1 as L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Further, if in addition (Hfp) holds, then � R+ eµx(v∞ 1 )3(x)dx < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' To see why the latter proposition may hold true, recall that f ≡ 0 on [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Hence, on this interval, the problem reduces to 1 2v′′ 1 (x) = λv1(x), x ∈ [1, L], v1(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' If we impose the condition v1(1) = 1, a direct computation shows that v1(x) = sinh(√2λ1(L−x)) sinh(√2λ1(L−1)) on [1, L] so that, for all x ∈ [1, ∞), v1(x) → v∞ 1 (x) = e−β(x−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The integrability condition then holds under the extra assumption (Hfp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In the following, we define h∞(x) := ceµxv∞ 1 (x), and ˜h∞(x) := ˜ce−µxv∞ 1 (x), (7) where ˜c := �� ∞ 0 e−µxv∞ 1 (x)dx �−1 and c := (˜c∥v∞ 1 ∥2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The constants c and ˜c are thought as Perron-Frobenius renormalisation constants (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' [AN72, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='185]), in the sense that h∞ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' ˜h∞) is a left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' right) eigenfunction associated to the maximal eigenvalue of the differential operator Lu = 1 2∂xxu + µ∂xu + r(x)u, (8) 5 L=10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='0 =50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='0 L=100normalised in such a way that � ∞ 0 ˜h∞(x)dx = 1 and � ∞ 0 h∞(x)˜h∞(x)dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' From this perspective, the function ˜h∞ should correspond to the stable configuration of the system and the function h∞ to the reproductive values of the individuals as a function of their positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Theorem 1 (Kolmogorov estimate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' As N → ∞, for all x, t > 0, NPx (ZtN > 0) → 2 Σ2th∞(x), where Σ2 2 := � R+ r(z)(h∞(z))2˜h∞(z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This theorem is the continuous analogous of Kolmogorov estimate for multi-type Galton-Watson process [AN72, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='187]: in our case, the variance in the offspring distribution is given by Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We now turn to the description of the genealogy and the Yaglom law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Intuitively, the next result states that the genealogy is asymptotically identical to the one of a critical Galton Watson [Lam18, HJR20, Joh19], whereas the marks are assigned independently according to ˜h∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let us now give a more precise description of our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' From now on, we condition on the event {ZtN > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' , vk) be k individuals chosen uniformly at random from NtN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Denote by dT (vi, vj) the time to the most recent common ancestor of vi and vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let xvi be the position of the ith individual at time tN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let U be a uniform r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' on [0, t] and θ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Define U θ such that ∀s ≤ t, P(U θ ≤ s) := (1 + θ)P(U ≤ s) 1 + θP(U ≤ s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (9) Let (U θ i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ∈ [k]) be k i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' copies of U θ and set ∀1 ≤ i < j ≤ k, U θ i,j = U θ j,i := max{U θ k : k ∈ {i, · · · , j − 1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, define the random distance matrix (Hi,j) := (Hi,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ̸= j ∈ [k]) such that for every bounded and continuous function ϕ : Rk2 → R, E � ϕ � (Hi,j) �� = k � ∞ 0 1 (1 + θ)2 � θ 1 + θ �k−1 E � ϕ � (U θ i,j) �� dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (10) Finally, (Wi) := (Wi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ∈ [k]) will denote a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' copies of a random variable W with law ˜h∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Theorem 2 (Yaglom law and limiting genealogies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Start with a single particle at x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Conditional on ZtN > 0, as N → ∞, (i) we have ZtN N → Σ2t 2 E, in distribution, where E is a standard exponential distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (ii) �� dT (vi,vj) N � , � xvi �� converges to the distribution of � (Hσi,σj), (Wσi) � where σ is a random uniform permutation {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' , k} and σ, (Hi,j) and (Wi) are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The random distance matrix (Hi,j) is the one obtained from a critical Galton Watson with finite second moment conditioned on surviving up to a large time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' See [Lam18, HJR20, Joh19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4 Notation Given two sequences of positive real numbers (aN) and (bN), we write aN ≪ bn if aN/bN → 0 as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We write aN ≲ bN if aN/bN is bounded in absolute value by a positive constant and aN ≍ bN if aN ≲ bN and bN ≲ aN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We write O(·) to refer to a quantity bounded in absolute value by a constant times what the quantity inside the parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Unless otherwise specified, these constants only depend on λ∞ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 6 2 Outline of the proof Our approach relies on a method of moments devised in [FRS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' To illustrate the approach, let us first think about the Yaglom law of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' To prove this result, one needs to show that the moments of ZtN/N converge to the moments of an exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' It turns out that this approach can be extended to genealogies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1, following the approach in [DGP11], we encode the genealogy at time tN as a random marked measured metric space (mmm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In turn, the moments of a random mmm are obtained by biasing the population by its kth moment and then picking k individuals uniformly at random (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='2, we introduce a limiting random mmm called the marked Coalescent Point Process (CPP) which corresponds to the limiting genealogy of a critical Galton-Watson process [Pop04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The remainder of the section is dedicated to the sketch of the proof for the convergence of the moments of our BBM to the moments of the marked CPP using the spinal decomposition introduced in [FRS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1 Marked Metric Spaces Let (E, dE) be a fixed complete separable metric space, referred to as the mark space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In our application, E = (0, ∞) is endowed with the usual distance on the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A marked metric measure space (mmm- space for short) is a triple [X, d, µ], where (X, d) is a complete separable metric space, and µ is a finite measure on X × E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' To define a topology on the set of mmm-spaces, for each k ≥ 1, we consider the map Rk : � (X × E)k → Rk2 + × Ek � (xi, ui);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ≤ k � �→ � d(xi, xj), ui;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i, j ≤ k � that maps k points in X×E to the matrix of pairwise distances and marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We denote by νk,X = µ⊗k◦R−1 k , the marked distance matrix distribution of [X, d, µ], which is the pushforward of µ⊗k by the map Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Note that µ is not necessarily a probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let k ≥ 1 and consider a measurable bounded test function ϕ: Rk2 + × Ek → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' One can define a functional Φ � X, d, µ � = ⟨νk,X, ϕ⟩ = � X×E ϕ � d(vi, vj), xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ̸= j ∈ [k] � k � i=1 µ(dvi ⊗ dxi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (11) Functionals of the previous form are called polynomials, and the set of all polynomials, obtained by varying k and ϕ, is denoted by Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, the moment of [X, d, µ] associated to Φ is defined as E(Φ � X, d, µ � ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let ϕ be of the form ϕ � di,j, xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ̸= j ∈ [k] � = � i,j ψi,j(di,j) � i ϕi(xi) where ψi,j, ϕi are bounded measurable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We say that Φ(X, d, µ) is a product polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We denote by ˜Π the set of product polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The moments of a random mmm can be rewritten as E � Φ � X, d, µ �� = E(|X|k) × 1 E(|X|k)E � |X|kϕ(d(vi, vj), Xvi, i ̸= j ∈ [k]) � , where (vi, Xvi) are k points sampled uniformly at random with their marks and |X| = µ(X ×E) is thought as the total size of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' As a consequence, the moments of a random mmm are obtained by biasing the population size by its kth moment and then picking k individuals uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The marked Gromov-weak (MGW) topology is the topology on mmm-spaces induced by Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A random mmm-space is a r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' with values in M – the set of (equivalence classes of) mmm-spaces – endowed with the Gromov-weak topology and the associated Borel σ-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, the marked Gromov-weak (MGW) topology is identical to the topology induced by the product polynomials ˜Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 7 Many properties of the marked Gromov-weak topology are derived in [DGP11] under the further assumption that µ is a probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In particular, the following result shows that Π forms a convergence determining class only when the limit satisfies a moment condition, which is a well-known criterion for a real variable to be identified by its moments, see for instance [Dur19, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This result was already stated for metric measure spaces without marks in [DG19, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='7] and was proved in [FRS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Suppose that [X, d, µ] is a random mmm-space verifying lim sup p→∞ E[µ(X × E)p]1/p p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (12) Then, for a sequence [Xn, dn, µn] of random mmm-spaces to converge in distribution for the marked Gromov-weak topology to [X, d, µ] it is sufficient that lim n→∞ E � Φ � Xn, dn, µn �� = E � Φ � X, d, µ �� for all Φ ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='2 Marked Brownian Coalescent Point Process (CPP) Let T > 0 and m be a measure on R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Assume that |m| := m(R+) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Consider P a PPP � dx x2 ⊗ dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Define YT = inf {y : (y, t) ∈ P, t ≥ T} , and dT (x, y) = sup{t : (z, t) ∈ P and x ≤ z ≤ y}, 0 < x < y < YT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The marked Brownian Coalescent Point Process (CPP) is defined as MT := � [0, YT ], dT , dv ⊗ m(dx) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This object is a natural extension of the standard Brownian CPP [Pop04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A direct computation shows that YT m(R+) (which can be thought as the population size at time T) is distributed as an exponential random variable with mean T|m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' If we think as the CPP as the genealogy of critical branching processes, this consistent with Yaglom’s law for critical branching processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Figure 2: Simulation of the unmarked Brownian CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' On the RHS, a vertical line of height x at location s represents an atom (x, s) of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' On the LHS, the tree corresponding to the right CPP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' the distance dT is the tree distance of the leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 8 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let K ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let (ϕi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ∈ [K]) and (ψi,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i, j ∈ [K]) be measurable bounded functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Consider an arbitrary product polynomial of the form ∀M = [X, d, µ], Ψ(M) := � � i,j ψi,j(dT (vi, vj)) � i ϕi(xi)µ(dvi ⊗ dxi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Then E [Ψ(MT )] = K!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='T KE �� i,j ψi,j(Uσi,σj) � �� i � m(dx)ϕi(x) �K , where (Ui;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ∈ [K − 1]) is a vector of uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' random variables on [0, T] and for i < j Ui,j = Uj,i = max{Uk : k = i, · · · , j − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The proof is identical to Proposition 4 in [FRS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='5 (Sampling from the CPP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let k ∈ N and sample k points (v1, · · · , vk) uniformly at random from the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let xv1 · · · , xvk be the corresponding types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Then ((d(vi, vj))i,j , (xvi)i) is identical in law to � (Hσi,σj), (Wσi) � where (Hi,j) is defined as in Theorem 2 (ii) and (Wi) are independent random variables with law m |m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The proof is identical to the one in the case of the unmarked CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' See [BFRS22, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3 Convergence of mmm Fix t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Recall that Nt refers the set of particles alive at time t in the BBM X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Set µt := � v∈Nt δv,xv, ∀v, v′ ∈ Nt, d(v, v′) = � t − |v ∧ v′| � , where v∧v′ denotes the MRCA of v and v′, and |v| denotes the generation of vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let Mt = [Nt, d, µt] be the resulting random mmm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, set ¯µt := 1 N � v∈NtN δv,xv, ∀v, v′ ∈ NtN, ¯d(v, v′) = � t − 1 N |v ∧ v′| � , and define the rescaled metric space ¯ Mt := [NtN, ¯d, ¯µt].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The main idea underlying Theorem 2 is to prove the convergence of ¯ Mt to a limiting CPP whose size and sampling structure coincides with (i) and (ii) in Theorem 2 — See Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Conditional on the event {ZtN > 0}, ( ¯ Mt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' N ∈ N) converges in distribution for the Gromov weak topology to a marked Brownian CPP with parameters (t, Σ2 2 ˜h∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The proof of the theorem relies on a cut-off procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Namely, let L := 1 µ − β log(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (13) Recall that XL refers to the BMM killed at 0 and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let µL t (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' ¯µL t ) be the empirical measure obtained by replacing N by N L in the definition of µt (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' ¯µt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let M L t be the mmm obtained from XL, that is M L t = [N L t , d, µL t ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' ¯ M L t is defined analogously to ¯ Mt (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' accelerating time by N and rescaling the empirical measure by 1/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We will proceed in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' For our choice of L, we will show that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' ¯ M L t converges to the limit described in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' ¯ M L t and ¯ Mt converge to the same limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The choice for L will be motivated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We start by motivating the fact that ¯ M L t converges to the desired limit using a spinal decomposition introduced in [FRS22] in a discrete time setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4 The K-spine Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The 1-spine is the stochastic process on [0, L] with generator 1 2∂xxu + v′ 1(x) v1(x)∂xu, u(0) = u(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In the following, qt(x, y) will denote the probability kernel of the 1-spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We can also directly determine the invariant distribution of the spine Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The 1-spine has a unique invariant measure given by Π(dx) = � v1(x) ||v1|| �2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let (U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=', UK−1) be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' random variables uniformly distributed on [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Define ∀ 1 ≤ i < j ≤ K − 1, d(i, j) = d(j, i) = max{Ui, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=', Uj−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (14) Let T be the unique planar ultrametric tree of depth t with K leaves labeled by {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=', K} such that the tree distance between the leaves is d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' See Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We then assign marks on this tree such that, on each branch of the tree, marks evolve according to the 1-spine (on [0, L]) and branch into independent particles at the branching points of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' See [FRS22] for a more formal definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' t 0 U1 U2 V1 V2 V3 t 0 x x0 ζv1 qt(x0, ·) qt−U1(ζv1, ·) Figure 3: K-spine with K = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Left panel: planar tree T generated from 2 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' uniform random variables (U1, U2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Right panel: branching 1-spines running along the branches of the tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The resulting planar marked ultrametric tree will be referred to as the K-spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We will denote by QK,t x the distribution of the K-spine rooted at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The t superscript refers to the depth of the underlying genealogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Note that QK,t x has an implicit dependence on N by our choice of L – see (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' To ease the notation, this dependence will be dropped in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In the following, B will denote the set consisting of the K − 1 branching points of the K-spine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' L will denote the set consisting of the K leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We will denote by ζv the mark (or the position) of the spine at a node v ∈ B ∪ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' For v ∈ B, |v| will denote the time component of the branching point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, (Vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ∈ [K]) is the enumeration of the leaves from left to right in the K-spine (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=', Vi is the leaf with label i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We will also need the accelerated version of the K-spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Definition 5 (Accelerated K-spine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Consider the 1-spine accelerated by N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' the transition kernel of the 1-spine is now given by qtN(x, y) ≡ qL tN(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We denote this kernel by ¯qt(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Consider the same planar structure as before, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=', the depth is t and the distance between points at time t is given by (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We denote by ¯QK,t x the distribution of the K-spine obtained by running accelerated spines along the branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' For any vertex v in the K-spine, ¯ζv will denote the mark of the vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 10 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='7 (Rescaled many-to-few).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let (ϕi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ∈ [K]) and (ψi,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i, j ∈ [K]) be measurable bounded functions and define ∀M = [X, d, µ], Ψ(M) = � � i,j 1(vi ̸= vj)ψi,j(d(vi, vj)) � i ϕ(xi)µ(dvi ⊗ dxi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Define h(t, x) := ce(λ∞ 1 −λ1)teµxv1(x) where c is the renormalisation constant in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Then Ex � Ψ( ¯ M L t ) � = 1 N K!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' h(0, x) tK−1 ¯QK,t x � ¯∆ � i,j ψi,j(Uσi,σj) � i ϕi(¯ζVσi ) � , where ¯∆ := � v∈B(U) r(¯ζv)h(|v|N, ¯ζv) � v∈L(U) 1 h(tN, ¯ζv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The second crucial result is the following convergence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let ( ˜ϕi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i ∈ [K]) and (ψi,j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' i, j ∈ [K]) be measurable bounded functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Assume further that the ˜ϕi’s are compactly supported in (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' As N → ∞, ¯QK,t x � ¯∆ � i,j ψi,j(Uσi,σj) � i ˜ϕi(¯ζVσi )h∞(¯ζVσi ) � → �Σ2 2 �K−1 E �� i,j ψi,j(Uσi,σj) � � i � R+ ˜ϕi(x)Π∞(dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let us now give a brief heuristics underlying the previous result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' By definition ¯QK,t x � ¯∆ � i,j ψi,j(Uσi,σj) � i ˜ϕi(¯ζVσi )h∞(¯ζVσi ) � = ¯QK,t x �� v∈B r(¯ζv)h(|v|N, ¯ζv) � i,j ψi,j(Uσi,σj) � i ˜ϕi(¯ζVσi ) h∞(¯ζVσi ) h(tN, ¯ζVσi ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The branching structure for the K-spine is binary a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' whereas the spines running along the branches are accelerated by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Recall that the invariant measure for the 1-spine is Π ≈ Π∞ as L → ∞ (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Moreover, we will show later on that under (Hfp), h(tN, x) ≈ h∞(x) for N large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' It is now reasonable to believe that, provided enough mixing, the RHS can be approximated by �� L 0 h∞(x)r(x)Π∞(dx) �K−1 E �� i,j ψi,j(Uσi,σj) � � i � ˜ϕi(x)Π∞(dx), assuming the values of the spine at the branching points and the leaves converge to a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' random variables with law Π∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This yields the content of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Challenge 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The previous argument relies on a k-mixing property of the 1-spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This analysis will be carried out in Section 4 using Sturm–Liouville theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='5 Limiting moments Let us now demonstrate the importance of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let ∀M = [X, d, µ], ˜Ψ(M) := � X � i,j ψi,j(d(vi, vj)) � i ˜ϕi(xi)h∞(xi) µ(dvi ⊗ dxi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 11 for any bounded measurable functions ˜ϕi, ψi,j such that the ˜ϕi are compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' From the many-to-few formulae, our result entails Ex � ˜Ψ( ¯ M L t ) � = Ex � � � i,j ψi,j( ¯d(vi, vj)) � i ˜ϕi(xi)h∞(xi) ¯µL t (dvi ⊗ dxi) � ≈ Ex � � � i,j 1(vi ̸= vj)ψi,j( ¯d(vi, vj)) � i ˜ϕi(xi)h∞(xi) ¯µL t (dvi ⊗ dxi) � ≈ 2h∞(x) NΣ2t × K!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' � tΣ2 2 �K E �� i,j ψi,j(Uσi,σj) � � i � R+ ˜ϕi(x)Π∞(dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let us formally take ψi,j ≡ 1 and ˜ϕi ≡ 1/h∞ in the previous expression (note that this is problematic since ˜ϕi is neither bounded nor compactly supported, see Challenge 4 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Then for large N, Ex �� 1 N ZL tN �K� ≈ 2h∞(x) NΣ2t × K!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' � tΣ2 2 �K � �� � exponential moments × � ∞ 0 ˜h∞(x)dx � �� � =1 , where we used the fact that ˜h∞ = Π∞/h∞ under our Perron–Frobenius renormalisation (see (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Now, the RHS coincides with the moments of a r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' with law � 1 − 2h∞(x) NΣ2t � δ0(dx) + 2h∞(x) NΣ2t exp � − 2x Σ2t � 2dx Σ2t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' If we identify the Dirac measure at 0 with the extinction probability, this suggests the Kolmogorov estimate and the Yaglom law exposed in Theorem 1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Further, if we replace ˜ϕi = ϕi/h∞ in Theorem 4 (again a problematic step), the previous estimates entail Ex � Ψ( ¯ M L t ) ��ZL tN > 0 � ≈ K!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='E �� i,j ψi,j(Uσi,σj) � � i � R+ ϕi(x) � tΣ2 2 � ˜h∞(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' where Ψ(M) is now an arbitrary polynomial of the form Ψ(M) := � � i,j ψi,j(d(vi, vj)) � i ϕi(xi)µ(dvi ⊗ dxi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' According to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4, this coincides with the moments of the Brownian CPP described in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Challenge 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The previous computation only suggests that the probability for the population to be o( 1 N ) is given by the Kolmogorov estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Intuitively, the Dirac mass above corresponds to a population whose size becomes invisible at the limit after rescaling the population by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' It thus remains to show that if the population is small compared to N then it must be extinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This will be carried out in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Challenge 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Going from Theorem 4 to the convergence of the M L tN requires to use test functions ex- ploding at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' To overcome this technical difficultly, we will impose an extra thinning of the population by killing all the particles close to the boundaries at time tN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This final technical step will be carried in Section 7 using some general property of the Gromov-weak topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='6 Choosing the cutoff L We now motivate our choice for L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' According to the previous arguments, we want to choose L large enough such that (i) The particles do not reach L with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This will imply that ¯ M L t and ¯ Mt coincide with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (ii) The 1-spine reaches equilibrium in a time o(N) regardless of its initial position on [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' This is needed to justify the calculations of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 12 (i) Hitting the right boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let E be a compact set in the vicinity of the boundary L (say [L − 2, L − 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Recall from the discussion after Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3 that for x ≥ 1, h∞(x) = c eµxv∞ 1 (x) ≈ c eβe(µ−β)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A direct application of the many-to-few lemma with K = 1 (many-to-one) and ϕ1(x) = 1x∈E implies that Ex � � � v∈N L tN 1xv∈E � � = � E h(0, x) h(tN, y)qtN(x, y)dy ≈ � E h∞(x) h∞(y) Π∞(dy) ≈ h∞(x)O( =N−α � �� � e− µ+β µ−β log(N)), where the last approximation holds under the assumption that E is a compact set close to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Integrating on [0, tN], this yields that the occupation time of the set E on the time interval [0, tN] is O(N 1−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Recall that the probability of survival is of order 1/N so that the occupation time of the conditioned process is O(N 2−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Recalling that α > 2 under (Hfp), this yields the desired estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (ii) Mixing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Recall from the discussion after Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='3 that v1(x) ≈ e−β(x−1) for x ≥ 1 and L large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' As a consequence, the 1-spine (see Definition 4) is well approximated by the diffusion dzt = −βdt + dwt for zt ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A good proxy for the mixing time is the first returning time at 1 which is of the order log(N) = o(N) for every x ∈ [1, L], as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A more refined analysis will be carried in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 3 The many-to-few theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1 The general case In this section, we consider a general BBM killed at the boundary of a regular open domain Ω ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Unless otherwise specified, we used the same notation as in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We will assume that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The generator of a single particle is given by a differential operator Lf(x) = 1 2 � i,j aij(x)∂xixjf(x) + � i bi(x)∂xif(x), x ∈ Ω, (15) f(x) = 0, x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We assume that (aij) is uniformly elliptic, which means that there exists a constant θ > 0 such that for all ξ ∈ Rd and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' x ∈ Ω, �d i,j=1 aij(x)ξi, ξj ≥ θ∥ξ∥2 (see [Eva10, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In addition, we assume that ai,j ∈ C1(Ω) and supx∈Ω |bi(x)| < ∞ for all 1 ≤ i, j ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' A particle at location x branches into two particles at rate r(x) (we only consider binary branching).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We denote by Nt the set of particles alive at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' For any pair of particles v, w ∈ Nt, d(u, w) will denote the time to their most recent common ancestor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, we define the random mmm space Mt = [Nt, d, µt], where µt = � v∈Nt δv,xv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We say that a function h(x) is harmonic if and only h satisfies the Dirichlet problem Bh(x) := Lh(x) + r(x)h(x) = 0, for every x ∈ Ω, h(x) = 0, for x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The 1-spine whose generator is given by the Doob h-transform of the differential operator B B(hf) h (x) = 1 2 � i,j aij(x) � ∂xixjf(x) + ∂xih h ∂xjf(x) + ∂xjh h ∂xif(x) � + � i bi(x)∂xif(x), x ∈ Ω, f(x) = 0, x ∈ ∂Ω, where the first equality is a direct consequence of the fact that h is harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We will denote by qt(x, y) the transition probability at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The K-spine distribution QK,t x is defined analogously to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 13 Our many-to-few formulae rely on a uniform planarisation of the BBM that we now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' At every time t > 0, every particle is now endowed with two marks (xv, pv) where pv ∈ ∪n∈N{0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' As before, xv denotes the position of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The planarisation marks pv are assigned recursively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We mark the root with ∅ and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' At every branching point v, we distribute the marks (pv, 0) and (pv, 1) uniformly among the two children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (pv, 0) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' (pv, 1)) is said to the left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' right) child of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The mark pv does not vary between two branching points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=', pv1 = pv2 if the trajectory connecting v1 and v2 does not encounter any branching points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let N pl t be the the set particles at time t in the planar BBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' For every K-uplet v1, v2 · · · , vK in N pl t , let U(⃗v) = [X(⃗v), d(⃗v), µ(⃗v)] be the planar (ultra-)mmm space induced by this set of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The space consists of the set of vertices ancestral to some vertex in ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' In particular, U(⃗v) is binary and made of K leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, the measure is given by the counting measure on the leaves µ(⃗v) = � ⃗v δv,(xv,pv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let us now introduce some definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Consider a planar ultra-mmm U made of finitely many leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We define σ(U) to be the diameter U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We say that U has no simultaneous branching iff for every distinct pair of leaves (v1, v2), (v3, v4), we have d(v1, v2) ̸= d(v3, v4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' For such a tree, there exists a unique element v ∈ U such that v = Argmax d(w, ¯w) where the maximum is taken over the pair of leaves in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The individual v is said to be the MRCA of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Moreover, we define T0(U) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' T1(U)) to be the left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' right) subtree attached to the MRCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, B(U) will refer to the set of branching points and L(U) will refer to the set of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Finally, xMRCA(U) will be the spatial position of the MRCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='1 (Many-to-one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' For every bounded continuous function f Ex � � v∈Nt f(xv) � = � Ω f(y)qt(x, y)h(x) h(y) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' One can readily check that pt(x, y) := qt(x, y) h(x) h(y) are the fundamental solutions of the same PDE � ∂tu(t, y) = L∗u(t, y), y ∈ Ω, u(t, y) = 0, y ∈ ∂Ω, where L∗ is the adjoint of the differential operator (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let K ∈ N and t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Define the measure RK,t x on the set of planar mmm so that for every bounded measurable function F, RK,t x (F) := Ex � � � � v1̸=···̸=vK, vi∈N pl t F(U(⃗v)) � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Then RK,t x (F) = K!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' h(x)tK−1 QK,t x (∆F) with ∆ = � v∈B(U) r(ζv)h(ζv) � v∈L(U) 1 h(ζv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' We will show the result by an induction of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' The case K = 1 is the many-to-one lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' Let us now consider F of the following product form F(U) = f(σ(U))ψ0(T0(U))ψ1(T1(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content=' 14 Then RK,t x (F) = K!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfuv3m/content/2301.01697v1.pdf'} +page_content='Ex � � � v1<··· 0, this construction yields the +Poisson-Voronoi tessellation, denoted here by Wd,ρ, – one of the most classical models studied in +stochastic geometry. We refer to the monographs [22, 27, 29] for more detailed information, applications +and further references on Voronoi tessellations and in particular the Poisson-Voronoi tessellation. +A number of random tessellations studied in stochastic geometry, such as the Poisson hyperplane +or the STIT tessellations, have the distinguished feature of being stable under intersections with lower- +dimensional affine subspaces. By this we mean that the intersection with an affine subspace of one of these +random tessellations is again a model of the same type within the intersecting subspace. For example, the +intersection of a Poisson hyperplane tessellation with an affine subspace L is again a Poisson hyperplane +tessellation within L. However, a similar property is not true for the Poisson-Voronoi tessellation. In fact, +it has been shown by Chiu, Van De Weygaert and Stoyan [6] that the intersection of the Poisson-Voronoi +tessellation with an affine subspace cannot be a Voronoi tessellation induced by any stationary point +process within the subspace. In other words, the sectional Poisson-Voronoi tessellation is necessarily a +‘non-Voronoi’ tessellation. However, besides a few mean values determined in [6, 20] further probabilistic +aMünster University, Germany. Email: gusakova@uni-muenster.de +bMünster University, Germany. Email: zakhar.kabluchko@uni-muenster.de +cRuhr University Bochum, Germany. Email: christoph.thaele@rub.de +1 +arXiv:2301.03297v1 [math.PR] 9 Jan 2023 + +or geometric information about the sectional Poisson-Voronoi tessellation seems not available in the +existing literature, although they are of importance for stereological applications (see [29, Chapter 11.5.4] +or [21, Section 14.4.6] as well as the references cited therein). It is one of the main purposes of this paper +to derive a precise description of the sectional Poisson-Voronoi tessellation and to study its typical cell. +We do this by establishing a connection with the so-called β-Voronoi tessellations, a random tessellation +model we recently introduced and studied in the series of papers [10, 7, 8, 9]. Their analysis in turn was +based on the connection with the class of beta random polytopes, which has already seen a number of +applications in stochastic geometry [11, 13, 14, 15]. +We study the problem of the sectional Poisson-Voronoi tessellation just explained in a more general +framework. In fact, the random tessellation we study is either +• a β-Voronoi tessellation Vd,β,γ in Rd with parameters β ≥ −1 and γ > 0, +• a β′-Voronoi tessellation V′ +d,β,γ in Rd with parameters β > d +2 + 1 and γ > 0, +• or a Gaussian-Voronoi tessellation Gd,λ in Rd with parameter λ > 0; +a description of all these models will be provided in Section 3. We remark that the classical Poisson- +Voronoi tessellation generated by a stationary Poisson point process in Rd with intensity ρ > 0 appears +in this framework as the β-Voronoi tessellation corresponding to the parameters β = −1 and γ = +π +d+1 +2 ρ/Γ( d+1 +2 ). Now, let L ⊂ Rd be an affine subspace of dimension 1 ≤ ℓ ≤ d − 1. We show in Theorem +4.1 below that +• the sectional tessellation Vd,β,γ ∩ L is a (β + d−ℓ +2 )-Voronoi tessellation in L with the same γ, +• the sectional tessellation V′ +d,β,γ ∩ L is a (β − d−ℓ +2 )′-Voronoi tessellation in L with the same γ, +• the sectional tessellation Gd,λ ∩ L is again a Gaussian-Voronoi tessellation in L with the same λ. +In particular, the intersection of the classical Poisson-Voronoi tessellation Wd,ρ with L turns out to be +a β-Voronoi tessellation within L with β = +d−ℓ +2 +− 1 and γ = π +d+1 +2 ρ/Γ( d+1 +2 ). +For clarity we should +remark that none of the random tessellations Vd,β,γ with β > −1 are actually Voronoi tessellations. +Our terminology is motivated by the fact that Vd,β,γ may be viewed as a deformation of the classical +Poisson-Voronoi tessellation (which corresponds to β = −1). +With the identification of the sectional Poisson-Voronoi tessellation at hand, in the second part of this +paper we study its geometric properties. More precisely, we determine in Theorem 5.1 its face intensities +in terms of quantities which have already appeared in the study of beta random polytopes [11, 15]. From +here on, we determine the expected volume, the expected intrinsic volumes as well as the expected f- +vector of the typical cell (and even more generally the typical k-face) of the sectional Poisson-Voronoi +tessellation. Moreover, generalizing earlier results of Miles [20] we consider the asymptotics, as d → ∞, of +several characteristics (such as the volume of the typical cell) of the sectional Poisson-Voronoi tessellation +and identify the limits with the corresponding characteristics of a suitable Gaussian-Voronoi tessellation. +The weak convergence on the level of tessellations is discussed as well using a coupling construction +similar to the one in [8]. +2 +Preliminaries on random tessellations +In this section we collect some definitions and facts about general stationary random tessellations in Rd. +For more detailed discussions we refer the reader to [27, Chapters 4 and 10] as well as [29, Chapter 10]. +A tessellation T in Rd is a countable, locally finite collection of d-dimensional polytopes, which cover +the space and have non-empty, disjoint interiors. The elements of T are called the cells of T. Given a +polytope c ⊂ Rd we denote by Fk(c) the set of its k-dimensional faces, 0 ≤ k ≤ d, where Fd(c) = {c}, +and let F(c) := �d +k=0 Fk(c). A tessellation T is called face-to-face if for any two of its cells c1, c2 ∈ T +one has that +c1 ∩ c2 ∈ (F(c1) ∩ F(c2)) ∪ {∅}, +2 + +that is, the intersection of two cells is either empty or a common face of both cells. For a face-to-face +tessellation T one defines Fk(T) = � +c∈T Fk(c) and F(T) = � +c∈T F(c). A face-to-face tessellation in Rd +is called normal if each k-dimensional face of the tessellation is contained in the boundary of precisely +d + 1 − k cells, for all k ∈ {0, 1, . . . , d − 1}. +We denote by T the set of all face-to-face tessellations in Rd, which is supplied with a measurable +structure as in [27, Chapter 10]. By a random tessellation we understand a particle process T in Rd +(in the usual sense of stochastic geometry, see [27, Section 4.1]) satisfying supp T ∈ T almost surely. It is +convenient to identify the random point process T with its support. A random tessellation is stationary, +provided that its distribution is invariant under all shifts in Rd and isotropic if its distribution is invariant +under all rotations in Rd. For a stationary random tessellation T and k ∈ {0, 1, . . . , d − 1} we define the +stationary particle process T (k) := � +F∈Fk(T ) δF of k-dimensional polytopes, which is referred to as the +process of k-faces. +Next, we recall the concept of a typical cell (and a typical k-face) of a stationary random tessellation +T ; see [27, Section 4.1,4.2], [27, page 450], [26, Section 4.3] for more details. Let C′ be the space of +non-empty compact subsets of Rd endowed with the Hausdorff metric. A centre function is a Borel +function z : C′ → Rd such that z(C + m) = z(C) + m for all C ∈ C′ and m ∈ Rd. The intensity of +k-faces of T is defined by +γk(T ) := E +� +F∈Fk(T ) +1(z(F) ∈ [0, 1]d), +k = 0, . . . , d. +These quantities are known to be independent of the choice of the centre function z. Assuming that +γk(T ) ∈ (0, ∞), the typical k-face of T with respect to the centre function z is the k-dimensional +random polytope whose distribution is given by +Pz +T ,k( · ) := +1 +γk(T ) E +� +F∈Fk(T ) +1(F − z(F) ∈ · )1(z(F) ∈ [0, 1]d). +In particular, for k = d we get the concept of the typical cell of T . It should be noted that translation- +invariant characteristics of the distribution of the typical k-face do not depend on the choice of z. More +precisely, if z and z′ are two centre functions, then Pz′ +T ,k is the push-forward of Pz +T ,k under the map +F �→ F − z′(F). +3 +Construction of β-, β′- and Gaussian-Voronoi tessellations +3.1 +General Laguerre tessellations +In this section we only briefly recall some facts about Laguerre tessellations and refer the reader to [7, +Sections 3.2–3.4] and [8, Sections 2.3 and 3.1] for further details. +We start by defining a general Laguerre tessellation. Given two points v, w ∈ Rd and h ∈ R we define +the power of w with respect to the pair (v, h) as +pow(w, (v, h)) := ∥w − v∥2 + h. +In this situation, h is referred to as the weight (or height) of the point v. Let X be a countable set of +marked points of the form (v, h) ∈ Rd × R. Then the Laguerre cell of (v, h) ∈ X is the set +C((v, h), X) := {w ∈ Rd : pow(w, (v, h)) ≤ pow(w, (v′, h′)) for all (v′, h′) ∈ X}. +The point v is called the nucleus of the cell C((v, h), X). Note that a Laguerre cell may be empty and +even if it is non-empty, it does not need to contain its nucleus. The collection of all non-empty Laguerre +cells of X is called the Laguerre diagram: +L(X) := {C((v, h), X): (v, h) ∈ X, C((v, h), X) ̸= ∅}. +3 + +In the special case when the heights h of all points are the same (say, h0 ∈ R) the above definition leads +to the classical Voronoi cell. More precisely, let Y be a countable set of points in Rd whose “marked” +version X is obtained by attaching a fixed weight h0 to each point. Then the Voronoi cell of v ∈ Y is +V (v, Y ) = C((v, h0), X) = {w ∈ Rd : ∥w − v∥ ≤ ∥w − v′∥ for all v′ ∈ Y }. +The collection of the Voronoi cells of all v ∈ Y is called the Voronoi diagram V(Y ). +It should +be mentioned that a Laguerre diagram is not necessarily a tessellation in Rd, at least as long as no +additional assumptions on the geometric properties of the set X are imposed. Such assumptions have +been described in detail in [18, 19, 25]. In the present article we are interested in random tessellations +built on Poisson point processes. More precisely, we consider a Poisson point process ξ in Rd × E, where +E ⊂ R is a Borel set (an interval), and the corresponding Laguerre diagram L(ξ). Lemmas 1 and 2 in +[7] (see also [8, Lemma 2.1]) provide sufficient conditions on ξ which ensure that, almost surely, L(ξ) is a +stationary random face-to-face normal tessellation in Rd. In the following we work under these conditions +and remark that they are automatically satisfied in the three cases we consider from Section 3.3 on. +3.2 +Laguerre tessellations via paraboloid growth processes +An alternative approach to the construction of Laguerre diagrams uses so-called paraboloid growth pro- +cesses with overlaps (or simply paraboloid growth process), which were first introduced in [1, 28] in order +to study the asymptotic geometry of random polytopes; see also [2, 3, 4, 5]. In this section we briefly +describe this rather useful construction and refer for more details to [8, Section 3.1]. Let +Π±,x := {(v′, h′) ∈ Rd × R: h′ = ±∥v′ − v∥2 + h} +be the upward (+) and downward (−) standard paraboloids with apex x := (v, h) ∈ Rd × R, denoted +as apex Π±,x := x. In case (v, h) = (0, 0) we simply write Π± = Π±,(0,0). Given a set A ⊂ Rd × R we put +A↓ : = {(v, h′) ∈ Rd × R: (v, h) ∈ A for some h ≥ h′}, +A↑ : = {(v, h′) ∈ Rd × R: (v, h) ∈ A for some h ≤ h′}. +Following the definition from [1], for a given Poisson point process ξ in Rd × R, we introduce the +paraboloid growth process Ψ(ξ): +Ψ(ξ) := +� +x∈ξ +Π↑ ++,x. +It should be noted that, in typical situations, the majority of paraboloids will be completely covered by +other paraboloids, implying that they do not "contribute" to the model and can thus be omitted without +loosing any information about the set Ψ(ξ). This leads to the definition of extreme points. A point x ∈ ξ +is called extreme in the paraboloid growth process Ψ(ξ) if and only if its associated paraboloid is not +fully covered by the paraboloids associated with other points of ξ, i.e., if +Π↑ ++,x ̸⊂ +� +y∈ξ,y̸=x +Π↑ ++,y. +We denote by ext(Ψ(ξ)) the set of all extreme points of the paraboloid growth process Ψ(ξ). Using the +paraboloid growth process we can construct a random diagram in Rd. Given a point x = (v, h) ∈ ξ define +the Ψ-cell of x as +CΨ(x, ξ) := +� +{w ∈ Rd : +� +(w, 0)↑ ∪ (w, 0)↓� +∩ bd Ψ(ξ) ∈ Π+,x}, +if x ∈ ext(Ψ(ξ)), +∅, +otherwise, +where bd A denotes the boundary of a set A. +In other words, w belongs to CΨ(x, ξ) if and only if +∥w − v∥2 + h ≤ ∥w − v′∥ + h′ for all (v′, h′) ∈ ξ. Thus, the Ψ-cell of an extreme point x of the paraboloid +growth process Ψ(ξ) is non-empty and coincides with the Laguerre cell C(x, ξ). Next, we construct the +diagram LΨ(ξ) as the collection of all non-empty Ψ-cells: +LΨ(ξ) := {CΨ(x, ξ): CΨ(x, ξ) ̸= ∅} = {CΨ(x, ξ): x ∈ ext(Ψ(ξ))}. +We directly have that LΨ(ξ) = L(ξ). +4 + +Figure 3.1: Left panel: Simulation of a β-Voronoi tessellation in R2 with β = 5. Middle panel: Simulation +of a β′-Voronoi tessellation in R2 with β = 2.5. Right panel: Simulation of a Gaussian-Voronoi tessellation +in R2. +3.3 +Three families of random tessellations +In this article we consider random tessellations in Rd build on the following three families of Poisson +point processes. For β > −1 and 0 < γ < ∞ we consider a Poisson point process ηd,β,γ in Rd × [0, +∞) +whose intensity measure has density +(v, h) �→ γ cd+1,βhβ, +cd+1,β := Γ +� d+1 +2 ++ β + 1 +� +π +d+1 +2 Γ(β + 1) +, +(3.1) +with respect to the Lebesgue measure on Rd × [0, +∞). Further, for β > d +2 + 1 and 0 < γ < ∞ we +consider a Poisson point process η′ +d,β,γ in Rd × (−∞, 0) with intensity measure having density +(v, h) �→ γ c′ +d+1,β(−h)−β, +c′ +d+1,β := +Γ (β) +π +d+1 +2 Γ(β − d+1 +2 ) +, +(3.2) +with respect to the Lebesgue measure on Rd × (−∞, 0). The constants cd+1,β and c′ +d+1,β in the above +definitions are introduced for convenience. For example, they make the statement of Theorem 4.1 below +more transparent. Finally, for λ > 0 and 0 < γ < ∞ we consider a Poisson point process ζd,λ,γ in Rd × R +whose intensity measure has density +(v, h) �→ γ eλh, +with respect to the Lebesgue measure on Rd × R. It was shown in [7, Lemma 3] and in [8, Section +3.3] that the Poisson point processes ηd,β,γ and ζd,λ,γ satisfy the sufficient conditions of Lemma 1 and +Lemma 2 in [7] and, hence, the corresponding Laguerre diagrams Vd,β,γ := L(ηd,β,γ), V′ +d,β,γ := L(η′ +d,β,γ) +and Gd,λ := L(ζd,λ,γ) are stationary random normal tessellations in Rd, which are called β-Voronoi, +β′-Voronoi and Gaussian-Voronoi tessellations, respectively. These tessellations have been studied +in [10, 7, 8, 9], where they were considered in Rd−1 instead of Rd. Simulations of these tessellations in +the plane are shown in Figure 3.1. Note that although the point process η′ +d,β,γ is well-defined in the range +β > d+1 +2 , the corresponding β′-Voronoi tessellation exists in the smaller range β > d +2 + 1 only, see [7, +Lemma 3]. +Remark 3.1. Note that changing the parameter γ amounts to shifting the Poisson point process ζd,λ,γ +along the height coordinate h. In particular, the distribution of the point process Gd,λ does not depend +on the choice of γ, which is reflected in our notation. +Remark 3.2. It will be convenient to extend the above definition of the β-Voronoi tessellation to the +case β = −1 (with arbitrary γ > 0) by defining Vd,−1,γ := Wd,r(d)γ to be the classical Poisson-Voronoi +5 + +tessellation constructed on the homogeneous Poisson point process on Rd with constant intensity r(d)γ, +where +r(d) := Γ +�d + 1 +2 +� +π− d+1 +2 . +A justification for this definition is given by the following proposition. For the necessary background on +point processes and weak convergence we refer to [24, Chapter 3]. +Proposition 3.3. As β ↓ −1, the Poisson process ηd,β,γ converges, weakly on the space of locally finite +integer-valued measures on Rd × [0, ∞), to the Poisson point process whose intensity measure is the +Lebesgue measure on Rd × {0} times r(d)γ. +Proof. Write β = −1 + ε with ε ↓ 0. Then the constant appearing in (3.1) satisfies +cd+1,β = Γ +� d+1 +2 ++ ε +� +π +d+1 +2 Γ(ε) += Γ +� d+1 +2 +� +π +d+1 +2 +ε(1 + o(1)), +as ε ↓ 0. It follows that for every c > 0 and every bounded Borel set B ⊂ Rd the number of points of the +Poisson point process ηd,β,γ appearing in B × [0, c] is Poisson-distributed with expectation +γλd(B)cd+1,β +� c +0 +h−1+εdh −→ +ε↓0 Γ +�d + 1 +2 +� +π− d+1 +2 γλd(B) = r(d)γλd(B), +where λd denotes the d-dimensional Lebesgue measure. Since the right-hand side does not depend on c, +it follows that, for every 0 < c1 < c2, the expected number of points of ηd,β,γ in B × [c1, c2] converges +to 0, as ε ↓ 0. Hence, the intensity measure of ηd,β,γ converges as β ↓ −1 to the Lebesgue measure +on Rd × {0} times r(d)γ, vaguely on the space Rd × [0, ∞). Then, the claim of the proposition follows +from [16, Theorem 16.16 (iv)] or [24, Propositions 3.6 and 3.19]. +4 +Affine sections of β, β′- and Gaussian-Voronoi tessellations +In this section we study the intersection of the d-dimensional random tessellations Vd,β,γ, V′ +d,β,γ and Gd,λ +with an affine subspace L ⊂ Rd of dimension ℓ ∈ {1, . . . , d − 1}. By stationarity and isotropy of these +tessellations, we may and will assume without loss of generality that L = Rℓ is the linear subspace of Rd +spanned by the first ℓ vectors of the standard orthonormal basis of Rd. The intersection of the tessellation +Vd,β,γ with Rℓ will be denoted by Vd,β,γ ∩ Rℓ. Similar convention is used for the tessellations V′ +d,β,γ and +Gd,λ. The following theorem identifies the distribution of Vd,β,γ ∩ Rℓ, V′ +d,β,γ ∩ Rℓ and Gd,λ ∩ Rℓ. +Theorem 4.1. Fix integers d ≥ 2 and 1 ≤ ℓ ≤ d − 1. +(i) For any β ≥ −1 and γ > 0, Vd,β,γ ∩ Rℓ has the same distribution as Vℓ,β+ d−ℓ +2 ,γ. +(ii) For any β > d +2 + 1 and γ > 0, V′ +d,β,γ ∩ Rℓ has the same distribution as V′ +ℓ,β− d−ℓ +2 ,γ. +(iii) For any λ > 0, Gd,λ ∩ Rℓ has the same distribution as Gℓ,λ. +Before we move on to the proof of Theorem 4.1 we would like to highlight the following special case +which deals with sections of the classical Poisson-Voronoi tessellation. Together with the results we obtain +below, this fully answers and resolves the problems raised in [6, 20, 21, 29]. +Corollary 4.2. Fix integers d ≥ 2 and 1 ≤ ℓ ≤ d − 1. Then for any ρ > 0 the intersection of the +d-dimensional Poisson-Voronoi tessellation Wd,ρ of intensity ρ with Rℓ has the same distribution as +V +ℓ, d−ℓ +2 −1,π +d+1 +2 +ρ/Γ( d+1 +2 ). +6 + +Proof of Theorem 4.1. Let us first consider the case ℓ = d−1 meaning that we intersect with a hyperplane. +Let ξ be one of the Poisson point processes ηd,β,γ, η′ +d,β,γ or ζd,λ,γ. The atoms of ξ live in the space +Rd+1 = Rd × R; a generic point in this space is denoted by (v, h) with v = (v1, . . . , vd) ∈ Rd being the +spatial coordinate and h ∈ R being the height coordinate. The Laguerre tessellation L(ξ) lives in the +space Rd defined by the equation h = 0. The linear hyperplane L ≡ Rd−1 ⊂ Rd ⊂ Rd+1 with which our +tessellations are intersected is given by the equations {vd = 0, h = 0}. +In order to prove the statement we will use the representation of the Laguerre tessellation L(ξ) in +terms of the paraboloid growth process Ψ(ξ) as described in Section 3.2. We extend the hyperplane +L ⊂ Rd by adding the height coordinate, namely we consider +L′ := {(v, h) = (v1, . . . , vd, h) ∈ Rd × R: vd = 0}. +For every point x = (v, h) ∈ Rd+1, the intersection of the d-dimensional paraboloid Π+,x with L′ is a +(d − 1)-dimensional upward paraboloid Π+,x′ ∩ L′ in L′ with apex given by +x′ = f(v, h) := (v1, . . . , vd−1, 0, h + v2 +d) ∈ L′. +If x runs through all atoms of ξ, then x′ runs through all atoms of the point process f(ξ) := {f(x): x ∈ ξ} +on L′, which is also a Poisson point process by the mapping theorem; see [17, Theorem 5.1]. +The +intersection of the Laguerre tessellation L(ξ) with L coincides with the Laguerre tessellation generated +by the point process f(ξ) (within L). +Now, we are going to identify the intensity measure µ of the Poisson point process f(ξ). To this end, +we consider the Poisson point processes ηd,β,γ, η′ +d,β,γ and ζd,λ,γ separately. +Case (i). +Let first β > −1. By the mapping theorem for Poisson point processes [17, Theorem 5.1], +f(ηd,β,γ) := {f(x): x ∈ ηd,β,γ} is a Poisson point process in Rd−1×[0, +∞) ⊂ L′. To compute its intensity +measure µ, we take some Borel set B ⊂ L, any s > 0 and observe that an atom (v, h) of ηd,β,γ is mapped +by f to B × [0, s] if and only if (v1, . . . , vd−1) ∈ B and h + v2 +d ≤ s. The latter condition means that h ≤ s +and |vd| ≤ +√ +s − h. It follows that the intensity measure µ satisfies +µ(B × [0, s]) = γ cd+1,β +� +Rd +� ∞ +0 +hβ1(f(v, h) ∈ B × [0, s]) dhdv += γ cd+1,β +� +Rd +� s +0 +hβ1(v ∈ B × [− +√ +s − h, +√ +s − h]) dhdv += 2γ cd+1,β λd−1(B) +� s +0 +hβ√ +s − h dh += γ Γ( d+1 +2 ++ β + 1) +π +d +2 Γ(β + 3 +2) +λd−1(B) sβ+ 3 +2 +β + 3 +2 +. +(4.1) +In the case β = −1, we let ηd,−1,γ be the Poisson point process on Rd × {0} (which is considered as a +subset of Rd+1) whose intensity with respect to the Lebesgue measure on Rd ×{0} is constant and equals +Γ( d+1 +2 )π− d+1 +2 γ. Thus, the heights of all points in ηd,−1,γ are 0. Then, the Laguerre tessellation generated +by ηd,−1,γ on Rd coincides with Vd,−1,γ by our convention described in Remark 3.2. To compute the +intensity measure µ of the Poisson point process f(ηd,−1,γ), we take some Borel set B ⊂ L, any s > 0 +and observe that an atom (v, 0) of ηd,−1,γ is mapped by f to B × [0, s] if and only if (v1, . . . , vd−1) ∈ B +and v2 +d ≤ s. It follows that the intensity measure µ satisfies +µ(B × [0, s]) = γ Γ( d+1 +2 ) +π +d+1 +2 +� +Rd 1(f(v, 0) ∈ B × [0, s]) dv += γ 2Γ( d+1 +2 ) +π +d+1 +2 +λd−1(B)√s. +(4.2) +7 + +By differentiating (4.1) and (4.2) with respect to s, it follows that for all β ≥ −1, the intensity +measure of f(ηd,β,γ) has density +(v′, 0, h) �→ γ cd,β+ 1 +2 hβ+ 1 +2 , +v′ ∈ Rd−1, h > 0, +with respect to the Lebesgue measure on L × [0, +∞). Consequently, the Laguerre tessellation generated +by f(ηd,β,γ) within L has the same distribution as Vd−1,β+ 1 +2 ,γ. +Case (ii). +Next, we deal with f(η′ +d,β,γ) := {f(x): x ∈ η′ +d,β,γ}. Let us first consider only those points +of f(η′ +d,β,γ) that have negative height and compute the intensity measure µ of these points. The points +with positive height coordinate have no influence on the resulting tessellation, as we will argue below. +To determine µ, we take some Borel set B ⊂ L, any s < 0 and observe that an atom (v, −g) of η′ +d,β,γ +(with g > 0) is mapped by f to B × (−∞, s] if and only if (v1, . . . , vd−1) ∈ B and v2 +d ≤ s + g. The latter +condition means that g ≥ −s and |vd| ≤ √s + g. It follows that the intensity measure µ satisfies +µ(B × (−∞, s]) = γ c′ +d+1,β +� +Rd +� ∞ +0 +g−β1(f(v, −g) ∈ B × [−s, ∞)) dgdv += γ c′ +d+1,β +� +Rd +� ∞ +−s +g−β1(v ∈ B × [−√s + g, √s + g]) dgdv += 2γ c′ +d+1,β λd−1(B) +� ∞ +−s +g−β√g + s dg += γ +Γ(β − 1 +2) +π +d +2 Γ(β − d+1 +2 ) +λd−1(B)(−s)−β+ 3 +2 +β − 3 +2 +. +Differentiating, we conclude that the intensity measure µ has density +(v′, 0, h) �→ γ c′ +d,β+ 1 +2 (−h)−β+ 1 +2 , +v′ ∈ Rd−1, h < 0, +with respect to the Lebesgue measure on L × (−∞, 0). So, by the mapping theorem [17, Theorem 5.1], +the restriction of f(η′ +d,β,γ) to L × (−∞, 0) is a Poisson point process with the same intensity measure +as η′ +d−1,β− 1 +2 ,γ. The Laguerre tessellation generated by this Poisson point process within L ≡ Rd−1 has +the same distribution as V′ +d−1,β− 1 +2 ,γ. +It remains to observe that adding the points of f(η′ +d,β,γ) with +positive height coordinate does not change the Laguerre tessellation. Indeed, every point in Rd−1 × {0} +is an accumulation point of η′ +d−1,β− 1 +2 ,γ, hence the lower boundary of the paraboloid growth process +Ψ(η′ +d−1,β− 1 +2 ,γ) is contained in Rd−1×(−∞, 0] and points with positive height coordinate have no influence +on the tessellation. +Case (iii). +Finally, we consider f(ζd,λ,γ), which is a Poisson point process in L′. +To compute its +intensity measure µ, we take some Borel set B ⊂ L, any s ∈ R and observe that an atom (v, h) of ζd,λ,γ +is mapped by f to B × (−∞, s] if and only if (v1, . . . , vd−1) ∈ B and h + v2 +d ≤ s. It follows that +µ(B × (−∞, s]) = γ +� +Rd +� +R +eλh1(f(v, h) ∈ B × (−∞, s)) dhdv += 2γ λd−1(B) +� s +−∞ +ehλ√ +s − h dh += γ λd−1(B) +√πeλs +λ3/2 . +Thus, the density of the intensity measure of f(ζd,λ,γ) is given by +(v′, 0, h) �→ γ +� +π/λ eλh, +v′ ∈ Rd−1, h ∈ R. +8 + +Hence, the Laguerre tessellation generated by f(ζd,λ,γ) within L has the same distribution as Gd−1,λ +(recall that the parameter γ does not influence the distribution of the Gaussian-Voronoi tessellation). +This proves the claim for ℓ = d − 1. For general 1 ≤ ℓ ≤ d − 2 we can inductively repeat the above +argument d − ℓ times. +5 +Sectional Poisson-Voronoi tessellations +5.1 +Face intensities and the expected volume of the typical cell +As we have shown in Corollary 4.2, the sectional Poisson-Voronoi tessellation Wd,ρ ∩ Rℓ can be identified +with a suitable β-Voronoi tessellation. This makes it possible to compute explicitly several functionals +of the sectional Poisson-Voronoi tessellation. We begin with a formula for the intensity of j-dimensional +faces. This quantity, denoted by γj(Wd,ρ ∩ Rℓ), has been defined in Section 2. +Theorem 5.1. Let d ≥ 2, 1 ≤ ℓ ≤ d − 1 and 0 ≤ j ≤ ℓ. Then, for any ρ > 0, we have +γj(Wd,ρ ∩ Rℓ) = ρ +ℓ +d 2Jℓ+1,ℓ−j+1( d−ℓ−1 +2 +)π +ℓ +2 +d(ℓ + 1) +Γ( (ℓ+1)(d−1) +2 ++ 1)Γ(ℓ + 1 − ℓ +d)Γ( d +2 + 1)ℓ+1− ℓ +d +Γ( (ℓ+1)(d−1)+1 +2 +)Γ( ℓ+2 +2 )Γ( d+1 +2 )ℓ+1 +, +wher +Jℓ+1,ℓ−j+1 +�d − ℓ − 1 +2 +� += +�ℓ + 1 +j +� +Γ( (d−1)(ℓ+1)+3 +2 +) +√π Γ( (d−1)(ℓ+1) +2 ++ 1) +� +∞ +−∞ +(cosh u)−(d−1)(ℓ+1)−2 +× +� +1 +2 + i Γ( d+1 +2 ) +√π Γ( d +2) +� u +0 +(cosh v)d−1dv +�j +du +(5.1) +and i = √−1 stands for the imaginary unit. +Proof. By Corollary 4.2 we have γj(Wd,ρ ∩ Rℓ) = γj(Vℓ, d−ℓ−2 +2 +,r), where r = r(d)ρ = π +d+1 +2 ρ/Γ( d+1 +2 ). The +formula for γj(Vℓ, d−ℓ−2 +2 +,r) can be obtained by combining [7, Theorem 6] (which we apply with parameters +d := ℓ + 1, β := d−ℓ−2 +2 +and j := ℓ − j) with [7, Proposition 3] (with parameters d := ℓ + 1, k := ℓ + 1 − j). +Note that [7, Theorem 6] refers to [7, Theorem 2] which has to be applied with parameters d := ℓ + 1, +s := 1, ν := 0, γ := r. +Remark 5.2. The quantities Jd+1,k(β) for general d ≥ 0, k ∈ {1, . . . , d + 1} and β ≥ −1 have a natural +geometric meaning. Namely, Jd+1,k(β) is equal to the expected sum of internal angles at its k-vertex +faces of a random beta-simplex, which is defined as the convex hull of d + 1 independent random points +with density proportional to (1 − ∥x∥2)β in the d-dimensional unit ball, see [7, Section 6.1] and [11] for +details. From this interpretation it directly follows that +J1,1(β) = J2,1(β) = J2,2(β) = 1, +J3,1(β) = 1 +2, +J3,2(β) = 3 +2, +J3,3(β) = 1, +Jℓ+1,ℓ(β) = ℓ + 1 +2 +, +Jℓ+1,ℓ+1(β) = 1, +for any ℓ ≥ 1 and β ≥ −1. Moreover, if we denote by Σd a regular d-dimensional simplex and by σk(Σd) +the internal angle sum at its k-vertex faces, then +Jd+1,k(∞) := lim +β→∞ Jd+1,k(β) = σk(Σd) +according to [7, Proposition 2]. +9 + +d = 2 +d = 3 +d = 4 +d = 5 +d = 6 +ℓ = 1 +π +4√ρ +3 +√ +3 +3 +√4πρ·Γ +� 5 +3 +� +15π3/2 +64· 4 +√8ρ·Γ +� 3 +4 +� +7 5 +√ +5 +3· 5 +� +648π2ρ·Γ +� 9 +5 +� +2835· 6 +√ +3·π3/2 +16384· 6 +√32ρ·Γ +� 5 +6 +� +ℓ = 2 +− +5· 3 +√ +4 +3 +� +3π5ρ2·Γ +� 7 +3 +� +24 +√ +2 +35√πρ +77·24/5 +5·153/5·π9/5·ρ2/5·Γ +� 13 +5 +� +50· 3 +√ +6 +143· 3 +√ρ·Γ +� 8 +3 +� +ℓ = 3 +− +− +280665·π3/2 +821248· 4 +√ +2·ρ3/4·Γ +� 13 +4 +� +56·153/5· 5√ +2/π +187·ρ3/5·Γ +� 17 +5 +� +17320875·√ +3/2·π +176201728·√ρ +ℓ = 4 +− +− +− +144848704·23/5 +156/5π8/5� +1692197−141120π2 +� +ρ4/5Γ +� 21 +5 +� +15·62/3 +13·ρ2/3·Γ +� 13 +3 +� +ℓ = 5 +− +− +− +− +6823504578515625·35/6·π3/2 +4912276871446528· 6 +√ +2·ρ5/6·Γ +� 31 +6 +� +Table 5.1: E vol(Zd,ℓ,ρ) for small values of d and ℓ. +As a corollary of Theorem 5.1 we can compute the expected volume of the typical cell Zd,ℓ,ρ of the +sectional Poisson-Voronoi tessellation Wd,ρ ∩ Rℓ. Note that the volume does not change under shifts, +which is why it does not matter how to choose the centre function in the definition of the typical cell. +For ℓ = 1 and ℓ = 2 this quantity has been studied by Miles [20] who showed that +E vol(Zd,1,ρ) = ρ− 1 +d +Γ(d − 1 +2)Γ( d+1 +2 )2 +(d − 1)!Γ(2 − 1 +d)Γ( d +2)Γ( d +2 + 1)1− 1 +d +, +E vol(Zd,2,ρ) = ρ− 2 +d +3d · Γ( 3d +2 − 1)Γ( d+1 +2 )3 +πΓ( 3d−1 +2 +)Γ(3 − 2 +d)Γ( d +2 + 1)3− 2 +d +, +see Formulas (4.1) and (4.4) in [20]. Our result generalizes this to arbitrary 1 ≤ ℓ ≤ d − 1; special cases +with small values of d and ℓ are summarized in Table 5.1. +Corollary 5.3. Let ρ > 0. Then, for any d ≥ 2 and 1 ≤ ℓ ≤ d − 1 we have +E vol(Zd,ℓ,ρ) = ρ− ℓ +d +d(ℓ + 1) +2Jℓ+1,1( d−ℓ−1 +2 +)π +ℓ +2 +Γ( (ℓ+1)(d−1)+1 +2 +) +Γ( (ℓ+1)(d−1) +2 ++ 1) +Γ( ℓ+2 +2 ) +Γ(ℓ + 1 − ℓ +d) +Γ( d+1 +2 )ℓ+1 +Γ( d +2 + 1)ℓ+1− ℓ +d +. +Proof. It is known from [27, Equation (10.4)] that E vol(Zd,ℓ,ρ) = γℓ(Wd,ρ ∩ Rℓ)−1. The right-hand side +is known from Theorem 5.1. +Remark 5.4. Corollary 5.3 stays true for ℓ = d where it gives the expected volume of a typical Poisson- +Voronoi cell to be E vol(Zd,d,ρ) = 1/ρ. The quantity Jℓ+1,1(− 1 +2) cancels with the Gamma-factors by +the formula given in [12, Theorem 3.9] and the Legendre duplication formula for the Gamma function. +Theorem 5.1 also stays true for ℓ = d and gives the intensity of j-faces in the Poisson-Voronoi tessellation; +see [11, Remark 2.10] for another formula. +In the next result we compute the limit of the intensity of j-dimensional faces in the d-dimensional +Poisson-Voronoi tessellation intersected with Rℓ in the regime when d → ∞ while ℓ ∈ N stays fixed. +Proposition 5.5. Fix some ℓ ∈ N and 0 ≤ j ≤ ℓ. +Let (ρd)d∈N be a positive sequence such that +lim +d→∞(ρd)1/d = κ > 0. Then, +lim +d→∞ γj(Wd,ρd ∩ Rℓ) = Jℓ+1,ℓ−j+1(∞)(κ2πe) +ℓ +2 +√ +ℓ + 1 +2(ℓ − 1)! +Γ( ℓ +2) +, +where Jℓ+1,ℓ−j+1(∞) is the sum of angles at (ℓ − j)-dimensional faces of a regular ℓ-dimensional simplex +Σℓ; see Remark 5.2. +Remark 5.6. For example, we may take ρd = ρ > 0 to be constant, in which case κ = 1. +10 + +Proof of Proposition 5.5. By Theorem 5.1, +lim +d→∞ γj(Wd,ρd ∩ Rℓ) = lim +d→∞ +2Jℓ+1,ℓ−j+1( d−ℓ−1 +2 +)π +ℓ +2 +d(ℓ + 1)ρ +− ℓ +d +d +Γ( (ℓ+1)(d−1) +2 ++ 1) +Γ( (ℓ+1)(d−1)+1 +2 +) +Γ(ℓ + 1 − ℓ +d) +Γ( ℓ+2 +2 ) +Γ( d +2 + 1)ℓ+1− ℓ +d +Γ( d+1 +2 )ℓ+1 += 2Jℓ+1,ℓ−j+1(∞)(κ2π) +ℓ +2 +ℓ + 1 +2Γ(ℓ) +Γ( ℓ +2) lim +d→∞ +1 +d +Γ( (ℓ+1)(d−1) +2 ++ 1) +Γ( (ℓ+1)(d−1)+1 +2 +) +Γ( d +2 + 1)ℓ+1− ℓ +d +Γ( d+1 +2 )ℓ+1 +. +By Stirling’s formula for the Gamma function, Γ(z) = +� +2π/z(z/e)z(1 + O(z−1)). Since lim +n→∞ +Γ(n)nz +Γ(n+z) = 1 +we get +lim +d→∞ γj(Wd,ρd ∩ Rℓ) = Jℓ+1,ℓ−j+1(∞)(κ2πe) +ℓ +2 +√ +ℓ + 1 +2Γ(ℓ) +Γ( ℓ +2) lim +d→∞ +� d +4π +� ℓ +2d = Jℓ+1,ℓ−j+1(∞)(κ2πe) +ℓ +2 +√ +ℓ + 1 +2Γ(ℓ) +Γ( ℓ +2) . +This completes the argument. +We now study the asymptotic behaviour of the expected volume of the typical cell in the sections of +fixed dimension ℓ of a high-dimensional Poisson-Voronoi tessellation. +Corollary 5.7. Let (ρd)d∈N be a positive sequence such that lim +d→∞(ρd)1/d = κ > 0. Then, for every ℓ ∈ N, +lim +d→∞ E vol(Zd,ℓ,ρd) = +√ +ℓ + 1 +Jℓ+1,1(∞)(κ2πe) +ℓ +2 +Γ( ℓ +2) +2(ℓ − 1)!, +where Jℓ+1,1(∞) is the sum of solid angles of the regular ℓ-dimensional simplex Σℓ at its vertices; see +Remark 5.2. +Proof. This follows from the fact that E vol(Zd,ℓ,ρd) = γℓ(Wd,ρd ∩ Rℓ)−1 (see [27, Equation (10.4)]) by +applying Proposition 5.5 with j = ℓ. +In the special cases ℓ = 1, 2 and for every fixed ρ > 0, Corollary 5.7 combined with the results of +Remark 5.2 yields, for any constant ρ > 0, the limit relations +lim +d→∞ E vol(Zd,1,ρ) = +1 +√ +2e, +and +lim +d→∞ E vol(Zd,2,ρ) = +√ +3 +eπ , +which were already known from the work of Miles [20, pp. 318, 319]. Moreover, for ℓ = 3 we get +lim +d→∞ E vol(Zd,3,ρ) = +� +4e3/2(3 arccos(1/3) − π) +�−1, +for example. +This follows from the fact that the solid angle at a vertex of a regular tetrahedron is +θ := +1 +4π(3 arccos(1/3) − π), implying that J4,1(∞) = σ1(Σ3) = 4θ = 3 +π arccos(1/3) − 1. +5.2 +Expected intrinsic volumes and f-vectors of typical k-faces +Together with the volume of the typical cell Zd,ℓ,ρ of the sectional Poisson-Voronoi tessellation we can +consider its intrinsic volumes. We recall from [27, p. 222] that the intrinsic volume Vm(K) of order +0 ≤ m ≤ d of a compact convex set K ⊂ Rd may be defined as +Vm(K) := +d! +m!(d − m)! +Γ( m +2 + 1)Γ( d−m +2 ++ 1) +Γ( d +2 + 1) +Eλm(K|L), +where L ⊂ Rd is a uniformly distributed random subspace of dimension m, K|L denotes the orthogonal +projection of K onto L and λm(K|L) its m-dimensional Lebesgue measure. In addition, instead of the +11 + +typical sectional cell we can consider for 1 ≤ k ≤ ℓ the typical k-face Z(k) +d,ℓ,ρ of the sectional Poisson- +Voronoi tessellation Wd,ρ ∩ Rℓ, see Section 2 or [27, page 450] for a formal definition. For example, for +k = ℓ we get back the typical cell, for k = ℓ − 1 the typical facet and for k = 1 the typical edge of +the sectional tessellation. Using the results from [27, pages 466-467] for general stationary and isotropic +random tessellations we conclude (by combining the last two formulas there) that +EVj(Z(k) +d,ℓ,ρ) = +ℓ! +j!(ℓ − j!) +Γ( j +2 + 1)Γ( ℓ−j +2 + 1) +Γ( ℓ +2 + 1) +γk−j(Wd,ρ ∩ Rℓ−j) +γk(Wd,ρ ∩ Rℓ) +, +(5.2) +where d ≥ 2, and 1 ≤ ℓ ≤ d − 1, 0 ≤ k ≤ ℓ and 0 ≤ j ≤ k. This expression can be made fully explicit in +view of Corollary 5.3: +EVj(Z(k) +d,ℓ,ρ) = +� +ρ +Γ( d +2 + 1) +�− j +d +(ℓ + 1)! +j!(ℓ − j + 1)! +Γ( j +2 + 1) +πj/2 +Jℓ−j+1,ℓ−k+1( d−ℓ+j−1 +2 +) +Jℓ+1,ℓ−k+1( d−ℓ−1 +2 +) +× Γ( (ℓ−j+1)(d−1) +2 ++ 1)Γ(ℓ − j + 1 − ℓ−j +d ) +Γ( (ℓ+1)(d−1) +2 ++ 1)Γ(ℓ + 1 − ℓ +d) +Γ( (ℓ+1)(d−1)+1 +2 +)Γ( d+1 +2 )j +Γ( (ℓ−j+1)(d−1)+1 +2 +)Γ( d +2 + 1)j . +(5.3) +For intersections of dimension ℓ = 2, Miles [20, Equations (4.4), (4.5) on p. 319] derived a formula for the +expected area and perimeter of the typical cell which are particular cases of the above formula. Using +Proposition 5.5 it is easy to derive the large d limit of (5.2). Namely, if lim +d→∞(ρd)1/d = κ > 0, then +lim +d→∞ EVj(Z(k) +d,ℓ,ρd) = +√ +ℓ + 1 +√ℓ − j + 1 · +Γ( j +2 + 1) +j!(κ2πe)j/2 · Jℓ−j+1,ℓ−k+1(∞) +Jℓ+1,ℓ−k+1(∞) . +For intersections of dimensions ℓ = 2 and 3 (and k = ℓ) we recover results of Miles [20, pp. 319, 320]. +Finally, we deal with the expected number of j-dimensional faces of the typical cell of the sectional +Voronoi tessellation Wd,ρ ∩ Rℓ, which we denote by Efj(Zd,ℓ,ρ), for ρ > 0, d ≥ 2, 1 ≤ ℓ ≤ d − 1 and +0 ≤ j ≤ ℓ−1. Using the fact that, with probability 1, each j-dimensional face of the sectional tessellation +is contained in the boundary of precisely ℓ − j + 1 of its cells (by normality of the tessellation), it follows +that +Efj(Zd,ℓ,ρ) = (ℓ − j + 1)γj(Wd,ρ ∩ Rℓ) +γℓ(Wd,ρ ∩ Rℓ). +We can now apply Corollary 5.3 to conclude that +Efj(Zd,ℓ,ρ) = (ℓ − j + 1)Jℓ+1,ℓ−j+1( d−ℓ−1 +2 +) +Jℓ+1,1( d−ℓ−1 +2 +) +, +independently of ρ. Clearly, Ef0(Zd,1,ρ) = 2 for any d ≥ 2. Also, Efj(Zd,2,ρ) = 6 for any d ≥ 2 and +0 ≤ j ≤ 1, since the sectional Voronoi tessellation is stationary and normal. Some non-trivial values for +space dimensions d = 4, 5, 6 are collected in Table 5.2. +Similarly, we can compute the expected number of j-dimensional faces of the typical k-dimensional +face Z(k) +d,ℓ,ρ of the sectional Poisson-Voronoi tessellation Wd,ρ ∩ Rℓ for d ≥ 2, 1 ≤ ℓ ≤ d − 1, 1 ≤ k ≤ ℓ and +0 ≤ j ≤ k − 1: +Efj(Z(k) +d,ℓ,ρ) = (k − j + 1)Jℓ+1,ℓ−j+1( d−ℓ−1 +2 +) +Jℓ+1,ℓ−k+1( d−ℓ−1 +2 +). +In the large d limit this becomes +lim +d→∞ Efj(Z(k) +d,ℓ,ρ) = (k − j + 1)Jℓ+1,ℓ−j+1(∞) +Jℓ+1,ℓ−k+1(∞). +Again, for ℓ = 2, 3 we recover results of Miles [20, p. 320]. +12 + +d = 4, ℓ = 3 +d = 5, ℓ = 3 +d = 5, ℓ = 4 +d = 6, ℓ = 3 +d = 6, ℓ = 4 +d = 6, ℓ = 5 +j = 0 +10 240 +401 +67 200π2 +26 741 +4 233 600π2 +1 692 197−141 120π2 +524 288 +21 509 +52 003 +400 +34 394 098 106 368 +37 477 698 299 +j = 1 +15 360 +401 +100 800π2 +26 741 +8 467 200π2 +1 692 197−141 120π2 +786 432 +21 509 +52 003 +200 +85 985 245 265 920 +37 477 698 299 +j = 2 +5 922 +401 +2 + 33 600π2 +26 741 +10 153 182+4 233 600π2 +1 692 197−141 120π2 +305 162 +21 509 +162 009 +1 000 +74 276 903 321 600 +37 477 698 299 +j = 3 +− +− +10 153 182 +1 692 197−141 120π2 +− +64 003 +2 000 +25 430 109 716 480 +37 477 698 299 +j = 4 +− +− +− +− +− +53 194 508 510 +707 126 383 +Table 5.2: Efj(Zd,ℓ,ρ) for small values of d, ℓ and j. +6 +Convergence to the Gaussian-Voronoi tessellation in high dimensions +In Sections 5.1 and 5.2 we computed explicitly several characteristics of the sectional Poisson-Voronoi +tessellation Wd,ρd∩Rℓ and the limits of these characteristics in the regime when d → ∞ and (ρd)1/d → κ > +0, while ℓ ∈ N stays fixed. It turns out that these limits coincide with the corresponding characteristics +of the tessellation Gℓ,λ with λ = κ2πe. For example, for the typical cell Z(Gℓ,λ) of the Gaussian-Voronoi +tessellation Gℓ,λ it is known from [8, Section 5] (where the special case λ = 1/2 has been considered) that +E vol(Z(Gℓ,λ)) = +√ +ℓ + 1 +Jℓ+1,1(∞)λ +ℓ +2 +Γ( ℓ +2) +2(ℓ − 1)!. +This formula coincides with the one obtained in Corollary 5.7 if we choose λ = κ2πe. In the next two +theorems we explain this and other similar coincidences by proving weak convergence of the corresponding +tessellations and the typical cells. +Theorem 6.1. Take any positive sequence (ρd)d∈N with limd→∞(ρd)1/d = κ > 0 and let ℓ ∈ N be +fixed. Then, as d → ∞, the sectional Poisson-Voronoi tessellation Wd,ρd ∩ Rℓ converges to Gℓ,κ2πe in the +following sense: It is possible to define all these random tessellations on the same probability space such +that for every ℓ-dimensional ball BR ⊂ Rℓ of radius R > 0 centred at the origin the probability that the +restrictions of Wd,ρd ∩ Rℓ and Gℓ,κ2πe to BR coincide, converges to 1, as d → ∞. +Remark 6.2. The skeleton of a (random) tessellation T is the (random) closed set skel(T ) = � +c∈T bd c, +where bd c denotes the topological boundary of the cell c. +The mode of convergence appearing in +Theorem 6.1 implies that the random closed set skel(Wd,ρd ∩ Rℓ) converges to the random closed set +skel(Gℓ,κ2πe) weakly as d → ∞; see [27, Chapter 2] for this concept. +Proof of Theorem 6.1. The proof of this theorem basically follows the same route as the proof of The- +orem 4.2 in [8], which is the reason why we leave out some details here. By Corollary 4.2, the sectional +tessellation Wd,ρd ∩ Rℓ has the same distribution as Vℓ,βd,γd with +βd = 1 +2(d − ℓ) − 1 +and +γd = π +d+1 +2 Γ +�d + 1 +2 +�−1 +ρd. +We will prove that, as d → ∞, the Poisson point processes ηℓ,βd,γd converge, after an appropriate vertical +shift, to ζℓ,κ2πe,1; see Section 3.3 for their definitions. The vertical shift Qd : Rℓ+1 → Rℓ+1 is given by +Qd(v, h) = (v, h − ad) with +ad = 1 +π +�πΓ( d−ℓ +2 ) +ρd +� +2 +d−ℓ−2 . +(6.1) +Note that applying such a vertical shift to a point process does not change the resulting Laguerre tessel- +lation since it amounts to shifting all paraboloids along the height coordinate. +13 + +In the following, we show that the intensity function of the Poisson point process ξℓ,d := Qd(ηℓ,βd,γd) +converges, as d → ∞, to the intensity function of ξℓ,∞ := ζℓ,κ2πe,1 uniformly on every compact subset of +Rℓ+1. Indeed, the intensity function of Qd(ηℓ,βd,γd) is given by +fd(v, h) = ρdπ +d−ℓ +2 +Γ( d−ℓ +2 )(h + ad) +d−ℓ−2 +2 +1 +� +h + ad > 0 +� += ρdπ +d−ℓ +2 +Γ( d−ℓ +2 )a +d−ℓ−2 +2 +d +� +1 + h +ad +� d−ℓ−2 +2 +1 +� +h + ad > 0 +� += +� +1 + h +ad +�ad· d−ℓ−2 +2ad 1 +� +h + ad > 0 +� +. +Stirling’s formula for the Gamma function and (6.1) yield +lim +d→∞ +d − ℓ − 2 +2ad += +1 +κ2π lim +d→∞ +d − ℓ + 2 +2Γ( d−ℓ +2 ) +2 +d−ℓ+2 += κ2πe lim +d→∞ +d − ℓ + 2 +(d − ℓ) +d−ℓ +d−ℓ+2 += κ2πe. +Note that, in particular, ad → ∞ as d → ∞. We conclude that limd→∞ fd(v, h) = eκ2πeh uniformly +as long as h stays bounded. By standard results [24, Propositions 3.6 and 3.19], this also implies weak +convergence of the corresponding Poisson point processes. +After we have shown the convergence of the point processes ξℓ,d to ξℓ,∞, as d → ∞, we explain the +procedure allowing to transfer this result to the convergence of the corresponding tessellations LΨ(ξℓ,d) +to LΨ(ξℓ,∞) as d → ∞. Note that LΨ(ξℓ,d) has the same distribution as Wd,ρd ∩ Rℓ and LΨ(ξℓ,∞) has the +same distribution as Gℓ,κ2πe (see Corollary 4.2 and Section 3.2). +We fix an ℓ-dimensional ball BR ⊂ Rℓ of radius R > 0 centred at the origin and for any ε > 0 we aim +to find a region K(R, ε) ⊂ Rℓ+1, independent of d, such that with probability at least 1−ε the restrictions +of the tessellations LΨ(ξℓ,d) and LΨ(ξℓ,∞) to BR are completely determined by the restrictions of the +point processes ξℓ,d and ξℓ,∞ to K(R, ε) for any d. To this end, we note that LΨ(ξℓ,d) may be regarded +as a vertical projection along the h-axis of the boundary of the corresponding paraboloid growth process +Ψ(ξℓ,d) (see Section 3.2). From this it follows that if the restrictions of the tessellations LΨ(ξℓ,d) and +LΨ(ξℓ,∞) to BR do not coincide, then the boundaries of the corresponding paraboloid hull processes +bd Ψ(ξℓ,d) and bd Ψ(ξℓ,∞) restricted to the cylinder BR × R do not coincide as well. The construction of +the region K(R, ε) can be now performed as follows. First, we consider the event E(T, r) that bd Ψ(ξℓ,d) +restricted to the cylinder BR ×R is completely determined by the restriction of bd Ψ(ξℓ,d) to the cylinder +BR+r ×(−∞, T] for some T, r > 0. By this we mean that for every paraboloid Π+,x with x ∈ ext(Ψ(ξℓ,d)) +the set Π+,x ∩ bd Ψ(ξℓ,d) either does not intersect BR × R or is included in BR+r × (−∞, T]. We have +that +1 − P(E(T, r)) ≤ c1(R + r)ℓ(ec2(4T−r2) + e−c3T c4), +r, d > c5, +(6.2) +where all constants c1, . . . , c5 are positive and independent of the parameters d, r and T. Since the proof +of this estimate follows exactly the same route as the proof of [8, Lemma 4.4] (estimate for T) and [8, +Lemma 4.5] (estimate for r), we decided to omit the technical details. In particular, (6.2) shows that +for any ε > 0 there is a choice of T0 := T(ε) and r0 := r(ε) such that P(E(T0, r0)) ≥ 1 − ε. The +same holds for Ψ(ξℓ,∞). Further, we note that if a paraboloid Π+,x with x ∈ ext(Ψ(ξℓ,d)) is such that +Π+,x ∩ bd Ψ(ξℓ,d) ⊂ BR+r0 × (−∞, T0], then +x ∈ {(v, h) ∈ Rℓ+1 : h ≤ T0, ∥v∥ ≤ R + r0 + +� +T0 − h} =: K(R, ε). +To complete the proof, it suffices to argue that there exists a coupling of ξℓ,d and ξℓ,∞ on a common +probability space such that the probability that the restrictions of these processes to the region K(R, ε) +do not coincide converges to 0, as d → ∞. In a suitable coupling, this probability is bounded above by a +constant multiple of the L1-norm of the difference of their intensity measures restricted to K(R, ε), see [23, +Theorem 3.2.2]. As we have shown above, the densities fd(v, h) converge to eκ2πeh as d → ∞ uniformly +14 + +on compact sets and hence pointwise. Also, by the inequality 1 + x ≤ ex, we have fd(v, h) ≤ ec6h for +some absolute constant c6 > 0. The fact that this upper bound is integrable over K(r, ε) has been shown +in [8, Equation (4.18)]. Thus, fd(v, h) → eκ2πeh with respect to the L1-norm on K(R, ε), which ensures +that the required coupling of the Poisson processes indeed exists. This completes the argument. +Our aim is now to prove the weak convergence of the typical cell of the sectional Poisson-Voronoi +tessellation to the typical cell of the Gaussian-Voronoi tessellation. Fix some ℓ ∈ N and let C be the +space of compact subsets of Rℓ endowed with the Hausdorff metric. Put C′ = C\{∅}. The typical cells +considered below are defined with respect to some fixed centre function z : C′ → Rℓ in the sense of +Section 2, additionally satisfying z(C) ∈ C for every C ∈ C′. +Theorem 6.3. Take any positive sequence (ρd)d∈N with limd→∞(ρd)1/d = κ > 0. Then, as d → ∞, the +distribution of the typical cell of the sectional Poisson-Voronoi tessellation Wd,ρd ∩ Rℓ converges to the +distribution of the typical cell of the Gaussian-Voronoi tessellation Gℓ,κ2πe weakly on C. +This theorem is a consequence of Theorem 6.1 and the following general result. +Proposition 6.4. Let (Tn)n∈N be a sequence of stationary random tessellations on Rℓ converging to a +stationary random tessellation T∞ on Rℓ in the following sense: All random tessellations are defined on +a common probability space and for every R > 0 we have lim +n→∞ P[An(R)] = 1, where An(R) is the event +that the restrictions of Tn and T∞ to [−R, R]ℓ coincide. More precisely, An(R) = A′ +n(R) ∩ A′′ +n(R) with +A′ +n(R) := {for every C ∈ Tn s.t. C ∩ [−R, R]ℓ ̸= ∅ there is C′ ∈ T∞ s.t. C ∩ [−R, R]ℓ = C′ ∩ [−R, R]ℓ}, +A′′ +n(R) := {for every C′ ∈ T∞ s.t. C′ ∩ [−R, R]ℓ ̸= ∅ there is C ∈ Tn s.t. C ∩ [−R, R]ℓ = C′ ∩ [−R, R]ℓ}. +Also, suppose that the cell intensity of Tn converges to that of T∞, that is lim +n→∞ γℓ(Tn) = γℓ(T∞), and that +all these intensities are finite. Then, the distribution of the typical cell of Tn converges to the distribution +of the typical cell of T∞ weakly on C: +Pz +Tn,ℓ −→ +n→∞Pz +T∞,ℓ, +weakly on C. +(6.3) +Proof. Let f : C → R be a bounded continuous function and recall that z(C) is the centre of a cell C. +Consider the random variables +ξn := +� +C∈Tn, z(C)∈[0,1]ℓ +f(C), +n ∈ N ∪ {∞}. +By definition, the distribution of the typical cell of Tn satisfies +� +C +f dPz +Tn,ℓ = +Eξn +γℓ(Tn), +n ∈ N ∪ {∞}. +Since lim +n→∞ γℓ(Tn) = γℓ(T∞), to prove (6.3) it suffices to check that +lim +n→∞ Eξn = Eξ∞. +(6.4) +In the following we shall define a “good” event Cn(R) on which ξn and ξ∞ are equal. For n ∈ N ∪ {∞} +and R > 3 consider the random event +Bn(R) := {∄C ∈ Tn : C ∩ [0, 1]ℓ ̸= ∅, C ̸⊂ [−R/2, R/2]ℓ}. +A cell C with the properties listed in this definition is called a “long cell” in the tessellation Tn. The +event Bn(R) occurs if there is no long cell. On the event An(R), each long cell C ∈ Tn corresponds to a +long cell C′ ∈ T∞ with the same restriction to [−R, R]ℓ. It follows that An(R) ∩ (Bn(R))c ⊂ (B∞(R))c. +15 + +Since all cells in T∞ are bounded almost surely, the maximal diameter of a cell in T∞ intersecting [0, 1]ℓ +is some almost surely finite random variable M. It follows that +lim +R→∞ sup +n∈N +P[An(R) ∩ (Bn(R))c] ≤ lim +R→∞ P[(B∞(R))c] ≤ lim +R→∞ P[M > (R/2) − 1] = 0. +Recall also that limn→∞ P[An(R)] = 1 for every fixed R > 0. Consider now the “good” event Cn(R) := +An(R) ∩ Bn(R) and the “bad” event Dn(R) := (Cn(R))c. It follows from the above that +lim +R→∞ lim sup +n→∞ P[Dn(R)] = 0. +(6.5) +Note that, on the event Cn(R), the sets {C ∈ Tn : z(C) ∈ [0, 1]ℓ} and {C′ ∈ T∞ : z(C′) ∈ [0, 1]ℓ} are +equal, implying that ξn = ξ∞. Indeed, every cell C ∈ Tn with z(C) ∈ [0, 1]ℓ is contained in [−R/2, R/2]ℓ +(since z(C) ∈ C and there is no long cell) and, consequently, C is also a cell of T∞ (since An(R) occurs). +Conversely, for every cell C′ ∈ T∞ with z(C′) ∈ [0, 1]ℓ there is a cell C ∈ Tn such that the restrictions +of C and C′ to [−R, R]ℓ coincide (since An(R) occurs) and C is not long (since Bn(R) occurs), which +implies that C = C′. It follows that +E[1(Cn(R)) ξn] = E[1(Cn(R)) ξ∞], +n ∈ N, R > 3. +To complete the proof of (6.4), it suffices to verify that +lim +R→∞ lim sup +n→∞ E[1(Dn(R)) ξn] = 0, +lim +R→∞ lim sup +n→∞ E[1(Dn(R)) ξ∞] = 0. +In view of (6.5), it suffices to check that the family {ξn : n ∈ N ∪ {∞}} is uniformly integrable. Now, +|ξn| ≤ ∥f∥∞ηn +with +ηn := +� +C∈Tn, z(C)∈[0,1]ℓ +1, +n ∈ N ∪ {∞}. +As we already observed above, outside the event Dn(R) we have ηn = η∞. From (6.5) it follows that +ηn → η∞ almost surely. On the other hand, we have Eηn = γℓ(Tn) → γℓ(T∞) = Eη∞ by assumption of +the proposition (and all these expectations are finite). These two properties together with ηn ≥ 0 imply +that the family {ηn : n ∈ N∪{∞}} is uniformly integrable. Indeed, if this were not the case, we could find +ε > 0 such that, after passing to a subsequence, E[ηn1(Qn)] > ε for some events Qn with P[Qn] < 1/2n. +Let Wk := ∪∞ +n=kQn. If K is sufficiently large, then E[η∞1(WK)] < ε/2, while E[ηn1(WK)] > ε for all +n ≥ K. Applying Fatou’s lemma to the variables (ηn1((WK)c))n≥K leads to a contradiction with the +assumption Eηn → Eη∞. Finally, the bound |ξn| ≤ ∥f∥∞ηn implies that the family {ξn : n ∈ N ∪ {∞}} +is uniformly integrable as well, and the proof is complete. +Acknowledgement +ZK and CT were supported by the DFG priority program SPP 2265 Random Geometric Systems. AG +and ZK were supported by the DFG under Germany’s Excellence Strategy EXC 2044 – 390685587, +Mathematics Münster: Dynamics - Geometry - Structure. +References +[1] P. Calka, T. Schreiber, and J. E. Yukich. Brownian limits, local limits and variance asymptotics for convex +hulls in the ball. Ann. Probab., 41(1):50–108, 2013. +[2] P. Calka and J. E. Yukich. Variance asymptotics for random polytopes in smooth convex bodies. Probab. +Theory Related Fields, 158(1-2):435–463, 2014. +[3] P. Calka and J. E. Yukich. Variance asymptotics and scaling limits for Gaussian polytopes. Probab. Theory +Related Fields, 163(1-2):259–301, 2015. +16 + +[4] P. Calka and J. E. Yukich. +Variance asymptotics and scaling limits for random polytopes. +Adv. Math., +304:1–55, 2017. +[5] P. Calka and J. E. Yukich. Convex hulls of perturbed random point sets. Ann. Appl. Probab., 31(4):1598–1632, +2021. +[6] S. N. Chiu, R. van de Weygaert, and D. Stoyan. The sectional Poisson-Voronoi tessellation is not a Voronoi +tessellation. Adv. in Appl. Probab., 28(2):356–376, 1996. +[7] A. Gusakova, Z. Kabluchko, and C. Thäle. +The β-Delaunay tessellation: Description of the model and +geometry of typical cells. Adv. in Appl. Probab., 54(4):1252–1290, 2022. +[8] A. Gusakova, Z. Kabluchko, and C. Thäle. The β-Delaunay tessellation II: The Gaussian limit tessellation. +Electron. J. Probab., (27):1 – 33, 2022. +[9] A. Gusakova, Z. Kabluchko, and C. Thäle. The β-Delaunay tessellation III: Kendall’s problem and limit +theorems in high dimensions. ALEA, Lat. Am. J. Probab. Math. Stat., 19:23 – 50, 2022. +[10] A. Gusakova, Z. Kabluchko, and C. Thäle. The β-Delaunay tessellation IV: Mixing properties and central +limit theorems. Stoch. and Dynamics, accepted. Preprint at: https://arxiv.org/abs/2108.09472, 2023. +[11] Z. Kabluchko. Angles of random simplices and face numbers of random polytopes. Adv. Math., 380:Paper +No. 107612, 68, 2021. +[12] Z. Kabluchko. +Recursive scheme for angles of random simplices, and applications to random polytopes. +Discrete Comput. Geom., 66:902–937, 2021. +[13] Z. Kabluchko, D. Temesvari, and C. Thäle. Expected intrinsic volumes and facet numbers of random beta- +polytopes. Math. Nachr., 292(1):79–105, 2019. +[14] Z. Kabluchko and C. Thäle. The typical cell of a Voronoi tessellation on the sphere. Discrete Comput. Geom., +66:1330–1350, 2021. +[15] Z. Kabluchko, C. Thäle, and D. Zaporozhets. Beta polytopes and Poisson polyhedra: f-vectors and angles. +Adv. Math., 374:Paper No. 107333, 63, 2020. +[16] O. Kallenberg. Foundations of Modern Probability. Probability and its Applications (New York). Springer- +Verlag, New York, second edition, 2002. +[17] G. Last and M. Penrose. Lectures on the Poisson Process, volume 7 of Institute of Mathematical Statistics +Textbooks. Cambridge University Press, Cambridge, 2018. +[18] C. Lautensack. Random Laguerre Tessellations. PhD thesis, 2007. +[19] C. Lautensack and S. Zuyev. Random Laguerre tessellations. Adv. in Appl. Probab., 40(3):630–650, 2008. +[20] R.E. Miles. Sectional Voronoi tessellations. Revista de la Union Mathematica Argentina, 29:301–327. +[21] W. Nagel. Stereology. In New perspectives in stochastic geometry, pages 451–475. Oxford Univ. Press, Oxford, +2010. +[22] A. Okabe, B. Boots, K. Sugihara, and S. N. Chiu. Spatial Tessellations: Concepts and Applications of Voronoi +Diagrams. Wiley Series in Probability and Statistics. John Wiley & Sons, Ltd., Chichester, second edition, +2000. With a foreword by D. G. Kendall. +[23] R.-D. Reiss. A course on point processes. Springer Series in Statistics. Springer-Verlag, New York, 1993. +[24] S. I. Resnick. Extreme Values, Regular Variation and Point Processes. Springer Series in Operations Research +and Financial Engineering. Springer, New York, 2008. Reprint of the 1987 original. +[25] M. Schlottmann. Periodic and quasi-periodic Laguerre tilings. International Journal of Modern Physics B, +7(6-07):1351–1363, January 1993. +[26] R. Schneider and W. Weil. +Stochastische Geometrie. +Teubner Skripten zur Mathematischen Stochastik. +(Teubner Texts on Mathematical Stochastics). B. G. Teubner, Stuttgart, 2000. +[27] R. Schneider and W. Weil. Stochastic and Integral Geometry. Probability and its Applications (New York). +Springer-Verlag, Berlin, 2008. +[28] T. Schreiber and J. E. Yukich. +Variance asymptotics and central limit theorems for generalized growth +processes with applications to convex hulls and maximal points. Ann. Probab., 36:363–396, 2008. +[29] D. Stoyan, W. S. Kendall, and J. Mecke. Stochastic Geometry and its Applications. Wiley Series in Probability +and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons, Ltd., Chichester, 1987. +With a foreword by D. G. Kendall. +17 + diff --git a/5NE1T4oBgHgl3EQfmgQS/content/tmp_files/load_file.txt b/5NE1T4oBgHgl3EQfmgQS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c156d4fe3cbbe199a6c3dcf34925c796af92da46 --- /dev/null +++ b/5NE1T4oBgHgl3EQfmgQS/content/tmp_files/load_file.txt @@ -0,0 +1,962 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf,len=961 +page_content='Sectional Voronoi tessellations: Characterization and high-dimensional limits Anna Gusakovaa, Zakhar Kabluchkob, and Christoph Thälec Abstract The intersections of beta-Voronoi, beta-prime-Voronoi and Gaussian-Voronoi tessellations in Rd with ℓ-dimensional affine subspaces, 1 ≤ ℓ ≤ d − 1, are shown to be random tessellations of the same type but with different model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In particular, the intersection of a classical Poisson-Voronoi tessellation with an affine subspace is shown to have the same distribution as a certain beta-Voronoi tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The geometric properties of the typical cell and, more generally, typical k-faces, of the sectional Poisson-Voronoi tessellation are studied in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It is proved that in high dimensions, that is as d → ∞, the intersection of the d-dimensional Poison-Voronoi tessellation with an affine subspace of fixed dimension ℓ converges to the ℓ-dimensional Gaussian-Voronoi tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Keywords: Beta-Voronoi tessellation, Gaussian-Voronoi tessellation, high-dimensional limit, Laguerre tessellation, Poisson point process, Poisson-Voronoi tessellation, sectional tessellation, stochastic geo- metry, typical cell MSC: 60D05, 60G55 1 Introduction The present paper is devoted to the study of affine sections of Poisson-Voronoi tessellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' To define these, let ζ ⊂ Rd be the set of atoms of a stationary point process in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' For each point x ∈ ζ we construct the Voronoi cell V (x, ζ) of x as the set of all points in Rd which are closer to x than to any other point of ζ: V (x, ζ) := {z ∈ Rd : ∥x − z∥ ≤ ∥y − z∥ for all y ∈ ζ \\ {x}}, where ∥ · ∥ stands for the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The Voronoi cell can be thought of as a zone of influence or attraction of the point x and it is known that each Voronoi cell is a convex polytope in Rd, with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The collection of all these polytopes is the Voronoi tessellation associated with ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' If ζ is a homogeneous Poisson point process with constant intensity ρ > 0, this construction yields the Poisson-Voronoi tessellation, denoted here by Wd,ρ, – one of the most classical models studied in stochastic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We refer to the monographs [22, 27, 29] for more detailed information, applications and further references on Voronoi tessellations and in particular the Poisson-Voronoi tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' A number of random tessellations studied in stochastic geometry, such as the Poisson hyperplane or the STIT tessellations, have the distinguished feature of being stable under intersections with lower- dimensional affine subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' By this we mean that the intersection with an affine subspace of one of these random tessellations is again a model of the same type within the intersecting subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' For example, the intersection of a Poisson hyperplane tessellation with an affine subspace L is again a Poisson hyperplane tessellation within L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' However, a similar property is not true for the Poisson-Voronoi tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In fact, it has been shown by Chiu, Van De Weygaert and Stoyan [6] that the intersection of the Poisson-Voronoi tessellation with an affine subspace cannot be a Voronoi tessellation induced by any stationary point process within the subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In other words, the sectional Poisson-Voronoi tessellation is necessarily a ‘non-Voronoi’ tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' However, besides a few mean values determined in [6, 20] further probabilistic aMünster University, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Email: gusakova@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='de bMünster University, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Email: zakhar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='kabluchko@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='de cRuhr University Bochum, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Email: christoph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='thaele@rub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='de 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='03297v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='PR] 9 Jan 2023 or geometric information about the sectional Poisson-Voronoi tessellation seems not available in the existing literature, although they are of importance for stereological applications (see [29, Chapter 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4] or [21, Section 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='6] as well as the references cited therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It is one of the main purposes of this paper to derive a precise description of the sectional Poisson-Voronoi tessellation and to study its typical cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We do this by establishing a connection with the so-called β-Voronoi tessellations, a random tessellation model we recently introduced and studied in the series of papers [10, 7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Their analysis in turn was based on the connection with the class of beta random polytopes, which has already seen a number of applications in stochastic geometry [11, 13, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We study the problem of the sectional Poisson-Voronoi tessellation just explained in a more general framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In fact, the random tessellation we study is either a β-Voronoi tessellation Vd,β,γ in Rd with parameters β ≥ −1 and γ > 0, a β′-Voronoi tessellation V′ d,β,γ in Rd with parameters β > d 2 + 1 and γ > 0, or a Gaussian-Voronoi tessellation Gd,λ in Rd with parameter λ > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' a description of all these models will be provided in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We remark that the classical Poisson- Voronoi tessellation generated by a stationary Poisson point process in Rd with intensity ρ > 0 appears in this framework as the β-Voronoi tessellation corresponding to the parameters β = −1 and γ = π d+1 2 ρ/Γ( d+1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Now, let L ⊂ Rd be an affine subspace of dimension 1 ≤ ℓ ≤ d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We show in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1 below that the sectional tessellation Vd,β,γ ∩ L is a (β + d−ℓ 2 )-Voronoi tessellation in L with the same γ, the sectional tessellation V′ d,β,γ ∩ L is a (β − d−ℓ 2 )′-Voronoi tessellation in L with the same γ, the sectional tessellation Gd,λ ∩ L is again a Gaussian-Voronoi tessellation in L with the same λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In particular, the intersection of the classical Poisson-Voronoi tessellation Wd,ρ with L turns out to be a β-Voronoi tessellation within L with β = d−ℓ 2 − 1 and γ = π d+1 2 ρ/Γ( d+1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' For clarity we should remark that none of the random tessellations Vd,β,γ with β > −1 are actually Voronoi tessellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Our terminology is motivated by the fact that Vd,β,γ may be viewed as a deformation of the classical Poisson-Voronoi tessellation (which corresponds to β = −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' With the identification of the sectional Poisson-Voronoi tessellation at hand, in the second part of this paper we study its geometric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' More precisely, we determine in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1 its face intensities in terms of quantities which have already appeared in the study of beta random polytopes [11, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' From here on, we determine the expected volume, the expected intrinsic volumes as well as the expected f- vector of the typical cell (and even more generally the typical k-face) of the sectional Poisson-Voronoi tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Moreover, generalizing earlier results of Miles [20] we consider the asymptotics, as d → ∞, of several characteristics (such as the volume of the typical cell) of the sectional Poisson-Voronoi tessellation and identify the limits with the corresponding characteristics of a suitable Gaussian-Voronoi tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The weak convergence on the level of tessellations is discussed as well using a coupling construction similar to the one in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 2 Preliminaries on random tessellations In this section we collect some definitions and facts about general stationary random tessellations in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' For more detailed discussions we refer the reader to [27, Chapters 4 and 10] as well as [29, Chapter 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' A tessellation T in Rd is a countable, locally finite collection of d-dimensional polytopes, which cover the space and have non-empty, disjoint interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The elements of T are called the cells of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Given a polytope c ⊂ Rd we denote by Fk(c) the set of its k-dimensional faces, 0 ≤ k ≤ d, where Fd(c) = {c}, and let F(c) := �d k=0 Fk(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' A tessellation T is called face-to-face if for any two of its cells c1, c2 ∈ T one has that c1 ∩ c2 ∈ (F(c1) ∩ F(c2)) ∪ {∅}, 2 that is, the intersection of two cells is either empty or a common face of both cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' For a face-to-face tessellation T one defines Fk(T) = � c∈T Fk(c) and F(T) = � c∈T F(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' A face-to-face tessellation in Rd is called normal if each k-dimensional face of the tessellation is contained in the boundary of precisely d + 1 − k cells, for all k ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' , d − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We denote by T the set of all face-to-face tessellations in Rd, which is supplied with a measurable structure as in [27, Chapter 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' By a random tessellation we understand a particle process T in Rd (in the usual sense of stochastic geometry, see [27, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1]) satisfying supp T ∈ T almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It is convenient to identify the random point process T with its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' A random tessellation is stationary, provided that its distribution is invariant under all shifts in Rd and isotropic if its distribution is invariant under all rotations in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' For a stationary random tessellation T and k ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' , d − 1} we define the stationary particle process T (k) := � F∈Fk(T ) δF of k-dimensional polytopes, which is referred to as the process of k-faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Next, we recall the concept of a typical cell (and a typical k-face) of a stationary random tessellation T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' see [27, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2], [27, page 450], [26, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Let C′ be the space of non-empty compact subsets of Rd endowed with the Hausdorff metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' A centre function is a Borel function z : C′ → Rd such that z(C + m) = z(C) + m for all C ∈ C′ and m ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The intensity of k-faces of T is defined by γk(T ) := E � F∈Fk(T ) 1(z(F) ∈ [0, 1]d), k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' These quantities are known to be independent of the choice of the centre function z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Assuming that γk(T ) ∈ (0, ∞), the typical k-face of T with respect to the centre function z is the k-dimensional random polytope whose distribution is given by Pz T ,k( · ) := 1 γk(T ) E � F∈Fk(T ) 1(F − z(F) ∈ · )1(z(F) ∈ [0, 1]d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In particular, for k = d we get the concept of the typical cell of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It should be noted that translation- invariant characteristics of the distribution of the typical k-face do not depend on the choice of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' More precisely, if z and z′ are two centre functions, then Pz′ T ,k is the push-forward of Pz T ,k under the map F �→ F − z′(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 3 Construction of β-, β′- and Gaussian-Voronoi tessellations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1 General Laguerre tessellations In this section we only briefly recall some facts about Laguerre tessellations and refer the reader to [7, Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4] and [8, Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We start by defining a general Laguerre tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Given two points v, w ∈ Rd and h ∈ R we define the power of w with respect to the pair (v, h) as pow(w, (v, h)) := ∥w − v∥2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In this situation, h is referred to as the weight (or height) of the point v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Let X be a countable set of marked points of the form (v, h) ∈ Rd × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Then the Laguerre cell of (v, h) ∈ X is the set C((v, h), X) := {w ∈ Rd : pow(w, (v, h)) ≤ pow(w, (v′, h′)) for all (v′, h′) ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The point v is called the nucleus of the cell C((v, h), X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Note that a Laguerre cell may be empty and even if it is non-empty, it does not need to contain its nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The collection of all non-empty Laguerre cells of X is called the Laguerre diagram: L(X) := {C((v, h), X): (v, h) ∈ X, C((v, h), X) ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 3 In the special case when the heights h of all points are the same (say, h0 ∈ R) the above definition leads to the classical Voronoi cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' More precisely, let Y be a countable set of points in Rd whose “marked” version X is obtained by attaching a fixed weight h0 to each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Then the Voronoi cell of v ∈ Y is V (v, Y ) = C((v, h0), X) = {w ∈ Rd : ∥w − v∥ ≤ ∥w − v′∥ for all v′ ∈ Y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The collection of the Voronoi cells of all v ∈ Y is called the Voronoi diagram V(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It should be mentioned that a Laguerre diagram is not necessarily a tessellation in Rd, at least as long as no additional assumptions on the geometric properties of the set X are imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Such assumptions have been described in detail in [18, 19, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In the present article we are interested in random tessellations built on Poisson point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' More precisely, we consider a Poisson point process ξ in Rd × E, where E ⊂ R is a Borel set (an interval), and the corresponding Laguerre diagram L(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Lemmas 1 and 2 in [7] (see also [8, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1]) provide sufficient conditions on ξ which ensure that, almost surely, L(ξ) is a stationary random face-to-face normal tessellation in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In the following we work under these conditions and remark that they are automatically satisfied in the three cases we consider from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3 on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2 Laguerre tessellations via paraboloid growth processes An alternative approach to the construction of Laguerre diagrams uses so-called paraboloid growth pro- cesses with overlaps (or simply paraboloid growth process), which were first introduced in [1, 28] in order to study the asymptotic geometry of random polytopes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' see also [2, 3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In this section we briefly describe this rather useful construction and refer for more details to [8, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Let Π±,x := {(v′, h′) ∈ Rd × R: h′ = ±∥v′ − v∥2 + h} be the upward (+) and downward (−) standard paraboloids with apex x := (v, h) ∈ Rd × R, denoted as apex Π±,x := x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In case (v, h) = (0, 0) we simply write Π± = Π±,(0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Given a set A ⊂ Rd × R we put A↓ : = {(v, h′) ∈ Rd × R: (v, h) ∈ A for some h ≥ h′}, A↑ : = {(v, h′) ∈ Rd × R: (v, h) ∈ A for some h ≤ h′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Following the definition from [1], for a given Poisson point process ξ in Rd × R, we introduce the paraboloid growth process Ψ(ξ): Ψ(ξ) := � x∈ξ Π↑ +,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It should be noted that, in typical situations, the majority of paraboloids will be completely covered by other paraboloids, implying that they do not "contribute" to the model and can thus be omitted without loosing any information about the set Ψ(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' This leads to the definition of extreme points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' A point x ∈ ξ is called extreme in the paraboloid growth process Ψ(ξ) if and only if its associated paraboloid is not fully covered by the paraboloids associated with other points of ξ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=', if Π↑ +,x ̸⊂ � y∈ξ,y̸=x Π↑ +,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We denote by ext(Ψ(ξ)) the set of all extreme points of the paraboloid growth process Ψ(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Using the paraboloid growth process we can construct a random diagram in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Given a point x = (v, h) ∈ ξ define the Ψ-cell of x as CΨ(x, ξ) := � {w ∈ Rd : � (w, 0)↑ ∪ (w, 0)↓� ∩ bd Ψ(ξ) ∈ Π+,x}, if x ∈ ext(Ψ(ξ)), ∅, otherwise, where bd A denotes the boundary of a set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In other words, w belongs to CΨ(x, ξ) if and only if ∥w − v∥2 + h ≤ ∥w − v′∥ + h′ for all (v′, h′) ∈ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Thus, the Ψ-cell of an extreme point x of the paraboloid growth process Ψ(ξ) is non-empty and coincides with the Laguerre cell C(x, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Next, we construct the diagram LΨ(ξ) as the collection of all non-empty Ψ-cells: LΨ(ξ) := {CΨ(x, ξ): CΨ(x, ξ) ̸= ∅} = {CΨ(x, ξ): x ∈ ext(Ψ(ξ))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We directly have that LΨ(ξ) = L(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 4 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1: Left panel: Simulation of a β-Voronoi tessellation in R2 with β = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Middle panel: Simulation of a β′-Voronoi tessellation in R2 with β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Right panel: Simulation of a Gaussian-Voronoi tessellation in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3 Three families of random tessellations In this article we consider random tessellations in Rd build on the following three families of Poisson point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' For β > −1 and 0 < γ < ∞ we consider a Poisson point process ηd,β,γ in Rd × [0, +∞) whose intensity measure has density (v, h) �→ γ cd+1,βhβ, cd+1,β := Γ � d+1 2 + β + 1 � π d+1 2 Γ(β + 1) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1) with respect to the Lebesgue measure on Rd × [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Further, for β > d 2 + 1 and 0 < γ < ∞ we consider a Poisson point process η′ d,β,γ in Rd × (−∞, 0) with intensity measure having density (v, h) �→ γ c′ d+1,β(−h)−β, c′ d+1,β := Γ (β) π d+1 2 Γ(β − d+1 2 ) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2) with respect to the Lebesgue measure on Rd × (−∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The constants cd+1,β and c′ d+1,β in the above definitions are introduced for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' For example, they make the statement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1 below more transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Finally, for λ > 0 and 0 < γ < ∞ we consider a Poisson point process ζd,λ,γ in Rd × R whose intensity measure has density (v, h) �→ γ eλh, with respect to the Lebesgue measure on Rd × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It was shown in [7, Lemma 3] and in [8, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3] that the Poisson point processes ηd,β,γ and ζd,λ,γ satisfy the sufficient conditions of Lemma 1 and Lemma 2 in [7] and, hence, the corresponding Laguerre diagrams Vd,β,γ := L(ηd,β,γ), V′ d,β,γ := L(η′ d,β,γ) and Gd,λ := L(ζd,λ,γ) are stationary random normal tessellations in Rd, which are called β-Voronoi, β′-Voronoi and Gaussian-Voronoi tessellations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' These tessellations have been studied in [10, 7, 8, 9], where they were considered in Rd−1 instead of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Simulations of these tessellations in the plane are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Note that although the point process η′ d,β,γ is well-defined in the range β > d+1 2 , the corresponding β′-Voronoi tessellation exists in the smaller range β > d 2 + 1 only, see [7, Lemma 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Note that changing the parameter γ amounts to shifting the Poisson point process ζd,λ,γ along the height coordinate h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In particular, the distribution of the point process Gd,λ does not depend on the choice of γ, which is reflected in our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It will be convenient to extend the above definition of the β-Voronoi tessellation to the case β = −1 (with arbitrary γ > 0) by defining Vd,−1,γ := Wd,r(d)γ to be the classical Poisson-Voronoi 5 tessellation constructed on the homogeneous Poisson point process on Rd with constant intensity r(d)γ, where r(d) := Γ �d + 1 2 � π− d+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' A justification for this definition is given by the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' For the necessary background on point processes and weak convergence we refer to [24, Chapter 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' As β ↓ −1, the Poisson process ηd,β,γ converges, weakly on the space of locally finite integer-valued measures on Rd × [0, ∞), to the Poisson point process whose intensity measure is the Lebesgue measure on Rd × {0} times r(d)γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Write β = −1 + ε with ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Then the constant appearing in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1) satisfies cd+1,β = Γ � d+1 2 + ε � π d+1 2 Γ(ε) = Γ � d+1 2 � π d+1 2 ε(1 + o(1)), as ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It follows that for every c > 0 and every bounded Borel set B ⊂ Rd the number of points of the Poisson point process ηd,β,γ appearing in B × [0, c] is Poisson-distributed with expectation γλd(B)cd+1,β � c 0 h−1+εdh −→ ε↓0 Γ �d + 1 2 � π− d+1 2 γλd(B) = r(d)γλd(B), where λd denotes the d-dimensional Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Since the right-hand side does not depend on c, it follows that, for every 0 < c1 < c2, the expected number of points of ηd,β,γ in B × [c1, c2] converges to 0, as ε ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Hence, the intensity measure of ηd,β,γ converges as β ↓ −1 to the Lebesgue measure on Rd × {0} times r(d)γ, vaguely on the space Rd × [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Then, the claim of the proposition follows from [16, Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='16 (iv)] or [24, Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 4 Affine sections of β, β′- and Gaussian-Voronoi tessellations In this section we study the intersection of the d-dimensional random tessellations Vd,β,γ, V′ d,β,γ and Gd,λ with an affine subspace L ⊂ Rd of dimension ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' , d − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' By stationarity and isotropy of these tessellations, we may and will assume without loss of generality that L = Rℓ is the linear subspace of Rd spanned by the first ℓ vectors of the standard orthonormal basis of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The intersection of the tessellation Vd,β,γ with Rℓ will be denoted by Vd,β,γ ∩ Rℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Similar convention is used for the tessellations V′ d,β,γ and Gd,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The following theorem identifies the distribution of Vd,β,γ ∩ Rℓ, V′ d,β,γ ∩ Rℓ and Gd,λ ∩ Rℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Fix integers d ≥ 2 and 1 ≤ ℓ ≤ d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' (i) For any β ≥ −1 and γ > 0, Vd,β,γ ∩ Rℓ has the same distribution as Vℓ,β+ d−ℓ 2 ,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' (ii) For any β > d 2 + 1 and γ > 0, V′ d,β,γ ∩ Rℓ has the same distribution as V′ ℓ,β− d−ℓ 2 ,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' (iii) For any λ > 0, Gd,λ ∩ Rℓ has the same distribution as Gℓ,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Before we move on to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1 we would like to highlight the following special case which deals with sections of the classical Poisson-Voronoi tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Together with the results we obtain below, this fully answers and resolves the problems raised in [6, 20, 21, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Fix integers d ≥ 2 and 1 ≤ ℓ ≤ d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Then for any ρ > 0 the intersection of the d-dimensional Poisson-Voronoi tessellation Wd,ρ of intensity ρ with Rℓ has the same distribution as V ℓ, d−ℓ 2 −1,π d+1 2 ρ/Γ( d+1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 6 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Let us first consider the case ℓ = d−1 meaning that we intersect with a hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Let ξ be one of the Poisson point processes ηd,β,γ, η′ d,β,γ or ζd,λ,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The atoms of ξ live in the space Rd+1 = Rd × R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' a generic point in this space is denoted by (v, h) with v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' , vd) ∈ Rd being the spatial coordinate and h ∈ R being the height coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The Laguerre tessellation L(ξ) lives in the space Rd defined by the equation h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The linear hyperplane L ≡ Rd−1 ⊂ Rd ⊂ Rd+1 with which our tessellations are intersected is given by the equations {vd = 0, h = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In order to prove the statement we will use the representation of the Laguerre tessellation L(ξ) in terms of the paraboloid growth process Ψ(ξ) as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We extend the hyperplane L ⊂ Rd by adding the height coordinate, namely we consider L′ := {(v, h) = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' , vd, h) ∈ Rd × R: vd = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' For every point x = (v, h) ∈ Rd+1, the intersection of the d-dimensional paraboloid Π+,x with L′ is a (d − 1)-dimensional upward paraboloid Π+,x′ ∩ L′ in L′ with apex given by x′ = f(v, h) := (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' , vd−1, 0, h + v2 d) ∈ L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' If x runs through all atoms of ξ, then x′ runs through all atoms of the point process f(ξ) := {f(x): x ∈ ξ} on L′, which is also a Poisson point process by the mapping theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' see [17, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The intersection of the Laguerre tessellation L(ξ) with L coincides with the Laguerre tessellation generated by the point process f(ξ) (within L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Now, we are going to identify the intensity measure µ of the Poisson point process f(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' To this end, we consider the Poisson point processes ηd,β,γ, η′ d,β,γ and ζd,λ,γ separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Case (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Let first β > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' By the mapping theorem for Poisson point processes [17, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1], f(ηd,β,γ) := {f(x): x ∈ ηd,β,γ} is a Poisson point process in Rd−1×[0, +∞) ⊂ L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' To compute its intensity measure µ, we take some Borel set B ⊂ L, any s > 0 and observe that an atom (v, h) of ηd,β,γ is mapped by f to B × [0, s] if and only if (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' , vd−1) ∈ B and h + v2 d ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The latter condition means that h ≤ s and |vd| ≤ √ s − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It follows that the intensity measure µ satisfies µ(B × [0, s]) = γ cd+1,β � Rd � ∞ 0 hβ1(f(v, h) ∈ B × [0, s]) dhdv = γ cd+1,β � Rd � s 0 hβ1(v ∈ B × [− √ s − h, √ s − h]) dhdv = 2γ cd+1,β λd−1(B) � s 0 hβ√ s − h dh = γ Γ( d+1 2 + β + 1) π d 2 Γ(β + 3 2) λd−1(B) sβ+ 3 2 β + 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1) In the case β = −1, we let ηd,−1,γ be the Poisson point process on Rd × {0} (which is considered as a subset of Rd+1) whose intensity with respect to the Lebesgue measure on Rd ×{0} is constant and equals Γ( d+1 2 )π− d+1 2 γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Thus, the heights of all points in ηd,−1,γ are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Then, the Laguerre tessellation generated by ηd,−1,γ on Rd coincides with Vd,−1,γ by our convention described in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' To compute the intensity measure µ of the Poisson point process f(ηd,−1,γ), we take some Borel set B ⊂ L, any s > 0 and observe that an atom (v, 0) of ηd,−1,γ is mapped by f to B × [0, s] if and only if (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' , vd−1) ∈ B and v2 d ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It follows that the intensity measure µ satisfies µ(B × [0, s]) = γ Γ( d+1 2 ) π d+1 2 � Rd 1(f(v, 0) ∈ B × [0, s]) dv = γ 2Γ( d+1 2 ) π d+1 2 λd−1(B)√s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2) 7 By differentiating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2) with respect to s, it follows that for all β ≥ −1, the intensity measure of f(ηd,β,γ) has density (v′, 0, h) �→ γ cd,β+ 1 2 hβ+ 1 2 , v′ ∈ Rd−1, h > 0, with respect to the Lebesgue measure on L × [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Consequently, the Laguerre tessellation generated by f(ηd,β,γ) within L has the same distribution as Vd−1,β+ 1 2 ,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Case (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Next, we deal with f(η′ d,β,γ) := {f(x): x ∈ η′ d,β,γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Let us first consider only those points of f(η′ d,β,γ) that have negative height and compute the intensity measure µ of these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The points with positive height coordinate have no influence on the resulting tessellation, as we will argue below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' To determine µ, we take some Borel set B ⊂ L, any s < 0 and observe that an atom (v, −g) of η′ d,β,γ (with g > 0) is mapped by f to B × (−∞, s] if and only if (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' , vd−1) ∈ B and v2 d ≤ s + g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The latter condition means that g ≥ −s and |vd| ≤ √s + g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It follows that the intensity measure µ satisfies µ(B × (−∞, s]) = γ c′ d+1,β � Rd � ∞ 0 g−β1(f(v, −g) ∈ B × [−s, ∞)) dgdv = γ c′ d+1,β � Rd � ∞ −s g−β1(v ∈ B × [−√s + g, √s + g]) dgdv = 2γ c′ d+1,β λd−1(B) � ∞ −s g−β√g + s dg = γ Γ(β − 1 2) π d 2 Γ(β − d+1 2 ) λd−1(B)(−s)−β+ 3 2 β − 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Differentiating, we conclude that the intensity measure µ has density (v′, 0, h) �→ γ c′ d,β+ 1 2 (−h)−β+ 1 2 , v′ ∈ Rd−1, h < 0, with respect to the Lebesgue measure on L × (−∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' So, by the mapping theorem [17, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1], the restriction of f(η′ d,β,γ) to L × (−∞, 0) is a Poisson point process with the same intensity measure as η′ d−1,β− 1 2 ,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The Laguerre tessellation generated by this Poisson point process within L ≡ Rd−1 has the same distribution as V′ d−1,β− 1 2 ,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It remains to observe that adding the points of f(η′ d,β,γ) with positive height coordinate does not change the Laguerre tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Indeed, every point in Rd−1 × {0} is an accumulation point of η′ d−1,β− 1 2 ,γ, hence the lower boundary of the paraboloid growth process Ψ(η′ d−1,β− 1 2 ,γ) is contained in Rd−1×(−∞, 0] and points with positive height coordinate have no influence on the tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Case (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Finally, we consider f(ζd,λ,γ), which is a Poisson point process in L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' To compute its intensity measure µ, we take some Borel set B ⊂ L, any s ∈ R and observe that an atom (v, h) of ζd,λ,γ is mapped by f to B × (−∞, s] if and only if (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' , vd−1) ∈ B and h + v2 d ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It follows that µ(B × (−∞, s]) = γ � Rd � R eλh1(f(v, h) ∈ B × (−∞, s)) dhdv = 2γ λd−1(B) � s −∞ ehλ√ s − h dh = γ λd−1(B) √πeλs λ3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Thus, the density of the intensity measure of f(ζd,λ,γ) is given by (v′, 0, h) �→ γ � π/λ eλh, v′ ∈ Rd−1, h ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 8 Hence, the Laguerre tessellation generated by f(ζd,λ,γ) within L has the same distribution as Gd−1,λ (recall that the parameter γ does not influence the distribution of the Gaussian-Voronoi tessellation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' This proves the claim for ℓ = d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' For general 1 ≤ ℓ ≤ d − 2 we can inductively repeat the above argument d − ℓ times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 5 Sectional Poisson-Voronoi tessellations 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1 Face intensities and the expected volume of the typical cell As we have shown in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2, the sectional Poisson-Voronoi tessellation Wd,ρ ∩ Rℓ can be identified with a suitable β-Voronoi tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' This makes it possible to compute explicitly several functionals of the sectional Poisson-Voronoi tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We begin with a formula for the intensity of j-dimensional faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' This quantity, denoted by γj(Wd,ρ ∩ Rℓ), has been defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Let d ≥ 2, 1 ≤ ℓ ≤ d − 1 and 0 ≤ j ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Then, for any ρ > 0, we have γj(Wd,ρ ∩ Rℓ) = ρ ℓ d 2Jℓ+1,ℓ−j+1( d−ℓ−1 2 )π ℓ 2 d(ℓ + 1) Γ( (ℓ+1)(d−1) 2 + 1)Γ(ℓ + 1 − ℓ d)Γ( d 2 + 1)ℓ+1− ℓ d Γ( (ℓ+1)(d−1)+1 2 )Γ( ℓ+2 2 )Γ( d+1 2 )ℓ+1 , wher Jℓ+1,ℓ−j+1 �d − ℓ − 1 2 � = �ℓ + 1 j � Γ( (d−1)(ℓ+1)+3 2 ) √π Γ( (d−1)(ℓ+1) 2 + 1) � +∞ −∞ (cosh u)−(d−1)(ℓ+1)−2 × � 1 2 + i Γ( d+1 2 ) √π Γ( d 2) � u 0 (cosh v)d−1dv �j du (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1) and i = √−1 stands for the imaginary unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2 we have γj(Wd,ρ ∩ Rℓ) = γj(Vℓ, d−ℓ−2 2 ,r), where r = r(d)ρ = π d+1 2 ρ/Γ( d+1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The formula for γj(Vℓ, d−ℓ−2 2 ,r) can be obtained by combining [7, Theorem 6] (which we apply with parameters d := ℓ + 1, β := d−ℓ−2 2 and j := ℓ − j) with [7, Proposition 3] (with parameters d := ℓ + 1, k := ℓ + 1 − j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Note that [7, Theorem 6] refers to [7, Theorem 2] which has to be applied with parameters d := ℓ + 1, s := 1, ν := 0, γ := r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The quantities Jd+1,k(β) for general d ≥ 0, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' , d + 1} and β ≥ −1 have a natural geometric meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Namely, Jd+1,k(β) is equal to the expected sum of internal angles at its k-vertex faces of a random beta-simplex, which is defined as the convex hull of d + 1 independent random points with density proportional to (1 − ∥x∥2)β in the d-dimensional unit ball, see [7, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1] and [11] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' From this interpretation it directly follows that J1,1(β) = J2,1(β) = J2,2(β) = 1, J3,1(β) = 1 2, J3,2(β) = 3 2, J3,3(β) = 1, Jℓ+1,ℓ(β) = ℓ + 1 2 , Jℓ+1,ℓ+1(β) = 1, for any ℓ ≥ 1 and β ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Moreover, if we denote by Σd a regular d-dimensional simplex and by σk(Σd) the internal angle sum at its k-vertex faces, then Jd+1,k(∞) := lim β→∞ Jd+1,k(β) = σk(Σd) according to [7, Proposition 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='d = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='d = 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='d = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='d = 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='d = 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='ℓ = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4√ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='√4πρ·Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='15π3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='64· 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='√8ρ·Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='7 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3· 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='648π2ρ·Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2835· 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3·π3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='16384· 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='√32ρ·Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='ℓ = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5· 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3π5ρ2·Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='35√πρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='77·24/5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5·153/5·π9/5·ρ2/5·Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� 13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='50· 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='143· 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='√ρ·Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='ℓ = 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='280665·π3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='821248· 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2·ρ3/4·Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� 13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='56·153/5· 5√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2/π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='187·ρ3/5·Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='17320875·√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3/2·π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='176201728·√ρ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='ℓ = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='144848704·23/5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='156/5π8/5� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1692197−141120π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='ρ4/5Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='15·62/3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='13·ρ2/3·Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� 13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='ℓ = 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='6823504578515625·35/6·π3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4912276871446528· 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2·ρ5/6·Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� 31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1: E vol(Zd,ℓ,ρ) for small values of d and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' As a corollary of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1 we can compute the expected volume of the typical cell Zd,ℓ,ρ of the sectional Poisson-Voronoi tessellation Wd,ρ ∩ Rℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Note that the volume does not change under shifts, which is why it does not matter how to choose the centre function in the definition of the typical cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' For ℓ = 1 and ℓ = 2 this quantity has been studied by Miles [20] who showed that E vol(Zd,1,ρ) = ρ− 1 d Γ(d − 1 2)Γ( d+1 2 )2 (d − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='Γ(2 − 1 d)Γ( d 2)Γ( d 2 + 1)1− 1 d , E vol(Zd,2,ρ) = ρ− 2 d 3d · Γ( 3d 2 − 1)Γ( d+1 2 )3 πΓ( 3d−1 2 )Γ(3 − 2 d)Γ( d 2 + 1)3− 2 d , see Formulas (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4) in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Our result generalizes this to arbitrary 1 ≤ ℓ ≤ d − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' special cases with small values of d and ℓ are summarized in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Let ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Then, for any d ≥ 2 and 1 ≤ ℓ ≤ d − 1 we have E vol(Zd,ℓ,ρ) = ρ− ℓ d d(ℓ + 1) 2Jℓ+1,1( d−ℓ−1 2 )π ℓ 2 Γ( (ℓ+1)(d−1)+1 2 ) Γ( (ℓ+1)(d−1) 2 + 1) Γ( ℓ+2 2 ) Γ(ℓ + 1 − ℓ d) Γ( d+1 2 )ℓ+1 Γ( d 2 + 1)ℓ+1− ℓ d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It is known from [27, Equation (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4)] that E vol(Zd,ℓ,ρ) = γℓ(Wd,ρ ∩ Rℓ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The right-hand side is known from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3 stays true for ℓ = d where it gives the expected volume of a typical Poisson- Voronoi cell to be E vol(Zd,d,ρ) = 1/ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The quantity Jℓ+1,1(− 1 2) cancels with the Gamma-factors by the formula given in [12, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='9] and the Legendre duplication formula for the Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1 also stays true for ℓ = d and gives the intensity of j-faces in the Poisson-Voronoi tessellation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' see [11, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='10] for another formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In the next result we compute the limit of the intensity of j-dimensional faces in the d-dimensional Poisson-Voronoi tessellation intersected with Rℓ in the regime when d → ∞ while ℓ ∈ N stays fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Fix some ℓ ∈ N and 0 ≤ j ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Let (ρd)d∈N be a positive sequence such that lim d→∞(ρd)1/d = κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Then, lim d→∞ γj(Wd,ρd ∩ Rℓ) = Jℓ+1,ℓ−j+1(∞)(κ2πe) ℓ 2 √ ℓ + 1 2(ℓ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Γ( ℓ 2) , where Jℓ+1,ℓ−j+1(∞) is the sum of angles at (ℓ − j)-dimensional faces of a regular ℓ-dimensional simplex Σℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' For example, we may take ρd = ρ > 0 to be constant, in which case κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 10 Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1, lim d→∞ γj(Wd,ρd ∩ Rℓ) = lim d→∞ 2Jℓ+1,ℓ−j+1( d−ℓ−1 2 )π ℓ 2 d(ℓ + 1)ρ − ℓ d d Γ( (ℓ+1)(d−1) 2 + 1) Γ( (ℓ+1)(d−1)+1 2 ) Γ(ℓ + 1 − ℓ d) Γ( ℓ+2 2 ) Γ( d 2 + 1)ℓ+1− ℓ d Γ( d+1 2 )ℓ+1 = 2Jℓ+1,ℓ−j+1(∞)(κ2π) ℓ 2 ℓ + 1 2Γ(ℓ) Γ( ℓ 2) lim d→∞ 1 d Γ( (ℓ+1)(d−1) 2 + 1) Γ( (ℓ+1)(d−1)+1 2 ) Γ( d 2 + 1)ℓ+1− ℓ d Γ( d+1 2 )ℓ+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' By Stirling’s formula for the Gamma function, Γ(z) = � 2π/z(z/e)z(1 + O(z−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Since lim n→∞ Γ(n)nz Γ(n+z) = 1 we get lim d→∞ γj(Wd,ρd ∩ Rℓ) = Jℓ+1,ℓ−j+1(∞)(κ2πe) ℓ 2 √ ℓ + 1 2Γ(ℓ) Γ( ℓ 2) lim d→∞ � d 4π � ℓ 2d = Jℓ+1,ℓ−j+1(∞)(κ2πe) ℓ 2 √ ℓ + 1 2Γ(ℓ) Γ( ℓ 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' This completes the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We now study the asymptotic behaviour of the expected volume of the typical cell in the sections of fixed dimension ℓ of a high-dimensional Poisson-Voronoi tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Let (ρd)d∈N be a positive sequence such that lim d→∞(ρd)1/d = κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Then, for every ℓ ∈ N, lim d→∞ E vol(Zd,ℓ,ρd) = √ ℓ + 1 Jℓ+1,1(∞)(κ2πe) ℓ 2 Γ( ℓ 2) 2(ℓ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=', where Jℓ+1,1(∞) is the sum of solid angles of the regular ℓ-dimensional simplex Σℓ at its vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' This follows from the fact that E vol(Zd,ℓ,ρd) = γℓ(Wd,ρd ∩ Rℓ)−1 (see [27, Equation (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4)]) by applying Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5 with j = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In the special cases ℓ = 1, 2 and for every fixed ρ > 0, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='7 combined with the results of Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2 yields, for any constant ρ > 0, the limit relations lim d→∞ E vol(Zd,1,ρ) = 1 √ 2e, and lim d→∞ E vol(Zd,2,ρ) = √ 3 eπ , which were already known from the work of Miles [20, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 318, 319].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Moreover, for ℓ = 3 we get lim d→∞ E vol(Zd,3,ρ) = � 4e3/2(3 arccos(1/3) − π) �−1, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' This follows from the fact that the solid angle at a vertex of a regular tetrahedron is θ := 1 4π(3 arccos(1/3) − π), implying that J4,1(∞) = σ1(Σ3) = 4θ = 3 π arccos(1/3) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2 Expected intrinsic volumes and f-vectors of typical k-faces Together with the volume of the typical cell Zd,ℓ,ρ of the sectional Poisson-Voronoi tessellation we can consider its intrinsic volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We recall from [27, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 222] that the intrinsic volume Vm(K) of order 0 ≤ m ≤ d of a compact convex set K ⊂ Rd may be defined as Vm(K) := d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' (d − m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Γ( m 2 + 1)Γ( d−m 2 + 1) Γ( d 2 + 1) Eλm(K|L), where L ⊂ Rd is a uniformly distributed random subspace of dimension m, K|L denotes the orthogonal projection of K onto L and λm(K|L) its m-dimensional Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In addition, instead of the 11 typical sectional cell we can consider for 1 ≤ k ≤ ℓ the typical k-face Z(k) d,ℓ,ρ of the sectional Poisson- Voronoi tessellation Wd,ρ ∩ Rℓ, see Section 2 or [27, page 450] for a formal definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' For example, for k = ℓ we get back the typical cell, for k = ℓ − 1 the typical facet and for k = 1 the typical edge of the sectional tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Using the results from [27, pages 466-467] for general stationary and isotropic random tessellations we conclude (by combining the last two formulas there) that EVj(Z(k) d,ℓ,ρ) = ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' (ℓ − j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=') Γ( j 2 + 1)Γ( ℓ−j 2 + 1) Γ( ℓ 2 + 1) γk−j(Wd,ρ ∩ Rℓ−j) γk(Wd,ρ ∩ Rℓ) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2) where d ≥ 2, and 1 ≤ ℓ ≤ d − 1, 0 ≤ k ≤ ℓ and 0 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' This expression can be made fully explicit in view of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3: EVj(Z(k) d,ℓ,ρ) = � ρ Γ( d 2 + 1) �− j d (ℓ + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' (ℓ − j + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Γ( j 2 + 1) πj/2 Jℓ−j+1,ℓ−k+1( d−ℓ+j−1 2 ) Jℓ+1,ℓ−k+1( d−ℓ−1 2 ) × Γ( (ℓ−j+1)(d−1) 2 + 1)Γ(ℓ − j + 1 − ℓ−j d ) Γ( (ℓ+1)(d−1) 2 + 1)Γ(ℓ + 1 − ℓ d) Γ( (ℓ+1)(d−1)+1 2 )Γ( d+1 2 )j Γ( (ℓ−j+1)(d−1)+1 2 )Γ( d 2 + 1)j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3) For intersections of dimension ℓ = 2, Miles [20, Equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5) on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 319] derived a formula for the expected area and perimeter of the typical cell which are particular cases of the above formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5 it is easy to derive the large d limit of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Namely, if lim d→∞(ρd)1/d = κ > 0, then lim d→∞ EVj(Z(k) d,ℓ,ρd) = √ ℓ + 1 √ℓ − j + 1 · Γ( j 2 + 1) j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' (κ2πe)j/2 · Jℓ−j+1,ℓ−k+1(∞) Jℓ+1,ℓ−k+1(∞) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' For intersections of dimensions ℓ = 2 and 3 (and k = ℓ) we recover results of Miles [20, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 319, 320].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Finally, we deal with the expected number of j-dimensional faces of the typical cell of the sectional Voronoi tessellation Wd,ρ ∩ Rℓ, which we denote by Efj(Zd,ℓ,ρ), for ρ > 0, d ≥ 2, 1 ≤ ℓ ≤ d − 1 and 0 ≤ j ≤ ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Using the fact that, with probability 1, each j-dimensional face of the sectional tessellation is contained in the boundary of precisely ℓ − j + 1 of its cells (by normality of the tessellation), it follows that Efj(Zd,ℓ,ρ) = (ℓ − j + 1)γj(Wd,ρ ∩ Rℓ) γℓ(Wd,ρ ∩ Rℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We can now apply Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3 to conclude that Efj(Zd,ℓ,ρ) = (ℓ − j + 1)Jℓ+1,ℓ−j+1( d−ℓ−1 2 ) Jℓ+1,1( d−ℓ−1 2 ) , independently of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Clearly, Ef0(Zd,1,ρ) = 2 for any d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Also, Efj(Zd,2,ρ) = 6 for any d ≥ 2 and 0 ≤ j ≤ 1, since the sectional Voronoi tessellation is stationary and normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Some non-trivial values for space dimensions d = 4, 5, 6 are collected in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Similarly, we can compute the expected number of j-dimensional faces of the typical k-dimensional face Z(k) d,ℓ,ρ of the sectional Poisson-Voronoi tessellation Wd,ρ ∩ Rℓ for d ≥ 2, 1 ≤ ℓ ≤ d − 1, 1 ≤ k ≤ ℓ and 0 ≤ j ≤ k − 1: Efj(Z(k) d,ℓ,ρ) = (k − j + 1)Jℓ+1,ℓ−j+1( d−ℓ−1 2 ) Jℓ+1,ℓ−k+1( d−ℓ−1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In the large d limit this becomes lim d→∞ Efj(Z(k) d,ℓ,ρ) = (k − j + 1)Jℓ+1,ℓ−j+1(∞) Jℓ+1,ℓ−k+1(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Again, for ℓ = 2, 3 we recover results of Miles [20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 320].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 12 d = 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' ℓ = 3 d = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' ℓ = 3 d = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' ℓ = 4 d = 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' ℓ = 3 d = 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' ℓ = 4 d = 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' ℓ = 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='j = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='10 240 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='401 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='67 200π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='26 741 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4 233 600π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1 692 197−141 120π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='524 288 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='21 509 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='52 003 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='34 394 098 106 368 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='37 477 698 299 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='j = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='15 360 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='401 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='100 800π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='26 741 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='8 467 200π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1 692 197−141 120π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='786 432 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='21 509 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='52 003 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='85 985 245 265 920 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='37 477 698 299 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='j = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5 922 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='401 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2 + 33 600π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='26 741 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='10 153 182+4 233 600π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1 692 197−141 120π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='305 162 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='21 509 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='162 009 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1 000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='74 276 903 321 600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='37 477 698 299 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='j = 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='10 153 182 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1 692 197−141 120π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='64 003 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2 000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='25 430 109 716 480 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='37 477 698 299 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='j = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='53 194 508 510 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='707 126 383 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2: Efj(Zd,ℓ,ρ) for small values of d, ℓ and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 6 Convergence to the Gaussian-Voronoi tessellation in high dimensions In Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2 we computed explicitly several characteristics of the sectional Poisson-Voronoi tessellation Wd,ρd∩Rℓ and the limits of these characteristics in the regime when d → ∞ and (ρd)1/d → κ > 0, while ℓ ∈ N stays fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It turns out that these limits coincide with the corresponding characteristics of the tessellation Gℓ,λ with λ = κ2πe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' For example, for the typical cell Z(Gℓ,λ) of the Gaussian-Voronoi tessellation Gℓ,λ it is known from [8, Section 5] (where the special case λ = 1/2 has been considered) that E vol(Z(Gℓ,λ)) = √ ℓ + 1 Jℓ+1,1(∞)λ ℓ 2 Γ( ℓ 2) 2(ℓ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='. This formula coincides with the one obtained in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='7 if we choose λ = κ2πe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In the next two theorems we explain this and other similar coincidences by proving weak convergence of the corresponding tessellations and the typical cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Take any positive sequence (ρd)d∈N with limd→∞(ρd)1/d = κ > 0 and let ℓ ∈ N be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Then, as d → ∞, the sectional Poisson-Voronoi tessellation Wd,ρd ∩ Rℓ converges to Gℓ,κ2πe in the following sense: It is possible to define all these random tessellations on the same probability space such that for every ℓ-dimensional ball BR ⊂ Rℓ of radius R > 0 centred at the origin the probability that the restrictions of Wd,ρd ∩ Rℓ and Gℓ,κ2πe to BR coincide, converges to 1, as d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The skeleton of a (random) tessellation T is the (random) closed set skel(T ) = � c∈T bd c, where bd c denotes the topological boundary of the cell c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The mode of convergence appearing in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1 implies that the random closed set skel(Wd,ρd ∩ Rℓ) converges to the random closed set skel(Gℓ,κ2πe) weakly as d → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' see [27, Chapter 2] for this concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The proof of this theorem basically follows the same route as the proof of The- orem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2 in [8], which is the reason why we leave out some details here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2, the sectional tessellation Wd,ρd ∩ Rℓ has the same distribution as Vℓ,βd,γd with βd = 1 2(d − ℓ) − 1 and γd = π d+1 2 Γ �d + 1 2 �−1 ρd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We will prove that, as d → ∞, the Poisson point processes ηℓ,βd,γd converge, after an appropriate vertical shift, to ζℓ,κ2πe,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3 for their definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The vertical shift Qd : Rℓ+1 → Rℓ+1 is given by Qd(v, h) = (v, h − ad) with ad = 1 π �πΓ( d−ℓ 2 ) ρd � 2 d−ℓ−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1) Note that applying such a vertical shift to a point process does not change the resulting Laguerre tessel- lation since it amounts to shifting all paraboloids along the height coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 13 In the following, we show that the intensity function of the Poisson point process ξℓ,d := Qd(ηℓ,βd,γd) converges, as d → ∞, to the intensity function of ξℓ,∞ := ζℓ,κ2πe,1 uniformly on every compact subset of Rℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Indeed, the intensity function of Qd(ηℓ,βd,γd) is given by fd(v, h) = ρdπ d−ℓ 2 Γ( d−ℓ 2 )(h + ad) d−ℓ−2 2 1 � h + ad > 0 � = ρdπ d−ℓ 2 Γ( d−ℓ 2 )a d−ℓ−2 2 d � 1 + h ad � d−ℓ−2 2 1 � h + ad > 0 � = � 1 + h ad �ad· d−ℓ−2 2ad 1 � h + ad > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Stirling’s formula for the Gamma function and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1) yield lim d→∞ d − ℓ − 2 2ad = 1 κ2π lim d→∞ d − ℓ + 2 2Γ( d−ℓ 2 ) 2 d−ℓ+2 = κ2πe lim d→∞ d − ℓ + 2 (d − ℓ) d−ℓ d−ℓ+2 = κ2πe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Note that, in particular, ad → ∞ as d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We conclude that limd→∞ fd(v, h) = eκ2πeh uniformly as long as h stays bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' By standard results [24, Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='19], this also implies weak convergence of the corresponding Poisson point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' After we have shown the convergence of the point processes ξℓ,d to ξℓ,∞, as d → ∞, we explain the procedure allowing to transfer this result to the convergence of the corresponding tessellations LΨ(ξℓ,d) to LΨ(ξℓ,∞) as d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Note that LΨ(ξℓ,d) has the same distribution as Wd,ρd ∩ Rℓ and LΨ(ξℓ,∞) has the same distribution as Gℓ,κ2πe (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We fix an ℓ-dimensional ball BR ⊂ Rℓ of radius R > 0 centred at the origin and for any ε > 0 we aim to find a region K(R, ε) ⊂ Rℓ+1, independent of d, such that with probability at least 1−ε the restrictions of the tessellations LΨ(ξℓ,d) and LΨ(ξℓ,∞) to BR are completely determined by the restrictions of the point processes ξℓ,d and ξℓ,∞ to K(R, ε) for any d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' To this end, we note that LΨ(ξℓ,d) may be regarded as a vertical projection along the h-axis of the boundary of the corresponding paraboloid growth process Ψ(ξℓ,d) (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' From this it follows that if the restrictions of the tessellations LΨ(ξℓ,d) and LΨ(ξℓ,∞) to BR do not coincide, then the boundaries of the corresponding paraboloid hull processes bd Ψ(ξℓ,d) and bd Ψ(ξℓ,∞) restricted to the cylinder BR × R do not coincide as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The construction of the region K(R, ε) can be now performed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' First, we consider the event E(T, r) that bd Ψ(ξℓ,d) restricted to the cylinder BR ×R is completely determined by the restriction of bd Ψ(ξℓ,d) to the cylinder BR+r ×(−∞, T] for some T, r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' By this we mean that for every paraboloid Π+,x with x ∈ ext(Ψ(ξℓ,d)) the set Π+,x ∩ bd Ψ(ξℓ,d) either does not intersect BR × R or is included in BR+r × (−∞, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' We have that 1 − P(E(T, r)) ≤ c1(R + r)ℓ(ec2(4T−r2) + e−c3T c4), r, d > c5, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2) where all constants c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' , c5 are positive and independent of the parameters d, r and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Since the proof of this estimate follows exactly the same route as the proof of [8, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4] (estimate for T) and [8, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5] (estimate for r), we decided to omit the technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In particular, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2) shows that for any ε > 0 there is a choice of T0 := T(ε) and r0 := r(ε) such that P(E(T0, r0)) ≥ 1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The same holds for Ψ(ξℓ,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Further, we note that if a paraboloid Π+,x with x ∈ ext(Ψ(ξℓ,d)) is such that Π+,x ∩ bd Ψ(ξℓ,d) ⊂ BR+r0 × (−∞, T0], then x ∈ {(v, h) ∈ Rℓ+1 : h ≤ T0, ∥v∥ ≤ R + r0 + � T0 − h} =: K(R, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' To complete the proof, it suffices to argue that there exists a coupling of ξℓ,d and ξℓ,∞ on a common probability space such that the probability that the restrictions of these processes to the region K(R, ε) do not coincide converges to 0, as d → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In a suitable coupling, this probability is bounded above by a constant multiple of the L1-norm of the difference of their intensity measures restricted to K(R, ε), see [23, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' As we have shown above, the densities fd(v, h) converge to eκ2πeh as d → ∞ uniformly 14 on compact sets and hence pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Also, by the inequality 1 + x ≤ ex, we have fd(v, h) ≤ ec6h for some absolute constant c6 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The fact that this upper bound is integrable over K(r, ε) has been shown in [8, Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='18)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Thus, fd(v, h) → eκ2πeh with respect to the L1-norm on K(R, ε), which ensures that the required coupling of the Poisson processes indeed exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' This completes the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Our aim is now to prove the weak convergence of the typical cell of the sectional Poisson-Voronoi tessellation to the typical cell of the Gaussian-Voronoi tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Fix some ℓ ∈ N and let C be the space of compact subsets of Rℓ endowed with the Hausdorff metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Put C′ = C\\{∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The typical cells considered below are defined with respect to some fixed centre function z : C′ → Rℓ in the sense of Section 2, additionally satisfying z(C) ∈ C for every C ∈ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Take any positive sequence (ρd)d∈N with limd→∞(ρd)1/d = κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Then, as d → ∞, the distribution of the typical cell of the sectional Poisson-Voronoi tessellation Wd,ρd ∩ Rℓ converges to the distribution of the typical cell of the Gaussian-Voronoi tessellation Gℓ,κ2πe weakly on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' This theorem is a consequence of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='1 and the following general result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Let (Tn)n∈N be a sequence of stationary random tessellations on Rℓ converging to a stationary random tessellation T∞ on Rℓ in the following sense: All random tessellations are defined on a common probability space and for every R > 0 we have lim n→∞ P[An(R)] = 1, where An(R) is the event that the restrictions of Tn and T∞ to [−R, R]ℓ coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' More precisely, An(R) = A′ n(R) ∩ A′′ n(R) with A′ n(R) := {for every C ∈ Tn s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' C ∩ [−R, R]ℓ ̸= ∅ there is C′ ∈ T∞ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' C ∩ [−R, R]ℓ = C′ ∩ [−R, R]ℓ}, A′′ n(R) := {for every C′ ∈ T∞ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' C′ ∩ [−R, R]ℓ ̸= ∅ there is C ∈ Tn s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' C ∩ [−R, R]ℓ = C′ ∩ [−R, R]ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Also, suppose that the cell intensity of Tn converges to that of T∞, that is lim n→∞ γℓ(Tn) = γℓ(T∞), and that all these intensities are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Then, the distribution of the typical cell of Tn converges to the distribution of the typical cell of T∞ weakly on C: Pz Tn,ℓ −→ n→∞Pz T∞,ℓ, weakly on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Let f : C → R be a bounded continuous function and recall that z(C) is the centre of a cell C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Consider the random variables ξn := � C∈Tn, z(C)∈[0,1]ℓ f(C), n ∈ N ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' By definition, the distribution of the typical cell of Tn satisfies � C f dPz Tn,ℓ = Eξn γℓ(Tn), n ∈ N ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Since lim n→∞ γℓ(Tn) = γℓ(T∞), to prove (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='3) it suffices to check that lim n→∞ Eξn = Eξ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4) In the following we shall define a “good” event Cn(R) on which ξn and ξ∞ are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' For n ∈ N ∪ {∞} and R > 3 consider the random event Bn(R) := {∄C ∈ Tn : C ∩ [0, 1]ℓ ̸= ∅, C ̸⊂ [−R/2, R/2]ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' A cell C with the properties listed in this definition is called a “long cell” in the tessellation Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The event Bn(R) occurs if there is no long cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' On the event An(R), each long cell C ∈ Tn corresponds to a long cell C′ ∈ T∞ with the same restriction to [−R, R]ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It follows that An(R) ∩ (Bn(R))c ⊂ (B∞(R))c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 15 Since all cells in T∞ are bounded almost surely, the maximal diameter of a cell in T∞ intersecting [0, 1]ℓ is some almost surely finite random variable M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It follows that lim R→∞ sup n∈N P[An(R) ∩ (Bn(R))c] ≤ lim R→∞ P[(B∞(R))c] ≤ lim R→∞ P[M > (R/2) − 1] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Recall also that limn→∞ P[An(R)] = 1 for every fixed R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Consider now the “good” event Cn(R) := An(R) ∩ Bn(R) and the “bad” event Dn(R) := (Cn(R))c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It follows from the above that lim R→∞ lim sup n→∞ P[Dn(R)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5) Note that, on the event Cn(R), the sets {C ∈ Tn : z(C) ∈ [0, 1]ℓ} and {C′ ∈ T∞ : z(C′) ∈ [0, 1]ℓ} are equal, implying that ξn = ξ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Indeed, every cell C ∈ Tn with z(C) ∈ [0, 1]ℓ is contained in [−R/2, R/2]ℓ (since z(C) ∈ C and there is no long cell) and, consequently, C is also a cell of T∞ (since An(R) occurs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Conversely, for every cell C′ ∈ T∞ with z(C′) ∈ [0, 1]ℓ there is a cell C ∈ Tn such that the restrictions of C and C′ to [−R, R]ℓ coincide (since An(R) occurs) and C is not long (since Bn(R) occurs), which implies that C = C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' It follows that E[1(Cn(R)) ξn] = E[1(Cn(R)) ξ∞], n ∈ N, R > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' To complete the proof of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='4), it suffices to verify that lim R→∞ lim sup n→∞ E[1(Dn(R)) ξn] = 0, lim R→∞ lim sup n→∞ E[1(Dn(R)) ξ∞] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' In view of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5), it suffices to check that the family {ξn : n ∈ N ∪ {∞}} is uniformly integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Now, |ξn| ≤ ∥f∥∞ηn with ηn := � C∈Tn, z(C)∈[0,1]ℓ 1, n ∈ N ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' As we already observed above, outside the event Dn(R) we have ηn = η∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' From (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='5) it follows that ηn → η∞ almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' On the other hand, we have Eηn = γℓ(Tn) → γℓ(T∞) = Eη∞ by assumption of the proposition (and all these expectations are finite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' These two properties together with ηn ≥ 0 imply that the family {ηn : n ∈ N∪{∞}} is uniformly integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Indeed, if this were not the case, we could find ε > 0 such that, after passing to a subsequence, E[ηn1(Qn)] > ε for some events Qn with P[Qn] < 1/2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Let Wk := ∪∞ n=kQn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' If K is sufficiently large, then E[η∞1(WK)] < ε/2, while E[ηn1(WK)] > ε for all n ≥ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Applying Fatou’s lemma to the variables (ηn1((WK)c))n≥K leads to a contradiction with the assumption Eηn → Eη∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Finally, the bound |ξn| ≤ ∥f∥∞ηn implies that the family {ξn : n ∈ N ∪ {∞}} is uniformly integrable as well, and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Acknowledgement ZK and CT were supported by the DFG priority program SPP 2265 Random Geometric Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' AG and ZK were supported by the DFG under Germany’s Excellence Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics - 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Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=', 31(4):1598–1632, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Chiu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' van de Weygaert, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Stoyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The sectional Poisson-Voronoi tessellation is not a Voronoi tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' in Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=', 28(2):356–376, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Gusakova, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Kabluchko, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Thäle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The β-Delaunay tessellation: Description of the model and geometry of typical cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' in Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=', 54(4):1252–1290, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Gusakova, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Kabluchko, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Thäle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The β-Delaunay tessellation II: The Gaussian limit tessellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=', (27):1 – 33, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Gusakova, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Kabluchko, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Thäle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The β-Delaunay tessellation III: Kendall’s problem and limit theorems in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' ALEA, Lat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=', 19:23 – 50, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Gusakova, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Kabluchko, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Thäle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The β-Delaunay tessellation IV: Mixing properties and central limit theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' and Dynamics, accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Preprint at: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='org/abs/2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content='09472, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' [11] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Kabluchko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Angles of random simplices and face numbers of random polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=', 380:Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' 107612, 68, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' [12] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Kabluchko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Recursive scheme for angles of random simplices, and applications to random polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Discrete Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=', 66:902–937, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' [13] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Kabluchko, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Temesvari, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Thäle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Expected intrinsic volumes and facet numbers of random beta- polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Nachr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=', 292(1):79–105, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' [14] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Kabluchko and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Thäle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' The typical cell of a Voronoi tessellation on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Discrete Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=', 66:1330–1350, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfmgQS/content/2301.03297v1.pdf'} +page_content=' [15] Z.' metadata={'source': 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A State of the Art +Review of large Generative AI models +Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an +Quantitative Methods Department, Universidad Pontificia Comillas, Madrid, Spain +201905616@alu.comillas.edu, ecgarrido@icade.comillas.edu +Abstract. During the last two years there has been a plethora of large +generative models such as ChatGPT or Stable Diffusion that have been +published. Concretely, these models are able to perform tasks such as +being a general question and answering system or automatically creat- +ing artistic images that are revolutionizing several sectors. Consequently, +the implications that these generative models have in the industry and +society are enormous, as several job positions may be transformed. For +example, Generative AI is capable of transforming effectively and cre- +atively texts to images, like the DALLE-2 model; text to 3D images, +like the Dreamfusion model; images to text, like the Flamingo model; +texts to video, like the Phenaki model; texts to audio, like the AudioLM +model; texts to other texts, like ChatGPT; texts to code, like the Codex +model; texts to scientific texts, like the Galactica model or even create +algorithms like AlphaTensor. This work consists on an attempt to de- +scribe in a concise way the main models are sectors that are affected by +generative AI and to provide a taxonomy of the main generative models +published recently. +1 +Introduction +Generative AI refers to artificial intelligence that can generate novel content, +rather than simply analyzing or acting on existing data like expert systems [23]. +In particular, expert systems contained knowledge bases and an inference engine +that generated content via an if-else rule database. However, modern generative +artificial intelligence contain a discriminator or transformer model trained on a +corpus or dataset that is able to map the input information into a latent high- +dimensional space and a generator model, that is able to generate an stochastic +behaviour creating novel content in every new trial even from the same prompts +as an input, performing unsupervised, semi-supervised or supervised learning, +depending on the particular methodology. Regarding the created content by the +model, generative artificial intelligence models are different from predictive ma- +chine learning systems, that merely perform a discrimination behaviour, solv- +ing classification or regression problems. In particular, these models are able +to discriminate information and generate information of the transformed input +information, or prompt. +The key aspect about generative models is that their architecture and the +data that they have been fed is enormous. For example, it is possible now to +arXiv:2301.04655v1 [cs.LG] 11 Jan 2023 + +2 +Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an +estimate the parameters of the model by feeding it the contents of the whole +Wikipedia, Github, social networks, Google images and more. Despite being fed +with an enormous size of data, thanks to the rise of computing we can design +deep neural networks [18], transformers [22] and other models such as genera- +tive adversarial networks [9] or variational autoencoders [23] whose capacity is +able to model the complexity of the data, without suffering from underfitting. +As they are able to modelize the high-dimensional probability distribution of +language or photos of a concrete or general domain, if they are complemented +by generative models that map the latent high-dimensional semantic space of +language of photos to a multimedia representation of text, audio or video we +can map any input format like texts to any output format like video. In this +sense, applications of this technology are endless, in the sense that we can train +a model to generate genuine different multimedia formats as video, audio or text +from different multimedia input formats, as for example, text. +We believe that it is necessary to provide a state-of-the-art review on the +most popular generative AI models as they are revolutionizing several indus- +tries like the art industry [2] or universities [16,30]. As models are now able to +generate genuine artistic content or large texts answering a prompt, these two +industries and other ones that we will detail throughout this manuscript will +need to readapt their activity to continue providing value. In this sense, gen- +erative AI models will not replace humans but enhance our content, being an +inspiration for artists or improving the content generated by professors. In order +to provide information for a professional working in any industry that can be +benefited by these models we have made the organization of the paper as the +following one. First, we will provide a taxonomy of the main generative models +that have appeared in the industry. Then, the following sections will analyze +each of the categories of the taxonomy. Finally, we finish the manuscript with a +conclusions and further work section. We do not study the technical aspects of +every model, such as transformers in detail as our purpose in this review is on the +applications of the models and the content that they generative but not on how +they work. For a detailed explanation of deep learning models and generative +models we recommend other references [18,23]. +2 +A Taxonomy of Generative AI models +Before analyzing each model in detail, we have tried to organize the current +generative artificial models into a taxonomy whose categories represent the main +mappings between each multimedia input and output type of data. The result +is the one that we have illustrated in Figure 1. We have discovered a total of 9 +categories, where each of the models that appear in Figure 1 will be described +in detail in the following section. Each of the covered models has been published +recently, as we illustrate in Figure 2, as our main concern in this manuscript is +to describe the latest advances in generative AI models. +Interestingly, only six organizations are behind the deployment of these mod- +els, as we illustrate in Figure 3. The main reason behind this fact is that in order + +State of the Art of Generative AI +3 +Fig. 1. A taxonomy of the most popular generative AI models that have recently +appeared classified according to their input and generated formats. +Fig. 2. Covered models by date of release. All models were released during 2022 except +LaMDA, which was released in 2021 and Muse, in 2023. + +Text-to- +Text-to- +Image-to- +image +3D +text +DALL-E +Stable +2 +Dreamfusion +Magic3D +Flamingo +Diffusion +VisualGPT +Imagen +Muse +Text.to- +Text.to- +Text-To- +Video +Audio +Text +Phenaki +Soundify +AudioLM +Whisper +ChatGPT3 +LaMDA +Jukebox +PEER +Speech From +Brain +Text.to. +Text-to +Other +Code +Science +Models +Codex +Alphacode +Galactica +Minerva +Alphatensor +GATO +Human Motion +Diffusion ModelMeta Al Speech From Brain +Muse +Whisper +PEER +Stable Diffusion +Hurnan Motion Diffusion +Model +Jukebox +Imagen +VisualGPT +Magic3D +Minerva +LaMDA +Soundify +GATO +Alphacode +Flamingo +DallI-E-2 +Galadtica +Alphatensor +Codex +Dreamfusion +ChatGPT3 +AudioL +04/01/2021 +14/04/2021 +23/07/2021 +31/10/2021 +08/02/2022 +19/05/2022 +27/08/2022 +05/12/20224 +Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an +to be able to estimate the parameters of these models, it is mandatory to have +an enormous computation power and a highly skilled and experienced team in +data science and data engineering. Consequently, only the companies shown on +Figure 3, with the help of acquired startups and collaborations with academia, +have been able to successfully deploy generative artificial intelligence models. +Fig. 3. Models by developer. In terms of major companies participating in startups, +note that Microsoft invested 1 billion dollars in OpenAI and helps them with the de- +velopment of models. As well, note that Google acquired Deepmind in 2014. In terms +of universities, note that VisualGPT was developed by KAUST, Carnegie Mellon Uni- +versity and Nanyang Technological University and that the Human Motion Diffusion +Model was developed by Tel Aviv University, Israel. As well, other projects are de- +veloped by a company in collaboration with a university. Concretely, this is the case +for Stable Diffsion (Runway, Stability AI and LMU MUNICH), Soundify (Runway and +Carnegie Mellon University) and DreamFusion (Google and UC Berkeley) + +GoogleResearch +openAI +DALL-E 2 +Chat +GPT3 +Imagen +Muse +Jukebox +Whisper +DreamFusion +Phenaki +Minerva +AudioLM +LaMDA +ODeepMnd +00MetaAl +Flamingo +Alphacode +PEER +Speech +From Brain +Alphatensor +GATO +Galactica +Rrunway +nVIDIA +Stable +Diffusion +Soundify +Magic3DState of the Art of Generative AI +5 +Now that we have provided and analyzed the latest generative artificial in- +telligence models, the following section will cover each of the categories of the +taxonomy presented in Figure 1 in detail. +3 +Generative AI models categories +In this section we will cover in detail the nine categories described in Figure 1 of +the previous section. For every category, we illustrate the details of the models +shown in Figure 1. +3.1 +Text-to-image models +We begin the review by considering the models whose input is a text prompt +and whose output is an image. +DALL·E 2 : DALL·E 2, created by OpenAI, is able to generate original, genuine +and realistic images and art from a prompt consisting on a text description [10]. +Luckily, it is possible to use the OPENAI API to get access to this model. In +particular, DALL·E 2 manages to combine concepts, attributes and different +styles. In order to do so, it uses the CLIP neural network. CLIP (Contrastive +Language-Image Pre-Training) is a neural network trained on a variety of (image, +text) pairs [25]. Using CLIP, that can be instructed in natural language to predict +the most relevant text snippet, given an image, the model has recently merged +as a successful representation learner for images. Concretely, CLIP embeddings +have several desirable properties: they are robust to image distribution shift, have +impressive zero-shot capabilities and have been fine-tuned to achieve state-of-the- +art results. In order to obtain a full generative model of images, the CLIP image +embedding decoder module is combined with a prior model, which generates +possible CLIP image embeddings from a given text caption. We illustrate an +image generated from a prompt in Figure 4 +Fig. 4. Image generated from the prompt ”A shiba inu wearing a beret and black +turtleneck”. + +6 +Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an +IMAGEN : Imagen is a text-to-image diffusion model [17] consisting on large +transformer language models. Critically, the main discovery observed with this +model made is that large language models, pre-trained on text-only corpora, are +very effective at encoding text for image synthesis [28]. Precisely, using Imagen, +it has been found out that increasing the size of the language model boosts both +sample fidelity and image-text alignment much more than increasing the size of +the image diffusion model. The model was created by Google and the API can +be found in their web page. For the evaluation of their model, Google created +Drawbench, a set of 200 prompts that support the evaluation and comparison of +text-to-image models. Most concretely, the model is based on a pretrained text +encoder (like BERT [12]) that performs a mapping from text to a sequence of +word embeddings and a cascade of conditional diffusion models that map these +embeddings to images of increasing resolutions. We show an image generated +from a prompt in Figure 5. +Fig. 5. Image generated from the prompt ”A cute corgi lives in a house made out of +sushi”. +Stable Diffusion : Stable Diffusion is a latent-diffusion model that is open- +source and has been developed by the CompVis group at LMU Munich. The +main difference of this model with respect to the other ones is the use of a +latent diffusion model and that it performs image modification as it can perform +operations in its latent space. For Stable Diffusion, we can use the API via their +website. More concretely, Stable Diffusion consists of two parts: the text encoder +and the image generator [17]. The image information creator works completely +in the latent space. This property makes it faster than previous diffusion models +that worked in a pixel space. We illustrate a Stable Diffusion image example in +Figure 7. + +State of the Art of Generative AI +7 +Fig. 6. Image generated from the prompt ”A cute corgi lives in a house made out of +sushi”. +Muse : This model is a Text-to-image transformer model that achieves state-of- +the-art image generation while being more efficient than diffusion or autoregres- +sive models [6]. Concretely, it is trained on a masked modelling task in discrete +token space. Consequently, it is more efficient because of the use of discrete +tokens and requiring fewer sampling iterations. Compared to Parti, a autore- +gressive model, Muse is more efficient because of parallel decoding. Muse is 10x +faster at inference time than Imagen-3B or Parti-3B and 3x faster than Stable +Diffusion v 1.4. Muse is also faster than than Stable Diffusion in spite of both +models working in the latent space of a VQGAN. We append a comparison of +the generated images by DALL·E 2, IMAGEN and Muse in Figure ??. +3.2 +Text-to-3D models +The models that have been described in the previous section deal with the map- +ping of text prompts to 2D images. However, for some industries like gaming, +it is necessary to generate 3D images. In this section, we briefly describe two +text-to-3D models: Dreamfusion and Magic3D. +Dreamfusion : DreamFusion is a text-to-3D model developed by Google Re- +search that uses a pretrained 2D text-to-image diffusion model to perform text- +to-3D synthesis [24]. In particular, Dreamfusion replaces previous CLIP tech- +niques with a loss derived from distillation of a 2D diffusion model. Concretely, +the diffusion model can be used as a loss within a generic continuous optimization +problem to generate samples. Critically, sampling in parameter space is much +harder than in pixels as we want to create 3D models that look like good images +when rendered from random angles. To solve the issue, this model uses a differ- +entiable generator. Other approaches are focused on sampling pixels, however, +this model instead focuses on creating 3D models that look like good images +when rendered from random angles. We illustrate in Figure 8 an example of + +8 +Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an +Fig. 7. Comparison of generated images by the DALL·E 2, IMAGEN and Muse models +with respect to the prompts that appear in the column of the left. The first column of +images contains the results generated by DALL·E 2, the second the results obtained +with IMAGEN and the third the images created by Muse. +an image created by Dreamfusion from one particular angle along with all the +variations that can be generated from additional text prompts. In order to see +the full animated image, we recommend to visit the web page of Dreamfusion. +Fig. 8. A 3D squirrel generated by Dreamfusion is shown at the left. Then, the other +images contain the modifications generated to the squirrel with text prompts like ”wear- +ing a jacket”. + +DALL-E 2 +Imagen +MUSE +A high contrast +portrait of a very +happy fuzzy panda +dressed as a chef +in a high end kit- +chen making dough. +There is a painting +of flowers on the +wall behind him. +Rainbow coloured +penguin.a DSLR +wearing +Bsup!! +onaroad +photo of +a leather +motorcycle +made of ice +jacket +a squirrelState of the Art of Generative AI +9 +Magic3D : This model is a text to 3D model made by NVIDIA Corporation. +While the Dreamfusion model achieves remarkable results, the method has two +problems: mainly, the long processing time and the low-quality of the generated +images. However, these problems are addressed by Magic3D using a two-stage +optimization framework [20]. Firstly, Magic3D builds a low-resolution diffusion +prior and, then, it accelerates with a sparse 3D hash grid structure. Using that, +a textured 3D mesh model is furthered optimized with an efficient differentiable +render. Comparatively, regarding human evaluation, the model achieves better +results, as 61.7% prefer this model to DreamFusion. As we can see in Figure 9, +Magic3D achieves much higher quality 3D shapes in both geometry and texture +compared to DreamFusion. +Fig. 9. 3D Images generated by Magic3D and Dreamfusion, where ”Ours” refer to +Magic3D. We can see a total of 8 text prompts and the images that both models +generate from that prompts. +3.3 +Image-to-Text models +Sometimes, it is also useful to obtain a text that describes an image, that is +precisely the inverse mapping to the one that has been analyzed in the previous +subsections. In this section, we analyze two models that perform this task, along +with others: Flamingo and VisualGPT. +Flamingo : A Visual Language Model created by Deepmind using few shot +learning on a wide range of open-ended vision and language tasks, simply by +being prompted with a few input/output examples [1]. Concretely, the input of +Flamingo contains visually conditioned autoregressive text generation models +able to ingest a sequence of text tokens interleaved with images and/or videos + +Ours +DreamFusion[33] +Ours +DreamFusion[33] +6 +a kingfisher birdi +car made out of sushi* +anicecreamsundae +a beautifully carved wooden knight chess piecei +a small saguaro cactusplantedinaclaypor +A very beautiful tiny human heart organic sculpture made of copper wire +and threaded pipes, very intricate, curved, Studio lighting. high resolution* +modelofan adorable cottage +with athatched +aripestrawberm10 +Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an +and produce text as output. A query is made to the model along with a photo +or a video and the model answers with a text answer. Some examples can be +observed in Figure 10. Flamingo models take advantage of two complementary +models: a vision model that analyzes visual scenes and a large language model +which performs a basic form of reasoning. The language model is trained on a +large amount of text data. +Fig. 10. Input prompts that contain images and text and output generated text re- +spones from Flamingo. Every column contains a single example where we can see how +Flamingo answers the question using the image from the text. +VisualGPT : VisualGPT is an image captioning model made by OpenAI [7]. +Concretely, VisualGPT leverages knowledge from the pretrained language model +GPT-2 [5]. In order to bridge the semantic gap between different modalities, a +novel encoder-decoder attention mechanism [33] is designed with an unsaturated +rectified gating function. Critically, the biggest advantage of this model is that +it does not need for as much data as other image-to-text models. In particular, +improving data efficiency in image captioning networks would enable quick data +curation, description of rare objects, and applications in specialized domains. +Most interestingly, the API of this model can be found on GitHub. We include +three examples of text prompts generated by the model with respect to three +images fed to the model in Figure 11. +3.4 +Text-to-Video models +As we have seen in the previous subsections, it is now possible to generate images +from text. Consequently, the next logical step is to generate videos, that are + +Question: What do you think +Question: What is odd about +Question: What country is +the capacities of these are? +this image? Explain why it is +this? Why do you think so? +Answer: +unusual. Answer: +Answer: +p Completion +The image is odd because +the elephant is in the back +It is Canada. I think so +The floppy disk is 1.44MB +of the truck. It is unusual +and the CD is 700MIB. +because the flag is the +because elephants are not +Canadian flag. +usually transported in the +back of a truck.State of the Art of Generative AI +11 +Fig. 11. Three examples of text prompts generated by the images shown on the left. +We also show the attention scores that the model assign to every word of the texts. +In the third image, we can see for example how the most discriminative information +about the image is the word ”cat” and ”television”. +sequences of images, from texts. In this section, we provide information about +two models that are able to perform this task: Phenaki and Soundify. +Phenaki : This model has been made by Google Research, and it is capable +of performing realistic video synthesis, given a sequence of textual prompts [34]. +Most interestingly, we can get access to the API of the model from GitHub. In +particular, Phenaki is the first model that can generate videos from open domain +time variable prompts. To address data issues, it performs joint training on a +large image-text pairs dataset as well as a smaller number of video-text exam- +ples can result in generalization beyond what is available in the video datasets. +This is mainly due to image-text datasets having billions of inputs while text- + +GT: the lady is sitting on the wood bench +ours +woman sitting +on +e +benchin +apark +attention +0.7 +0.78 +0.82 +0.760.80.96 +0.80.690.85 +GT: a laptop with a keyboard and mouse are on this desk +ours +alaptop sitting +on +adeskwithamouse +attention +0.7 +0.78 +0.81 +0.70.7 +0.920.850.640.76 +GT: a cat is sitting in front of a television +Ours +sitting +frontof +television +attention 0.80.860.80.83 +0.7 0.720.60.71 +0.93 +GT: a number of people sitting on a snowy surface with skis +Ours +couple +of +people sitting +on +snowy suriace +attention +0.8 +0.87 +0.71 +0.85 +0.91 +0.76 0.71 +0.940.9512 +Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an +video datasets are much smaller. As well, limitations come from computational +capabilities for videos of variable length. +The model has three parts: the C-ViViT encoder, the training transformer +and the video generator. The encoder gets a compressed representation of videos. +First tokens are transformed into embeddings. This is followed by the temporal +transformer, then the spatial transformer. After the output of the spatial trans- +former, they apply a single linear projection without activation to map the tokens +back to pixel space. Consequently, the model generates temporally coherent and +diverse videos conditioned on open domain prompts even when the prompt is a +new composition of concepts. The videos can be minutes long, while the model +is trained on 1.4 second videos. Below we show in Figure 12 and in Figure 13 +some examples of the creation of a video through a series of text prompts and +from a series of text prompts and an image. +Fig. 12. Sequence of images created by the Phenaki model given four different prompts. + +1st prompt:"A photorealistic teddy bear is swimming in the ocean at San Francisco" +2nd prompt:“The teddy bear goes under water +3rd prompt: "The teddy bear keeps swimming under the water with colorful fishes' +4rd prompt:"A panda bear is swimming under water"State of the Art of Generative AI +13 +Fig. 13. Sequences of images created by the Phenaki model given an image and the +prompt. We can see how the model is able to manipulate the given image according to +the text prompt. +Soundify : In video editing, sound in half of the story. But, for professional +video editing, the problems come from finding suitable sounds, aligning sounds, +video and tuning parameters [21]. In order to solve this issue, Soundify is a +system developed by Runway that matches sound effects to video. This system +uses quality sound effects libraries and CLIP (a neural network with zero-shot +image classification capabilities cited before). Concretely, the system has three +parts: classification, synchronization, and mix. The classification matches effects +to a video by classifying sound emitters within. To reduce the distinct sound +emitters, the video is split based on absolute color histogram distances. In the +synchronization part, intervals are identified comparing effects label with each +frame and pinpointing consecutive matches above a threshold. In the mix part, +effects are split into around one-second chunks. Critically, chunks are stitched +via crossfades. +3.5 +Text-to-Audio models +As we have seen in the previous subsection, images are not the only important +non-structured data format. For videos, for music and in lots of contexts, audio +can be critical. Consequently, we analyze in this subsection three models whose +input information is text and whose output information is audio. +AudioLM : This model has been made by Google for high-quality audio gener- +ation with long-term consistency. In particular, AudioLM maps the input audio +into a sequence of discrete tokens and casts audio generation as language mod- +eling task in this representation space [4]. By training on large corpora of raw + +Given Image +Prompt: "Camera zooms quickly into the eye of the cat" +Given Image +I Prompt: "A white cat touches the camera with the paw" +Given Image +Prompt: "A white cat yawns loudly"14 +Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an +audio waveforms, AudioLM learns to generate natural and coherent continua- +tions given short prompts. The approach can be extended beyond speech by +generating coherent piano music continuations, despite being trained without +any symbolic representation of music. As with the other models, the API can +be found through GitHub. Audio signals involve multiple scales of abstractions. +When it comes to audio synthesis, multiple scales make achieving high audio +quality while displaying consistency very challenging. This gets achieved by this +model by combining recent advances in neural audio compression, self-supervised +representation learning and language modelling. +In terms of subjective evaluation, raters were asked to listen to a sample of +10 seconds and decide whether it is human speech or a synthetic continuation. +Based on 1000 ratings collected, the rate is 51.2%, which is not statistically +significant from assigning labels at random. This tells us that humans cannot +differentiate between synthetic and real samples. +Jukebox : This is a model, developed by OpenAI, that generates music with +singing in the raw audio domain [13]. Once again, its API can be found in +GitHub. Previously, earlier models in the text-to-music genre generated music +symbolically in the form of a pianoroll which specifies timing, pitch and velocity. +The challenging aspect is the non-symbolic approach where music is tried to be +produced directly as a piece of audio. In fact, the space of raw audio is extremely +high dimensional which makes the problem very challenging. Consequently, the +key issue is that modelling that raw audio produces long-range dependencies, +making it computationally challenging to learn the high-level semantics of music. +In order to solve this issue, this model tries to solve it by means of a hi- +erarchical VQ-VAE architecture to compress audio into a discrete space [14], +with a loss function designed to retain the most amount of information. This +model produces songs from very different genres such as rock, hip-hop and jazz. +However, the model is just limited to English songs. Concretely, its dataset for +training is from 1.2 million songs from LyricWiki. The VQ-VAE has 5 billion +parameters and is trained on 9-second audio clips for 3 days. +Whisper : This model is an Audio-to-Text converter developed by OpenAI. It +achieves several tasks in this field: multi-lingual speech recognition, translation +and language identification [26]. As in previous cases, its API can be found +in the GitHub website. The goal of a speech recognition system should be to +work reliably out of the box in a broad range of environments without requiring +supervised fine-tuning of a decoder for every deployment distribution. This is +hard because of the lack of a high-quality pre-trained decoder. +Concretely, this model is trained on 680,000 hours of labeled audio data. +This data is collected from the internet, which results in a very diverse dataset +covering a broad distribution of audio from many different environments, record- +ings setups, speakers and languages. The model makes sure that the dataset is +only from human voice as machine learning voice would impair the model. Files + +State of the Art of Generative AI +15 +are broken in 30 second segments paired with the subset of the transcript that +occurs within that time segment. +The model has an encoder-deccoder transformer, as this architecture has +been validated to scale reliably. We can observe the model’s architecture char- +acteristics through the figure below. We can see the different types of data and +the learning sequence. +3.6 +Text-to-Text models +The previous models all convert a non-structured data type into another one. +But, regarding text, it is very useful to convert text into another text in order to +satisfy tasks as general question and answering. The following four models treat +text and also output texts to satisfy different needs. +ChatGPT : The popular ChatGPT is a model by OpenAI which interacts +in a conversational way. As it is widely known, the model answers follow-up +questions, challenges incorrect premises and reject inappropriate requests. More +concretely, the algorithm behind ChatGPT is based on a transformer. However, +the training is made through Reinforcement Learning for Human Feedback. In +particular, an initial model is trained using supervised fine-tuning: human AI +trainers would provide conversations in which they played both sides, the user +and an AI assistant. Then, those people would be given the model-written re- +sponses to help them compose their response. This dataset was mixed to that of +InstructGPT [3], which was transformed into a dialogue format. A demo can be +found in their website and the API may also be found in OpenAI’s website. We +summarize the main steps of ChatGPT training in Figure 14, available in the +ChatGPT demo’s website. Finally, ChatGPT is also able to generate code and +simple mathematics. +LaMDA : LaMDA is a language model for dialog applications [32]. Unlike +most other language models, LaMDA was trained on dialogue. It is a family +of transformer-based neural language models specialized for dialog which have +up to 137B parameters and are pre-trained on 1.56T words of public dialog +data and web text. Fine-tuning can enable for safety and factual grounding of +the model. Only 0.001% of training data was used for fine-tuning, which is a +great achievement of the model. In particular, dialog modes take advantage of +Transformers’ ability to present long-term dependencies in text. Concretely, they +are generally very well-suited for model scaling. Consequently, LaMDA makes +use of a single model to perform multiple tasks: it generates several responses, +which are filtered for safety, grounded on an external knowledge source and re- +ranked to find the highest-quality response. We illustrate in Figure 15 an example +of a dialog with the model. +PEER : Collaborative language model developed by Meta AI research trained +on edit histories to cover the entire writing process [29]. It is based on four + +16 +Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an +Fig. 14. Training steps of ChatGPT, combining supervised learning with reinforcement +learning. +Fig. 15. Example of a dialog made with LaMDA. +steps: Plan, Edit, Explain and Repeat. These steps are repeated until the text +is in a satisfactory state that requires no further updates. The model allow to +decompose the task of writing a paper into multiple easier subtasks. As well, +the model allows humans to intervene at any time and steer the model in any +direction. +It is mainly trained on Wikipedia edit histories. The approach is a self- +training, using models to infill missing data and then train other models on this +synthetic data. The downside of this comes from comments being very noisy +and a lack of citations, which tries to be compensated by a retrieval system +which does not always work. The framework is based on an iterative process. + +Step1 +Step 2 +Step 3 +Collectdemonstrationdata +Collectcomparisondataand +Optimizeapolicyagainstthe +and traina supervised policy. +train a reward model. +rewardmodelusingthePPO +reinforcementlearningalgorithm. +A prompt is +A prompt and +Anewpromptis +sampledfromour +Explain reinforcement +several model +Explain reinforcement +sampled from +Write a story +promptdataset. +learningtoa6 yearold. +outputsare +learning to a 6 year old. +the dataset. +about otters. +sampled. +Explainr +agent is +The PPO model is +PPO +Alabeler +0 +initialized from the +demonstratesthe +supervised policy +desired output +teach +behavior. +We give treats and +punishments to teach. +Alabelerranksthe +Thepolicygenerates +Once upon a time +outputsfrombest +an output. +SFT +to worst. +0-0-0.0 +This datais used to +The reward model +RM +fine-tune GPT-3.5 +calculates a reward +with supervised +learning. +RM +forthe output. +This data is used +to train our +The reward is used +reward model. +to update the +0000 +rk +policy using PPO.LaMDA Everest +LaMDA: +Hi, I'm Mount Everest. What would you like to know about me? +User: +Why do you think people climb you? +LaMDA: +Because I represent a very high achievement to climb me. Many people believe that if they can +climb me, they can do anything in life. +User: +Who was the first person to do that? +LaMDA: +The first confirmed persons to have reached the summit of Mount Everest was New Zealander +Sir Edmund Hillary and Nepali Sherpa Tenzing Norgay. +https://en.wikipedia.org/wiki/Mount_Everest +User: +How old was Hillary then? +LaMIDA: +He was 33 years old when he first reached the top of me in 1953! +https://en.wikipedia.org/wiki/Edmund_Hillary +User: +When people fail, is it dangerous? +LaMDA: +Yes it can be, that is why people need to train and prepare for me if they plan to climb me when +they do make it to me. I'm very hard to climb and can be dangerous when there is a whiteout +Which means everything is white with very strong winds!State of the Art of Generative AI +17 +The entire process of formulating a plan, collecting documents, performing an +edit and explaining it can be repeated multiple times until arriving at a sequence +of texts. For the training, a DeepSpeed transformer is used. +Meta AI Speech from Brain : Model developed by Meta AI to help people +unable to communicate through speech, typing or gestures [11]. Previous tech- +niques relied on invasive brain-recording techniques which require neurosurgical +interventions. This model tries to decode language directly from noninvasive +brain recordings. This would provide a safer, more scalable solution that could +benefit many more people. The challenge with this proposed method come from +noise and differences in each person’s brain and where the sensors are placed. +A deep learning model is trained with contrastive learning and used to max- +imally align noninvasive brain recordings and speech sounds. A self-supervised +learning model called wave2vec 2.0. is used to identify the complex representa- +tions of speech in the brains of volunteers listening to audiobooks. The two nonin- +vasive technologies used to measure neuronal activity are electroencephalography +and magnetoencephalography. +Training data comes from four opensource datasets which represent 150 hours +of recordings of 169 volunteers listening to audiobooks. EEG and MEG record- +ings are inserted into a brain model, which consists of a standard deep convolu- +tional network with residual connections. These recordings are what comes from +individuals’ brains. This model then has both a speech model for sound and a +brain model for MEG data. +Results show that several components of the algorithm were beneficial to +decoding performance. As well, analysis shows that the algorithm improves as +EEG and MEG recordings increase. This research shows that self-supervised +trained AI can decode perveived speech despite noise and variability in that data. +The biggest limitation of this research is that it focuses on speech perception, +but the ultimate goal would be to extend this work to speech production. +3.7 +Text-to-Code models +Although we have covered text-to-text models, not all texts follows the same +syntax. An special type of text is code. In programming, it is essential to know +how to convert an idea into code. In order to do so, Codex and Alphacode models +help. +Codex : AI system created by OpenAI which translates text to code. It is +a general-purpose programming model, as it can be applied to basically any +programming task [8]. Programming can be broken down into two parts: breaking +a problem down into simpler problems and mapping those problems into existing +code (libraries, APIs, or functions) that already exist. The second part is the +most time-barring part for programmers, and it is where Codex excels the most. +The data collected for training was collected in May 2020 from public software +repositories hosted on GitHub, containing 179GB of unique Python files under 1 + +18 +Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an +MB. The model is fine-tuned from GPT-3, which already contains strong natural +language representations. The demo and the API can be found in Open AI’s +website. +Alphacode : Other language models have demonstrated an impressive ability +to generate code, but these systems still perform poorly when evaluated on more +complex, unseen problems. However, Alphacode is a system for code generation +for problems that require for deeper reasoning [19]. Three components are key for +this achievement: having an extensive dataset for training and evaluation, large +and efficient transformer based architectures and a large-scale model sampling. +In terms of training, the model is firstly pre-trained through GitHub repos- +itories amounting to 715.1 GB of code. This is a much more extensive dataset +than Codex’s pre training dataset. For the training to be better, a fine-tuning +dataset is introduced from the Codeforces plataform. Through this platform, +Codecontests are conducted, for the validation phase, in which we better the per- +formance of the model. Regarding the transformer-based architecture, they use +an encoder-decoder transformer architecture. Compared to decoder-only archi- +tectures commonly used, this architecture allows for a bidirectional description +and extra flexibility. As well, they use a shallow encoder and a deep encoder +to further the model’s efficiency. To reduce the cost of sampling, multi-query +attention is used. +3.8 +Text-to-Science models +Even scientific texts are being targeted by generative AI, as the Galactica and +Minerva models have shown. Although there is a long way to manage success in +this field, it is critical to study the first attempts towards automatic scientific +text generation. +Galactica : Galactica is a new large model for automatically organizing science +developed by Meta AI and Papers with Code. The main advantage of the model +is the ability to train on it for multiple epochs without overfitting, where up- +stream and downstream performance improves with use of repeated tokens. The +dataset design is critical to the approach as all of it is processed in a common +markdown format to blend knowledge between sources. Citations are processed +via a certain token that allows researchers to predict a citation given any in- +put context. The capability of the model of predicting citations improves with +scale and the model becomes better at the distribution of citations. In addition, +the model can perform multi-modal tasks involving SMILES chemical formulas +and protein sequences. Concretely, Galactica uses a transformer architecture in +a decoder-only setup with GeLU activation for all model sizes. +Minerva : Language model capable of solving mathematical and scientific ques- +tions using step-by-step reasoning. Minerva has a very clear focus on the collec- +tion of training data for this purpose. It solves quantitative reasoning problems, + +State of the Art of Generative AI +19 +makes models at scale and employs best-in-class inference techniques. Concretely, +Minerva solves these problems by generating solutions step-by-step, this means +including calculations and symbolic manipulation without having the need for +external tools such a calculator. +3.9 +Other models +We would like to finish our review by covering additional models that do not fit +any of the categories mentioned previously. +Alphatensor, created by the research company Deepmind, is a completely +revolutionary model in the industry for its ability to discover new algorithms +[15]. In the published example, Alpha Tensor creates a more efficient algorithm +for matrix multiplication, which is very important, as improving the efficiency +of algorithms affects a lot of computations, from neural networks to scientific +computing routines. +The methodology is based on a deep reinforcement learning approach in +which the agent, AlphaTensor is trained to play a single-player game where the +objective is finding tensor decomposisitions within a finite factor space. At each +step of the TensorGame, the player selects how to combine different entries of the +matrices to multiply. A score is assigned based on the number of selected oper- +ations required to reach the correct multiplication result. To solve TensorGame, +an agent, AlphaTensor was developed. AlphaTensor uses a specialized neural +network architecture to exploit symmetries using synthetic training games. +GATO is a single generalist agent made by Deepmind. It works as a multi- +modal, multi-task, multi-embodiment generalist policy [27]. The same network +with the same weights can carry very different capabilities from playing Atari, +caption images, chatting, stacking blocks and many more. There are many bene- +fits from using a single neural sequence model across all tasks. It reduces the need +for hand crafting policy models with their own inductive biases. It increases the +amount and diversity of training data. This general agent is successful at many +tasks and can be adapted with little extra data to succeed at an even larger +number of tasks. r training at the operating point of model scale that allows +real-time control of real-world robots, currently around 1.2B parameters in the +case of GATO. +Other published generative AI models are able to generate human motion +[31] or, in the case of ChatBCG, slides using ChatGPT as a surrogate model. +4 +Conclusions and further work +Through this paper, we can observe the capabilities which generative artificial +intelligence has. We have seen a great deal of creativity as well as personalization +in tasks such as text-to-image or in tasks such as text-to-audio. They also are +accurate in text-to-science or text-to-code tasks. This can help economies in a +major way as it can help optimize creative and non-creative tasks. + +20 +Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an +However, because of the way that they are constructed at the moment, these +models face a number of limitations. In terms of dataset, finding data for some +of the models found such as the text-to-science or the text-to-audio is very hard, +making it very time-consuming to train the model. In particular, datasets and +parameters have to be enormous, making it harder to train. One of the biggest +issues with models is trying solutions out of the problems in the dataset, with +which models have more trouble solving. As well, in terms of computation, a lot +of time and computation capacity is necessary in order to run them. Many days +and advanced computers are needed in order to run the models. +In addition, these models face bias from the data which needs to be controlled. +Galactica model tries to control this issue through a layer of no bias, but it still +a major issue for Generative Artificial Intelligence. +With the Minerva model, we can see that the model knows the steps which it +needs to take to solve an equation. This is groundbreaking as one of the biggest +limitations with these models is that the models do not understand exactly +what they are doing. Moreover, it’s still an industry starting; thus accuracy is +still an issue. Text-to-video models for example are only represented by Phenaki +because how hard it is to produce accurate videos. Text-to-science models do +find some accuracy but that accuracy is still way behind to what it should be +for professionals to actually rely on this technology on a day-to-day basis. +Furthermore, these models need to be constrained because of a lack of un- +derstanding of ethics. Phenaki on its paper even acknowledges that a system +like text-to-video can be used to create deep-fakes. Lastly, we are still in a phase +where we are discovering what exactly the purpose of this intelligence will be. +There has been articles comparing Google to ChatGPT3, which is totally inexact +as ChatGPT3 does not update its information in real time. We should be aware +about the limitations of these models to try and improve them in the following +years. +References +1. Alayrac, J.-B., Donahue, J., Luc, P., Miech, A., Barr, I., Hasson, Y., +Lenc, K., Mensch, A., Millican, K., Reynolds, M., et al. Flamingo: a visual +language model for few-shot learning. arXiv preprint arXiv:2204.14198 (2022). +2. 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Human motion diffusion model. arXiv preprint arXiv:2209.14916 (2022). +32. Thoppilan, R., De Freitas, D., Hall, J., Shazeer, N., Kulshreshtha, A., +Cheng, H.-T., Jin, A., Bos, T., Baker, L., Du, Y., et al. Lamda: Language +models for dialog applications. arXiv preprint arXiv:2201.08239 (2022). +33. Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, +A. N., Kaiser, �L., and Polosukhin, I. Attention is all you need. Advances in +neural information processing systems 30 (2017). +34. Villegas, R., Babaeizadeh, M., Kindermans, P.-J., Moraldo, H., Zhang, +H., Saffar, M. T., Castro, S., Kunze, J., and Erhan, D. Phenaki: Variable +length video generation from open domain textual description. +arXiv preprint +arXiv:2210.02399 (2022). + diff --git a/9NE3T4oBgHgl3EQfqwrj/content/tmp_files/load_file.txt b/9NE3T4oBgHgl3EQfqwrj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c9a4084eb584c1736fb8c652fce5084885b24a04 --- /dev/null +++ b/9NE3T4oBgHgl3EQfqwrj/content/tmp_files/load_file.txt @@ -0,0 +1,919 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf,len=918 +page_content='ChatGPT is not all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' A State of the Art Review of large Generative AI models Roberto Gozalo-Brizuela, Eduardo C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Garrido-Merch´an Quantitative Methods Department, Universidad Pontificia Comillas, Madrid, Spain 201905616@alu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='comillas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='edu, ecgarrido@icade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='comillas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='edu Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' During the last two years there has been a plethora of large generative models such as ChatGPT or Stable Diffusion that have been published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Concretely, these models are able to perform tasks such as being a general question and answering system or automatically creat- ing artistic images that are revolutionizing several sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Consequently, the implications that these generative models have in the industry and society are enormous, as several job positions may be transformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' For example, Generative AI is capable of transforming effectively and cre- atively texts to images, like the DALLE-2 model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' text to 3D images, like the Dreamfusion model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' images to text, like the Flamingo model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' texts to video, like the Phenaki model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' texts to audio, like the AudioLM model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' texts to other texts, like ChatGPT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' texts to code, like the Codex model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' texts to scientific texts, like the Galactica model or even create algorithms like AlphaTensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This work consists on an attempt to de- scribe in a concise way the main models are sectors that are affected by generative AI and to provide a taxonomy of the main generative models published recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 1 Introduction Generative AI refers to artificial intelligence that can generate novel content, rather than simply analyzing or acting on existing data like expert systems [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In particular, expert systems contained knowledge bases and an inference engine that generated content via an if-else rule database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' However, modern generative artificial intelligence contain a discriminator or transformer model trained on a corpus or dataset that is able to map the input information into a latent high- dimensional space and a generator model, that is able to generate an stochastic behaviour creating novel content in every new trial even from the same prompts as an input, performing unsupervised, semi-supervised or supervised learning, depending on the particular methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Regarding the created content by the model, generative artificial intelligence models are different from predictive ma- chine learning systems, that merely perform a discrimination behaviour, solv- ing classification or regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In particular, these models are able to discriminate information and generate information of the transformed input information, or prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The key aspect about generative models is that their architecture and the data that they have been fed is enormous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' For example, it is possible now to arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='04655v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='LG] 11 Jan 2023 2 Roberto Gozalo-Brizuela, Eduardo C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Garrido-Merch´an estimate the parameters of the model by feeding it the contents of the whole Wikipedia, Github, social networks, Google images and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Despite being fed with an enormous size of data, thanks to the rise of computing we can design deep neural networks [18], transformers [22] and other models such as genera- tive adversarial networks [9] or variational autoencoders [23] whose capacity is able to model the complexity of the data, without suffering from underfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' As they are able to modelize the high-dimensional probability distribution of language or photos of a concrete or general domain, if they are complemented by generative models that map the latent high-dimensional semantic space of language of photos to a multimedia representation of text, audio or video we can map any input format like texts to any output format like video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In this sense, applications of this technology are endless, in the sense that we can train a model to generate genuine different multimedia formats as video, audio or text from different multimedia input formats, as for example, text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We believe that it is necessary to provide a state-of-the-art review on the most popular generative AI models as they are revolutionizing several indus- tries like the art industry [2] or universities [16,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' As models are now able to generate genuine artistic content or large texts answering a prompt, these two industries and other ones that we will detail throughout this manuscript will need to readapt their activity to continue providing value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In this sense, gen- erative AI models will not replace humans but enhance our content, being an inspiration for artists or improving the content generated by professors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In order to provide information for a professional working in any industry that can be benefited by these models we have made the organization of the paper as the following one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' First, we will provide a taxonomy of the main generative models that have appeared in the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Then, the following sections will analyze each of the categories of the taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Finally, we finish the manuscript with a conclusions and further work section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We do not study the technical aspects of every model, such as transformers in detail as our purpose in this review is on the applications of the models and the content that they generative but not on how they work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' For a detailed explanation of deep learning models and generative models we recommend other references [18,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 2 A Taxonomy of Generative AI models Before analyzing each model in detail, we have tried to organize the current generative artificial models into a taxonomy whose categories represent the main mappings between each multimedia input and output type of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The result is the one that we have illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We have discovered a total of 9 categories, where each of the models that appear in Figure 1 will be described in detail in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Each of the covered models has been published recently, as we illustrate in Figure 2, as our main concern in this manuscript is to describe the latest advances in generative AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Interestingly, only six organizations are behind the deployment of these mod- els, as we illustrate in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The main reason behind this fact is that in order State of the Art of Generative AI 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' A taxonomy of the most popular generative AI models that have recently appeared classified according to their input and generated formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Covered models by date of release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' All models were released during 2022 except LaMDA, which was released in 2021 and Muse, in 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Text-to- Text-to- Image-to- image 3D text DALL-E Stable 2 Dreamfusion Magic3D Flamingo Diffusion VisualGPT Imagen Muse Text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='to- Text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='to- Text-To- Video Audio Text Phenaki Soundify AudioLM Whisper ChatGPT3 LaMDA Jukebox PEER Speech From Brain Text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Text-to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Other ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Code ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Science ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Models ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Codex ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Alphacode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Galactica ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Minerva ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Alphatensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='GATO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Human Motion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Diffusion ModelMeta Al Speech From Brain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Muse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Whisper ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='PEER ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Stable Diffusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Hurnan Motion Diffusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Jukebox ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Imagen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='VisualGPT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Magic3D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Minerva ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='LaMDA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Soundify ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='GATO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Alphacode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Flamingo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='DallI-E-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Galadtica ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Alphatensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Codex ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Dreamfusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='ChatGPT3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='AudioL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='04/01/2021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='14/04/2021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='23/07/2021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='31/10/2021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='08/02/2022 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='19/05/2022 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='27/08/2022 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='05/12/20224 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='Roberto Gozalo-Brizuela,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Eduardo C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Garrido-Merch´an to be able to estimate the parameters of these models, it is mandatory to have an enormous computation power and a highly skilled and experienced team in data science and data engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Consequently, only the companies shown on Figure 3, with the help of acquired startups and collaborations with academia, have been able to successfully deploy generative artificial intelligence models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Models by developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In terms of major companies participating in startups, note that Microsoft invested 1 billion dollars in OpenAI and helps them with the de- velopment of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' As well, note that Google acquired Deepmind in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In terms of universities, note that VisualGPT was developed by KAUST, Carnegie Mellon Uni- versity and Nanyang Technological University and that the Human Motion Diffusion Model was developed by Tel Aviv University, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' As well, other projects are de- veloped by a company in collaboration with a university.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Concretely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' this is the case for Stable Diffsion (Runway,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Stability AI and LMU MUNICH),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Soundify (Runway and Carnegie Mellon University) and DreamFusion (Google and UC Berkeley) GoogleResearch openAI DALL-E 2 Chat GPT3 Imagen Muse Jukebox Whisper DreamFusion Phenaki Minerva AudioLM LaMDA ODeepMnd 00MetaAl Flamingo Alphacode PEER Speech From Brain Alphatensor GATO Galactica Rrunway nVIDIA Stable Diffusion Soundify Magic3DState of the Art of Generative AI 5 Now that we have provided and analyzed the latest generative artificial in- telligence models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' the following section will cover each of the categories of the taxonomy presented in Figure 1 in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 3 Generative AI models categories In this section we will cover in detail the nine categories described in Figure 1 of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' For every category, we illustrate the details of the models shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='1 Text-to-image models We begin the review by considering the models whose input is a text prompt and whose output is an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' DALL·E 2 : DALL·E 2, created by OpenAI, is able to generate original, genuine and realistic images and art from a prompt consisting on a text description [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Luckily, it is possible to use the OPENAI API to get access to this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In particular, DALL·E 2 manages to combine concepts, attributes and different styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In order to do so, it uses the CLIP neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' CLIP (Contrastive Language-Image Pre-Training) is a neural network trained on a variety of (image, text) pairs [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Using CLIP, that can be instructed in natural language to predict the most relevant text snippet, given an image, the model has recently merged as a successful representation learner for images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Concretely, CLIP embeddings have several desirable properties: they are robust to image distribution shift, have impressive zero-shot capabilities and have been fine-tuned to achieve state-of-the- art results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In order to obtain a full generative model of images, the CLIP image embedding decoder module is combined with a prior model, which generates possible CLIP image embeddings from a given text caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We illustrate an image generated from a prompt in Figure 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Image generated from the prompt ”A shiba inu wearing a beret and black turtleneck”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 6 Roberto Gozalo-Brizuela, Eduardo C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Garrido-Merch´an IMAGEN : Imagen is a text-to-image diffusion model [17] consisting on large transformer language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Critically, the main discovery observed with this model made is that large language models, pre-trained on text-only corpora, are very effective at encoding text for image synthesis [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Precisely, using Imagen, it has been found out that increasing the size of the language model boosts both sample fidelity and image-text alignment much more than increasing the size of the image diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The model was created by Google and the API can be found in their web page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' For the evaluation of their model, Google created Drawbench, a set of 200 prompts that support the evaluation and comparison of text-to-image models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Most concretely, the model is based on a pretrained text encoder (like BERT [12]) that performs a mapping from text to a sequence of word embeddings and a cascade of conditional diffusion models that map these embeddings to images of increasing resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We show an image generated from a prompt in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Image generated from the prompt ”A cute corgi lives in a house made out of sushi”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Stable Diffusion : Stable Diffusion is a latent-diffusion model that is open- source and has been developed by the CompVis group at LMU Munich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The main difference of this model with respect to the other ones is the use of a latent diffusion model and that it performs image modification as it can perform operations in its latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' For Stable Diffusion, we can use the API via their website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' More concretely, Stable Diffusion consists of two parts: the text encoder and the image generator [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The image information creator works completely in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This property makes it faster than previous diffusion models that worked in a pixel space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We illustrate a Stable Diffusion image example in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' State of the Art of Generative AI 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Image generated from the prompt ”A cute corgi lives in a house made out of sushi”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Muse : This model is a Text-to-image transformer model that achieves state-of- the-art image generation while being more efficient than diffusion or autoregres- sive models [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Concretely, it is trained on a masked modelling task in discrete token space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Consequently, it is more efficient because of the use of discrete tokens and requiring fewer sampling iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Compared to Parti, a autore- gressive model, Muse is more efficient because of parallel decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Muse is 10x faster at inference time than Imagen-3B or Parti-3B and 3x faster than Stable Diffusion v 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Muse is also faster than than Stable Diffusion in spite of both models working in the latent space of a VQGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We append a comparison of the generated images by DALL·E 2, IMAGEN and Muse in Figure ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='. 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='2 Text-to-3D models The models that have been described in the previous section deal with the map- ping of text prompts to 2D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' However, for some industries like gaming, it is necessary to generate 3D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In this section, we briefly describe two text-to-3D models: Dreamfusion and Magic3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Dreamfusion : DreamFusion is a text-to-3D model developed by Google Re- search that uses a pretrained 2D text-to-image diffusion model to perform text- to-3D synthesis [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In particular, Dreamfusion replaces previous CLIP tech- niques with a loss derived from distillation of a 2D diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Concretely, the diffusion model can be used as a loss within a generic continuous optimization problem to generate samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Critically, sampling in parameter space is much harder than in pixels as we want to create 3D models that look like good images when rendered from random angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' To solve the issue, this model uses a differ- entiable generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Other approaches are focused on sampling pixels, however, this model instead focuses on creating 3D models that look like good images when rendered from random angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We illustrate in Figure 8 an example of 8 Roberto Gozalo-Brizuela, Eduardo C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Garrido-Merch´an Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Comparison of generated images by the DALL·E 2, IMAGEN and Muse models with respect to the prompts that appear in the column of the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The first column of images contains the results generated by DALL·E 2, the second the results obtained with IMAGEN and the third the images created by Muse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' an image created by Dreamfusion from one particular angle along with all the variations that can be generated from additional text prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In order to see the full animated image, we recommend to visit the web page of Dreamfusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' A 3D squirrel generated by Dreamfusion is shown at the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Then, the other images contain the modifications generated to the squirrel with text prompts like ”wear- ing a jacket”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' DALL-E 2 Imagen MUSE A high contrast portrait of a very happy fuzzy panda dressed as a chef in a high end kit- chen making dough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' There is a painting of flowers on the wall behind him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Rainbow coloured penguin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='a DSLR wearing Bsup!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' onaroad photo of a leather motorcycle made of ice jacket a squirrelState of the Art of Generative AI 9 Magic3D : This model is a text to 3D model made by NVIDIA Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' While the Dreamfusion model achieves remarkable results, the method has two problems: mainly, the long processing time and the low-quality of the generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' However, these problems are addressed by Magic3D using a two-stage optimization framework [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Firstly, Magic3D builds a low-resolution diffusion prior and, then, it accelerates with a sparse 3D hash grid structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Using that, a textured 3D mesh model is furthered optimized with an efficient differentiable render.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Comparatively, regarding human evaluation, the model achieves better results, as 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='7% prefer this model to DreamFusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' As we can see in Figure 9, Magic3D achieves much higher quality 3D shapes in both geometry and texture compared to DreamFusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 3D Images generated by Magic3D and Dreamfusion, where ”Ours” refer to Magic3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We can see a total of 8 text prompts and the images that both models generate from that prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='3 Image-to-Text models Sometimes, it is also useful to obtain a text that describes an image, that is precisely the inverse mapping to the one that has been analyzed in the previous subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In this section, we analyze two models that perform this task, along with others: Flamingo and VisualGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Flamingo : A Visual Language Model created by Deepmind using few shot learning on a wide range of open-ended vision and language tasks, simply by being prompted with a few input/output examples [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Concretely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' the input of Flamingo contains visually conditioned autoregressive text generation models able to ingest a sequence of text tokens interleaved with images and/or videos Ours DreamFusion[33] Ours DreamFusion[33] 6 a kingfisher birdi car made out of sushi* anicecreamsundae a beautifully carved wooden knight chess piecei a small saguaro cactusplantedinaclaypor A very beautiful tiny human heart organic sculpture made of copper wire and threaded pipes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' very intricate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' curved,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Studio lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' high resolution* modelofan adorable cottage with athatched aripestrawberm10 Roberto Gozalo-Brizuela, Eduardo C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Garrido-Merch´an and produce text as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' A query is made to the model along with a photo or a video and the model answers with a text answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Some examples can be observed in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Flamingo models take advantage of two complementary models: a vision model that analyzes visual scenes and a large language model which performs a basic form of reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The language model is trained on a large amount of text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Input prompts that contain images and text and output generated text re- spones from Flamingo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Every column contains a single example where we can see how Flamingo answers the question using the image from the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' VisualGPT : VisualGPT is an image captioning model made by OpenAI [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Concretely, VisualGPT leverages knowledge from the pretrained language model GPT-2 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In order to bridge the semantic gap between different modalities, a novel encoder-decoder attention mechanism [33] is designed with an unsaturated rectified gating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Critically, the biggest advantage of this model is that it does not need for as much data as other image-to-text models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In particular, improving data efficiency in image captioning networks would enable quick data curation, description of rare objects, and applications in specialized domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Most interestingly, the API of this model can be found on GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We include three examples of text prompts generated by the model with respect to three images fed to the model in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='4 Text-to-Video models As we have seen in the previous subsections, it is now possible to generate images from text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Consequently, the next logical step is to generate videos, that are Question: What do you think Question: What is odd about Question: What country is the capacities of these are?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' this image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Explain why it is this?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Why do you think so?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Answer: unusual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Answer: Answer: p Completion The image is odd because the elephant is in the back It is Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' I think so The floppy disk is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='44MB of the truck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' It is unusual and the CD is 700MIB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' because the flag is the because elephants are not Canadian flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' usually transported in the back of a truck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='State of the Art of Generative AI 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Three examples of text prompts generated by the images shown on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We also show the attention scores that the model assign to every word of the texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In the third image, we can see for example how the most discriminative information about the image is the word ”cat” and ”television”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' sequences of images, from texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In this section, we provide information about two models that are able to perform this task: Phenaki and Soundify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Phenaki : This model has been made by Google Research, and it is capable of performing realistic video synthesis, given a sequence of textual prompts [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Most interestingly, we can get access to the API of the model from GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In particular, Phenaki is the first model that can generate videos from open domain time variable prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' To address data issues, it performs joint training on a large image-text pairs dataset as well as a smaller number of video-text exam- ples can result in generalization beyond what is available in the video datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This is mainly due to image-text datasets having billions of inputs while text- GT: the lady is sitting on the wood bench ours woman sitting on e benchin apark attention 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='760.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='690.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='85 GT: a laptop with a keyboard and mouse are on this desk ours alaptop sitting on adeskwithamouse attention 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='76 GT: a cat is sitting in front of a television Ours sitting frontof television attention 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='93 GT: a number of people sitting on a snowy surface with skis Ours couple of people sitting on snowy suriace attention 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='940.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='9512 Roberto Gozalo-Brizuela, Eduardo C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Garrido-Merch´an video datasets are much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' As well, limitations come from computational capabilities for videos of variable length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The model has three parts: the C-ViViT encoder, the training transformer and the video generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The encoder gets a compressed representation of videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' First tokens are transformed into embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This is followed by the temporal transformer, then the spatial transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' After the output of the spatial trans- former, they apply a single linear projection without activation to map the tokens back to pixel space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Consequently, the model generates temporally coherent and diverse videos conditioned on open domain prompts even when the prompt is a new composition of concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The videos can be minutes long, while the model is trained on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='4 second videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Below we show in Figure 12 and in Figure 13 some examples of the creation of a video through a series of text prompts and from a series of text prompts and an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Sequence of images created by the Phenaki model given four different prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 1st prompt:"A photorealistic teddy bear is swimming in the ocean at San Francisco" 2nd prompt:“The teddy bear goes under water 3rd prompt: "The teddy bear keeps swimming under the water with colorful fishes\' 4rd prompt:"A panda bear is swimming under water"State of the Art of Generative AI 13 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Sequences of images created by the Phenaki model given an image and the prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We can see how the model is able to manipulate the given image according to the text prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Soundify : In video editing, sound in half of the story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' But, for professional video editing, the problems come from finding suitable sounds, aligning sounds, video and tuning parameters [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In order to solve this issue, Soundify is a system developed by Runway that matches sound effects to video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This system uses quality sound effects libraries and CLIP (a neural network with zero-shot image classification capabilities cited before).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Concretely, the system has three parts: classification, synchronization, and mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The classification matches effects to a video by classifying sound emitters within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' To reduce the distinct sound emitters, the video is split based on absolute color histogram distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In the synchronization part, intervals are identified comparing effects label with each frame and pinpointing consecutive matches above a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In the mix part, effects are split into around one-second chunks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Critically, chunks are stitched via crossfades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='5 Text-to-Audio models As we have seen in the previous subsection, images are not the only important non-structured data format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' For videos, for music and in lots of contexts, audio can be critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Consequently, we analyze in this subsection three models whose input information is text and whose output information is audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' AudioLM : This model has been made by Google for high-quality audio gener- ation with long-term consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In particular, AudioLM maps the input audio into a sequence of discrete tokens and casts audio generation as language mod- eling task in this representation space [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' By training on large corpora of raw Given Image Prompt: "Camera zooms quickly into the eye of the cat" Given Image I Prompt: "A white cat touches the camera with the paw" Given Image Prompt: "A white cat yawns loudly"14 Roberto Gozalo-Brizuela, Eduardo C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Garrido-Merch´an audio waveforms, AudioLM learns to generate natural and coherent continua- tions given short prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The approach can be extended beyond speech by generating coherent piano music continuations, despite being trained without any symbolic representation of music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' As with the other models, the API can be found through GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Audio signals involve multiple scales of abstractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' When it comes to audio synthesis, multiple scales make achieving high audio quality while displaying consistency very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This gets achieved by this model by combining recent advances in neural audio compression, self-supervised representation learning and language modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In terms of subjective evaluation, raters were asked to listen to a sample of 10 seconds and decide whether it is human speech or a synthetic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Based on 1000 ratings collected, the rate is 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='2%, which is not statistically significant from assigning labels at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This tells us that humans cannot differentiate between synthetic and real samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Jukebox : This is a model, developed by OpenAI, that generates music with singing in the raw audio domain [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Once again, its API can be found in GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Previously, earlier models in the text-to-music genre generated music symbolically in the form of a pianoroll which specifies timing, pitch and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The challenging aspect is the non-symbolic approach where music is tried to be produced directly as a piece of audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In fact, the space of raw audio is extremely high dimensional which makes the problem very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Consequently, the key issue is that modelling that raw audio produces long-range dependencies, making it computationally challenging to learn the high-level semantics of music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In order to solve this issue, this model tries to solve it by means of a hi- erarchical VQ-VAE architecture to compress audio into a discrete space [14], with a loss function designed to retain the most amount of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This model produces songs from very different genres such as rock, hip-hop and jazz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' However, the model is just limited to English songs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Concretely, its dataset for training is from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='2 million songs from LyricWiki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The VQ-VAE has 5 billion parameters and is trained on 9-second audio clips for 3 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Whisper : This model is an Audio-to-Text converter developed by OpenAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' It achieves several tasks in this field: multi-lingual speech recognition, translation and language identification [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' As in previous cases, its API can be found in the GitHub website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The goal of a speech recognition system should be to work reliably out of the box in a broad range of environments without requiring supervised fine-tuning of a decoder for every deployment distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This is hard because of the lack of a high-quality pre-trained decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Concretely, this model is trained on 680,000 hours of labeled audio data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This data is collected from the internet, which results in a very diverse dataset covering a broad distribution of audio from many different environments, record- ings setups, speakers and languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The model makes sure that the dataset is only from human voice as machine learning voice would impair the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Files State of the Art of Generative AI 15 are broken in 30 second segments paired with the subset of the transcript that occurs within that time segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The model has an encoder-deccoder transformer, as this architecture has been validated to scale reliably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We can observe the model’s architecture char- acteristics through the figure below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We can see the different types of data and the learning sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='6 Text-to-Text models The previous models all convert a non-structured data type into another one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' But, regarding text, it is very useful to convert text into another text in order to satisfy tasks as general question and answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The following four models treat text and also output texts to satisfy different needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' ChatGPT : The popular ChatGPT is a model by OpenAI which interacts in a conversational way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' As it is widely known, the model answers follow-up questions, challenges incorrect premises and reject inappropriate requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' More concretely, the algorithm behind ChatGPT is based on a transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' However, the training is made through Reinforcement Learning for Human Feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In particular, an initial model is trained using supervised fine-tuning: human AI trainers would provide conversations in which they played both sides, the user and an AI assistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Then, those people would be given the model-written re- sponses to help them compose their response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This dataset was mixed to that of InstructGPT [3], which was transformed into a dialogue format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' A demo can be found in their website and the API may also be found in OpenAI’s website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We summarize the main steps of ChatGPT training in Figure 14, available in the ChatGPT demo’s website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Finally, ChatGPT is also able to generate code and simple mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' LaMDA : LaMDA is a language model for dialog applications [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Unlike most other language models, LaMDA was trained on dialogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' It is a family of transformer-based neural language models specialized for dialog which have up to 137B parameters and are pre-trained on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='56T words of public dialog data and web text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Fine-tuning can enable for safety and factual grounding of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='001% of training data was used for fine-tuning, which is a great achievement of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In particular, dialog modes take advantage of Transformers’ ability to present long-term dependencies in text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Concretely, they are generally very well-suited for model scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Consequently, LaMDA makes use of a single model to perform multiple tasks: it generates several responses, which are filtered for safety, grounded on an external knowledge source and re- ranked to find the highest-quality response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We illustrate in Figure 15 an example of a dialog with the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' PEER : Collaborative language model developed by Meta AI research trained on edit histories to cover the entire writing process [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' It is based on four 16 Roberto Gozalo-Brizuela, Eduardo C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Garrido-Merch´an Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Training steps of ChatGPT, combining supervised learning with reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Example of a dialog made with LaMDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' steps: Plan, Edit, Explain and Repeat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' These steps are repeated until the text is in a satisfactory state that requires no further updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The model allow to decompose the task of writing a paper into multiple easier subtasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' As well, the model allows humans to intervene at any time and steer the model in any direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' It is mainly trained on Wikipedia edit histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The approach is a self- training, using models to infill missing data and then train other models on this synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The downside of this comes from comments being very noisy and a lack of citations, which tries to be compensated by a retrieval system which does not always work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The framework is based on an iterative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Step1 Step 2 Step 3 Collectdemonstrationdata Collectcomparisondataand Optimizeapolicyagainstthe and traina supervised policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' train a reward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' rewardmodelusingthePPO reinforcementlearningalgorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' A prompt is A prompt and Anewpromptis sampledfromour Explain reinforcement several model Explain reinforcement sampled from Write a story promptdataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' learningtoa6 yearold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' outputsare learning to a 6 year old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' about otters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Explainr agent is The PPO model is PPO Alabeler 0 initialized from the demonstratesthe supervised policy desired output teach behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We give treats and punishments to teach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Alabelerranksthe Thepolicygenerates Once upon a time outputsfrombest an output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' SFT to worst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 0-0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='0 This datais used to The reward model RM fine-tune GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='5 calculates a reward with supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' RM forthe output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This data is used to train our The reward is used reward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' to update the 0000 rk policy using PPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content="LaMDA Everest LaMDA: Hi, I'm Mount Everest." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' What would you like to know about me?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' User: Why do you think people climb you?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' LaMDA: Because I represent a very high achievement to climb me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Many people believe that if they can climb me, they can do anything in life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' User: Who was the first person to do that?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' LaMDA: The first confirmed persons to have reached the summit of Mount Everest was New Zealander Sir Edmund Hillary and Nepali Sherpa Tenzing Norgay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='org/wiki/Mount_Everest User: How old was Hillary then?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' LaMIDA: He was 33 years old when he first reached the top of me in 1953!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='org/wiki/Edmund_Hillary User: When people fail, is it dangerous?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' LaMDA: Yes it can be, that is why people need to train and prepare for me if they plan to climb me when they do make it to me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=" I'm very hard to climb and can be dangerous when there is a whiteout Which means everything is white with very strong winds!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='State of the Art of Generative AI 17 The entire process of formulating a plan, collecting documents, performing an edit and explaining it can be repeated multiple times until arriving at a sequence of texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' For the training, a DeepSpeed transformer is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Meta AI Speech from Brain : Model developed by Meta AI to help people unable to communicate through speech, typing or gestures [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Previous tech- niques relied on invasive brain-recording techniques which require neurosurgical interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This model tries to decode language directly from noninvasive brain recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This would provide a safer, more scalable solution that could benefit many more people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The challenge with this proposed method come from noise and differences in each person’s brain and where the sensors are placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' A deep learning model is trained with contrastive learning and used to max- imally align noninvasive brain recordings and speech sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' A self-supervised learning model called wave2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' is used to identify the complex representa- tions of speech in the brains of volunteers listening to audiobooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The two nonin- vasive technologies used to measure neuronal activity are electroencephalography and magnetoencephalography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Training data comes from four opensource datasets which represent 150 hours of recordings of 169 volunteers listening to audiobooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' EEG and MEG record- ings are inserted into a brain model, which consists of a standard deep convolu- tional network with residual connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' These recordings are what comes from individuals’ brains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This model then has both a speech model for sound and a brain model for MEG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Results show that several components of the algorithm were beneficial to decoding performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' As well, analysis shows that the algorithm improves as EEG and MEG recordings increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This research shows that self-supervised trained AI can decode perveived speech despite noise and variability in that data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The biggest limitation of this research is that it focuses on speech perception, but the ultimate goal would be to extend this work to speech production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='7 Text-to-Code models Although we have covered text-to-text models, not all texts follows the same syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' An special type of text is code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In programming, it is essential to know how to convert an idea into code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In order to do so, Codex and Alphacode models help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Codex : AI system created by OpenAI which translates text to code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' It is a general-purpose programming model, as it can be applied to basically any programming task [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Programming can be broken down into two parts: breaking a problem down into simpler problems and mapping those problems into existing code (libraries, APIs, or functions) that already exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The second part is the most time-barring part for programmers, and it is where Codex excels the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The data collected for training was collected in May 2020 from public software repositories hosted on GitHub, containing 179GB of unique Python files under 1 18 Roberto Gozalo-Brizuela, Eduardo C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Garrido-Merch´an MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The model is fine-tuned from GPT-3, which already contains strong natural language representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The demo and the API can be found in Open AI’s website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Alphacode : Other language models have demonstrated an impressive ability to generate code, but these systems still perform poorly when evaluated on more complex, unseen problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' However, Alphacode is a system for code generation for problems that require for deeper reasoning [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Three components are key for this achievement: having an extensive dataset for training and evaluation, large and efficient transformer based architectures and a large-scale model sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In terms of training, the model is firstly pre-trained through GitHub repos- itories amounting to 715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='1 GB of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This is a much more extensive dataset than Codex’s pre training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' For the training to be better, a fine-tuning dataset is introduced from the Codeforces plataform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Through this platform, Codecontests are conducted, for the validation phase, in which we better the per- formance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Regarding the transformer-based architecture, they use an encoder-decoder transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Compared to decoder-only archi- tectures commonly used, this architecture allows for a bidirectional description and extra flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' As well, they use a shallow encoder and a deep encoder to further the model’s efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' To reduce the cost of sampling, multi-query attention is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='8 Text-to-Science models Even scientific texts are being targeted by generative AI, as the Galactica and Minerva models have shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Although there is a long way to manage success in this field, it is critical to study the first attempts towards automatic scientific text generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Galactica : Galactica is a new large model for automatically organizing science developed by Meta AI and Papers with Code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The main advantage of the model is the ability to train on it for multiple epochs without overfitting, where up- stream and downstream performance improves with use of repeated tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The dataset design is critical to the approach as all of it is processed in a common markdown format to blend knowledge between sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Citations are processed via a certain token that allows researchers to predict a citation given any in- put context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The capability of the model of predicting citations improves with scale and the model becomes better at the distribution of citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In addition, the model can perform multi-modal tasks involving SMILES chemical formulas and protein sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Concretely, Galactica uses a transformer architecture in a decoder-only setup with GeLU activation for all model sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Minerva : Language model capable of solving mathematical and scientific ques- tions using step-by-step reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Minerva has a very clear focus on the collec- tion of training data for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' It solves quantitative reasoning problems, State of the Art of Generative AI 19 makes models at scale and employs best-in-class inference techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Concretely, Minerva solves these problems by generating solutions step-by-step, this means including calculations and symbolic manipulation without having the need for external tools such a calculator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='9 Other models We would like to finish our review by covering additional models that do not fit any of the categories mentioned previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Alphatensor, created by the research company Deepmind, is a completely revolutionary model in the industry for its ability to discover new algorithms [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In the published example, Alpha Tensor creates a more efficient algorithm for matrix multiplication, which is very important, as improving the efficiency of algorithms affects a lot of computations, from neural networks to scientific computing routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The methodology is based on a deep reinforcement learning approach in which the agent, AlphaTensor is trained to play a single-player game where the objective is finding tensor decomposisitions within a finite factor space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' At each step of the TensorGame, the player selects how to combine different entries of the matrices to multiply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' A score is assigned based on the number of selected oper- ations required to reach the correct multiplication result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' To solve TensorGame, an agent, AlphaTensor was developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' AlphaTensor uses a specialized neural network architecture to exploit symmetries using synthetic training games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' GATO is a single generalist agent made by Deepmind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' It works as a multi- modal, multi-task, multi-embodiment generalist policy [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' The same network with the same weights can carry very different capabilities from playing Atari, caption images, chatting, stacking blocks and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' There are many bene- fits from using a single neural sequence model across all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' It reduces the need for hand crafting policy models with their own inductive biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' It increases the amount and diversity of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This general agent is successful at many tasks and can be adapted with little extra data to succeed at an even larger number of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' r training at the operating point of model scale that allows real-time control of real-world robots, currently around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content='2B parameters in the case of GATO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Other published generative AI models are able to generate human motion [31] or, in the case of ChatBCG, slides using ChatGPT as a surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 4 Conclusions and further work Through this paper, we can observe the capabilities which generative artificial intelligence has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' We have seen a great deal of creativity as well as personalization in tasks such as text-to-image or in tasks such as text-to-audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' They also are accurate in text-to-science or text-to-code tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This can help economies in a major way as it can help optimize creative and non-creative tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' 20 Roberto Gozalo-Brizuela, Eduardo C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Garrido-Merch´an However, because of the way that they are constructed at the moment, these models face a number of limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In terms of dataset, finding data for some of the models found such as the text-to-science or the text-to-audio is very hard, making it very time-consuming to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In particular, datasets and parameters have to be enormous, making it harder to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' One of the biggest issues with models is trying solutions out of the problems in the dataset, with which models have more trouble solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' As well, in terms of computation, a lot of time and computation capacity is necessary in order to run them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Many days and advanced computers are needed in order to run the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' In addition, these models face bias from the data which needs to be controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Galactica model tries to control this issue through a layer of no bias, but it still a major issue for Generative Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' With the Minerva model, we can see that the model knows the steps which it needs to take to solve an equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' This is groundbreaking as one of the biggest limitations with these models is that the models do not understand exactly what they are doing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Moreover, it’s still an industry starting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' thus accuracy is still an issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Text-to-video models for example are only represented by Phenaki because how hard it is to produce accurate videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Text-to-science models do find some accuracy but that accuracy is still way behind to what it should be for professionals to actually rely on this technology on a day-to-day basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Furthermore, these models need to be constrained because of a lack of un- derstanding of ethics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Phenaki on its paper even acknowledges that a system like text-to-video can be used to create deep-fakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} +page_content=' Lastly, we are still in a phase where we are discovering what exactly the purpose of this intelligence will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE3T4oBgHgl3EQfqwrj/content/2301.04655v1.pdf'} 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Matthias Samwald +1 +1) Institute of Artificial Intelligence, Medical University of Vienna, Vienna, Austria +2) Section for Cognitive Systems, Technical University of Denmark, Lyngby, Denmark +3) School of Electrical Engineering and Computer Science, The Royal Institute of Technology +(KTH), Stockholm, Sweden +4) Department of Clinical Immunology, Copenhagen University Hospital, Copenhagen, Denmark +* equal contribution +Corresponding author: Matthias Samwald (matthias.samwald [at] meduniwien.ac.at) +Abstract +Large language models (LLMs) such as GPT-3 and ChatGPT have recently demonstrated +impressive results across a wide range of tasks. LLMs are still limited, however, in that they +frequently fail at complex reasoning, their reasoning processes are opaque, they are prone to +‘hallucinate’ facts, and there are concerns about their underlying biases. Letting models verbalize +reasoning steps as natural language, a technique known as chain-of-thought prompting, has +recently been proposed as a way to address some of these issues. Here we present the first release +of ThoughtSource, a meta-dataset and so�ware library for chain-of-thought (CoT) reasoning. The +goal of ThoughtSource is to improve future artificial intelligence systems by facilitating +qualitative understanding of CoTs, enabling empirical evaluations, and providing training data. +This first release of ThoughtSource integrates six scientific/medical, three general-domain and +five math word question answering datasets. +Background & Summary +The most recent generation of large language models (LLMs) has produced impressive results +across a wide range of tasks. Examples of such models include T0  +1, GPT-3  +2, InstructGPT  +3 and +ChatGPT (https://openai.com/blog/chatgpt/). These models demonstrated remarkable ability to + +ThoughtSource: a central hub for large language model reasoning data | 2 +generate text that is both realistic and coherent, as well as good performance on a broad +spectrum of tasks, despite not explicitly being trained on them +3. +However, despite this ability, LLMs are also limited in several ways. They o�en fail to produce +accurate predictions due to their inability to accomplish complex reasoning, such as solving +mathematical problems or question answering tasks requiring multi-hop reasoning. +Furthermore, they tend to be black boxes, making it difficult to understand how and why +predictions are generated. These limitations severely limit the application domains of LLMs and +have the potential to cause harm, as lack of explainability and robustness can lead to critical +failures and biases when these models are deployed in practice. +One recently proposed method for enabling complex reasoning and generating explanations with +LLMs is to force models to explicitly verbalize reasoning steps as natural language, a technique +known as chain-of-thought prompting  +4,5. This method improved performance on a variety of +tasks and sparked the active development of further refinements +6, such as decomposing +problems and structuring reasoning (e.g., least-to-most prompting  +7, ReAct  +8, self-ask  +9, maieutic +prompting  +10, successive prompting  +11) and/or extending LLM capabilities by leveraging external +services for tasks like information retrieval (e.g., self-ask  +9, IRCoT  +12, DSP +13). The terminology +surrounding these rapidly evolving techniques is not settled, hence in this document, we refer to +all approaches that result in a linear sequence of reasoning steps as 'chain-of-thought' (CoT). +Meta-datasets (datasets of datasets) that are easily accessible and standardized have proven useful +for training and evaluating versatile LLMs. Examples include SuperGLUE +14 for general-domain +language model tasks, BigBIO  +15 and BLURB  +16 for biomedical tasks, or Pile  +17 and ROOTS  +18 as +text corpora for LLM pre-training. Datasets can be complemented by tools such as +PromptSource, which was used to convert a large number of datasets into prompts fit for training +and interrogating LLMs. PromptSource facilitated training the highly performant T0 model  +1. +Here we present the first release of ThoughtSource, a meta-dataset and so�ware library for +chain-of-thought reasoning in LLMs (https://github.com/OpenBioLink/ThoughtSource). The +goals of ThoughtSource are to: +— Facilitate qualitative understanding of CoTs generated by LLMs under various conditions +(e.g., across tasks, models and prompts). +— Enable empirical and quantitative evaluation. +— Provide a library of diverse CoT training data for improving performance, robustness, +explainability and value-alignment of future LLM-based AI systems. + +ThoughtSource: a central hub for large language model reasoning data | 3 +Methods +We selected NLP benchmarks for question answering and natural language inference for which +pre-existing data for constructing CoTs was available. For some of the datasets, one or multiple +additional datasets were used as sources for additional CoTs, allowing for the comparison of +different CoT generation methodologies. We created data loader scripts compatible with the +Hugging Face datasets library  +19 for all datasets. Additionally, we collected metadata of attributes +such as descriptions, websites and licenses. We contacted dataset providers and encouraged them +to choose an open source/open data license if licensing information was unavailable or unclear. +We implemented two kinds of schemas: 1) source dataset schemas, which are unique to each +dataset and provide data close to their original format; and 2) a standardized ThoughtSource +schema, which maps all datasets into a common format. The ThoughtSource schema was created +by extending the question answering schema of the BigBIO project  +15. +We implemented tailored algorithms for converting each dataset because the collected datasets +provide explanations in different ways, such as math expressions or structured graph-based +explanations. Furthermore, we performed preprocessing such as capitalization and punctuation +correction. To recover standard formatted text from pre-tokenized datasets, we reversed the +tokenization. This preprocessing was performed only on data in the ThoughtSource schema, +while data in the Source schemas was le� in their original formatting. All code for running these +conversions is available in our Github repository. +We developed a suite of Python libraries and tools for generating novel CoTs and answers by +calling LLM APIs, as well as tools for evaluating, comparing and annotating datasets. We built +upon the LangChain library (https://github.com/hwchase17/langchain/) for interfacing with a +wide variety of external LLM APIs. +Data Records +This first release of ThoughtSource integrates six scientific/medical, three general-domain and +five math word question answering datasets (Table 1). For every dataset except for PubmedQA +and MedQA we provide ‘reference CoTs’. We created these reference CoTs by converting +rationales provided by original datasets into reasoning chains. These rationales, depending on +the dataset, were created by human experts or obtained from crowdsourcing. For MedQA, +MedMCQA and PubmedQA, we generated and integrated CoTs with the AI systems +text-davinci-002  +3 and code-davinci-002  +20 (described in detail by co-authors Liévin et al. in a +separate manuscript +21). Furthermore, we extended CommonsenseQA and StrategyQA with +AI-generated CoTs created by few-shot +4 and zero-shot +5 prompting. Since current LLM models +are still prone to errors, it should be noted that AI-generated CoTs may contain faulty reasoning. + +ThoughtSource: a central hub for large language model reasoning data | 4 +Table 1: Integrated datasets. For some core datasets, additional datasets were used as sources for +additional CoTs. +Dataset +License +Scientific and medical question answering +WorldTree V2  +22 +AI2 Mercury license +EntailmentBank +23 +CC BY 4.0 +OpenBookQA +24 +Apache License 2.0 +MedQA (USMLE ) +25 +Core dataset +MIT +CoT source: few-shot from Liévin et al. +21 +CC-BY 4.0 +MedMCQA +26 +Core dataset +MIT +CoT source: few-shot from Liévin et al. +21 +CC-BY 4.0 +PubmedQA +27 +Core dataset +MIT +CoT source: few-shot from Liévin et al. +21 +CC-BY 4.0 +General-domain question answering +CommonsenseQA +28 +Core dataset +MIT +CoT source: ECQA ³ +Community Data +License Agreements +Sharing license 1.0 +CoT source: few-shot from Wei et al . +4, zero-shot from +Kojima et al . +5 +Unspecified +StrategyQA +29 +Core dataset +MIT +CoT source: few-shot from Wei et al . +4, zero-shot from +Kojima et al . +5 +Unspecified +QED +30 +CC BY-SA 3.0 +Math word problems +AQUA-RAT +31 +Apache 2.0 +ASDiv +32 +CC BY-NC 4.0 +GSM8K +33 +MIT +MAWPS +34 +MIT +SVAMP +35 +MIT + +ThoughtSource: a central hub for large language model reasoning data | 5 +Scientific/medical question answering datasets +WorldTree V2  +22 is one of the most detailed multi-hop science question answering datasets +available. Finding the right multiple-choice answers requires a multi-hop inference combining +between 1 and 16 facts (average: 6). It contains explanations created by experts in the form of +multiple facts. We concatenated these facts and applied a set of rules to improve style and +grammaticality to yield reference CoTs that are close to natural language. +EntailmentBank  +23 contains open-domain science exam questions and answers, along with +systematic explanations that show how the correct answer is reached through a series of steps. +These steps are organized into a tree structure, known as an entailment tree, which starts with +known facts and progresses through intermediate conclusions until the final answer is reached. +These entailment trees are also serialized into text-based proofs by traversing the trees. We +applied a set of rules to improve style and grammaticality in these proofs to yield reference CoTs +that are close to natural language. +OpenBookQA  +24 contains questions modeled a�er open-book exams of elementary-level science. +They require multi-step reasoning, commonsense knowledge, and a diverse application of core +science facts to find the correct answer. The dataset provides over 1,300 core science facts and a +mapping to all of the questions. By design, questions in OpenBookQA are answered incorrectly +by both retrieval-based and word co-occurrence algorithms. The dataset contains a single-fact +explanation of the correct answer for each question, which we adopted to create reference CoTs. +MedQA  +25 is a free-form multiple-choice OpenQA dataset containing questions from medical +board exams in the US (USMLE), Mainland China and Taiwan. We imported the +English-language USMLE subset. Reference CoTs are not provided. +MedMCQA  +26 is a multiple-choice question answering dataset containing real-world medical +entrance exam questions from the All India Institute of Medical Sciences (AIIMS PG) and +National Eligibility cum Entrance Test (NEET PG). Answer rationales authored by human +experts were integrated as reference CoTs. +PubmedQA  +27 is a question answering dataset containing biomedical questions extracted from +PubMed abstracts that can be answered with yes/no/maybe answers. In addition to the short +answer, each question comes with a longer answer, which can be used as reference CoT. +For MedQA, MedMCQA and PubmedQA we added CoTs generated with the AI systems +text-davinci-002  +3 and code-davinci-002  +20. + +ThoughtSource: a central hub for large language model reasoning data | 6 +General-domain question answering datasets +CommonsenseQA  +28 is a collection of multiple-choice questions that test a wide range of general +knowledge. We created reference CoTs for the train and validation set derived from the +crowd-sourced ECQA dataset³. We also added AI-generated reasoning chains generated with +few-shot +4 and zero-shot +5 prompting, which are available for the validation split. +StrategyQA  +29 is a question answering dataset that tests the ability to reason through +open-domain questions and provide Yes/No answers. Each example includes a question, a +decomposition of the question into reasoning steps, and evidence paragraphs from Wikipedia. +The dataset was created through a crowdsourcing process to gather creative and diverse +questions. Human-generated freetext reasoning chains are part of the train split of the original +dataset and were used as CoTs. The dataset also includes relevant paragraphs from Wikipedia, +but these were not included in our CoTs. We extended the StrategyQA dataset with AI-generated +CoTs created through few-shot +4 and zero-shot +5 prompting, which are available for the train split. +QED  +30 is a collection of expert-annotated structured explanations for answers to questions, built +upon a subset of the Google Natural Questions dataset. Given a question and a passage from +Wikipedia, QED uses linguistic information to represent explanations as a series of interpretable +steps, such as referential equality, sentencehood, and entailment. Structured reasoning chains by +experts are provided for all examples. To create reference CoTs, we extracted the sentence that +entails the answer; statements about referential equality in QED were converted to natural +language and added as additional steps in the CoTs (e.g. "The noun phrase […] in the sentence and +the noun phrase […] in the question refer to the same thing."). +Math word problem datasets +Algebra Question Answering with Rationales (AQUA-RAT)  +31 is a large-scale multiple-choice +dataset containing algebraic word problems. Each problem consists of a question with five +possible answers and a rationale, a step-by-step natural language explanation of the solution. We +used natural language explanations as reference CoTs. +Academia Sinica Diverse (ASDiv) math word problem (MWP) dataset  +32 aims to provide more +diverse language patterns and problem types than previous datasets. It covers most of the math +topics taught in elementary school. Each MWP is labeled with its grade level (for indicating +difficulty), the needed math operation (e.g. division) and includes a short explanation of the +solution. ASDiv contains explanations of answers in the form of nested math expressions using +common operators such as addition, subtraction, division and multiplication. We generated +reference CoTs by converting these math expressions into natural language explanation chains +using a rule-based method. + +ThoughtSource: a central hub for large language model reasoning data | 7 +Grade School Math 8K (GSM8K)  +33 contains grade school math word problems. Despite their +conceptual simplicity, these problems are more challenging to process than earlier datasets due +to their linguistic diversity. The creators of GSM8K instructed crowd workers to write solutions +to problems in free text format, which we used as reference CoTs in ThoughtSource, omitting +any additional arithmetic specifications. +Math Word Problems (MAWPS)  +34 is an online platform that provides a collection of math word +problems. The problems have simple one- or two-line explanations for their solutions. MAWPS +includes datasets from various sources, offers tools for automatically creating datasets with +specific characteristics as well as the possibility to tune lexical and template overlap. We +converted explanatory math expressions to reference CoTs with an approach similar to the one +used for ASDiv. +Simple Variations on Arithmetic Math Word Problems (SVAMP)  +35 was created by applying +carefully chosen variations to examples from existing datasets, such as ASDiv and MAWPS. +These variations make it difficult for language models to solve the problems using simple +heuristics, and instead require a deeper understanding and reasoning ability. We converted math +expressions to reference CoTs with an approach similar to the one used for ASDiv. +Dataset schema +Tables 2–5 provide descriptions and datatypes of the various fields in the ThoughtSource schema. +Any performed sample task leads to a generated CoT and answer to the question. Annotations +can be added programmatically or through an annotator tool. +Table 2: Fields of the ‘sample’ object. +Field +Description +Datatype +id +Unique identifier of object +string +ref_id +Identifier of external objects such as documents or other +resources +string +question +Question of task +string +type +Type of the question answering task, currently one of +[“multiplechoice”, “text”, “number”, “collection”] +string +choices +Set of multiple options containing the gold answer +list(string) +context +Additional context for answering the question +string +cot +Reference CoT, o�en human-generated. +list(string) +answer +Gold answer of task. Can contain multiple elements if type is +collection +list(string) +generated_cot +List of generated_cot objects +list(generated_cot_object) + +ThoughtSource: a central hub for large language model reasoning data | 8 +Table 3: Fields of the ‘generated_cot’ object. +Field +Description +Datatype +id +Unique identifier of object +string +templates_version +Version of the fragments.json file +string +instruction +Identifier of the cot trigger fragment stored in +fragments.json +string +cot_trigger +Identifier of the cot trigger fragment stored in +fragments.json +string +cot_trigger_template +Template to specify structure of prompt text +string +prompt_text +Full text of prompt used for the CoT generation step +string +answers +List of generated answer objects +list(answer_object) +cot +Generated chain-of-thought +string +author +Name of the author +string +date +Date of the chain-of-thought generation +string +api_service +Identification of the used api service +string +model +Identification of the used language model +string +comment +Comment +string +annotation +List of annotation objects +list(annotation_object) +Table 4: Fields of the ‘answer’ object. +Field +Description +Datatype +id +Unique identifier of object +string +answer_extraction +Identifier of the answer extraction fragment stored in +fragments.json +string +cot_trigger_template +Template to specify structure of prompt text +string +answer_extraction +_text +Full text of prompt used for the answer extraction step +string +answer +Extracted answer +string +correct_answer +True if the extracted answer is equal to the gold answer, +else false +bool +Table 5: Fields of the ‘annotation’ object. +Field +Description +Datatype +author +Name of the author +string +date +Date of the creation of the annotation +string +key +Specifies the label of the annotation +string +value +Specifies the value of the annotation +string + +ThoughtSource: a central hub for large language model reasoning data | 9 +Table 6 shows the example counts, CoT counts and answer types of each dataset. The majority of +datasets in the current collection are of the multiple choice answer type. The medical dataset +MedMCQA is the largest among all datasets. +Table 6: Statistics and answer types for all datasets. Note that generated CoTs are not available for all +examples, and multiple CoTs might have been generated for any given example. +[Link: Notebook used for generating stats and graphs] +Dataset ID +Examples +Reference +CoTs +Examples w. +generated +CoTs +Generated +CoTs +Answer type +aqua +97,975 +97,975 +0 +0 +multiple choice +asdiv +1218 +1218 +0 +0 +number +commonsense_qa +12,102 +10,962 +1221 +2437 +multiple choice +entailment_bank +1840 +1840 +0 +0 +text +gsm8k +8792 +8792 +0 +0 +number +mawps +1921 +1921 +0 +0 +number +med_qa (USMLE) +12,723 +0 +1273 +133,660 +multiple choice +medmc_qa +193,155 +161,558 +1000 +104,987 +multiple choice +open_book_qa +5957 +5957 +0 +0 +multiple choice +pubmed_qa +1000 +0 +500 +2500 +multiple choice +qed +6175 +6175 +0 +0 +collection +strategy_qa +2780 +2290 +2289 +4532 +bool +svamp +1000 +1000 +0 +0 +number +worldtree +4367 +4365 +0 +0 +multiple choice +We analyzed the distribution of question and CoT field lengths (Fig. 1). MedQA has the longest +median question length, while PubMedQA has the longest median CoT length. Several datasets +contain outlier CoTs with extremely long text lengths. Context fields were only filled for the +PubmedQA and QED datasets, with mean context lengths of 116 and 56 tokens, respectively. + +ThoughtSource: a central hub for large language model reasoning data | 10 +Figure 1: Distribution of question and CoT field lengths. +Technical Validation +The datasets were reviewed by three team members and issues were tracked on the issue tracker +of the associated GitHub repository. +To characterize potential overlaps and relations between datasets, we calculated mutual n-gram +overlap using n=3. (Fig. 2) . To quantify the overlap between two sets of n-grams we use the +Szymkiewicz–Simpson coefficient (overlap coefficient), which can be interpreted as the +proportion of n-grams of the smaller dataset that are contained in the bigger dataset. + +worldtree +svamp +工白 +strategy_qa +百T +qed +pubmed_qa +open_book_qa +H +Dataset +medmc_qa +med_qa +mawps +gsm8k +entailment_bank +HH +commonsense_qa +asdiv +enbe +0 +50 +100 +150 +200 +250 +300 +Numberoftokens inquestionworldtree +svamp +strategy_qa +qed +pubmed_qa +open_book_qa +Dataset +medmc_qa +med_qa +mawps +gsm8k +entailment_bank +commonsense_qa +asdiv +enbe +H +0 +50 +100 +150 +200 +250 +300 +350 +Number of tokens in CoTThoughtSource: a central hub for large language model reasoning data | 11 +There is an overlap of 1.0 between the set of questions in WorldTree v2 and EntailmentBank. The +QA pairs in EntailmentBank were taken from the WorldTree v2 dataset  +23, so all the questions in +EntailmentBank are a subset of WorldTree v2. +Furthermore, there is significant overlap between the questions contained in ASDiv and SVAMP +and those in ASDiv and MAWPS. ASDiv and SVAMP have overlapped questions because a subset +of examples from ASDiv was used as seed examples for the creation of SVAMP. For MAWPS and +ASDiv, questions were crawled from web resources. The overlap could be due to examples being +crawled from the same web resources. +Besides overlaps in questions, we also identified overlaps in CoTs. WorldTree v2 provided an +initial pool of atomic facts that the annotators could use to construct an explanation tree in +EntailmentBank (in addition to creating their own facts). This explains the high overlap of +n-grams of CoTs in WorldTree v2 and EntailmentBank. Similarly, a subset of WorldTree v2 facts +was used for the creation of explanations in OpenbookQA. +Figure 2: n-gram overlap in questions and CoTs. Overlap is measured by mutual n-gram overlap using +n=3, values <0.01 are omitted. +Usage Notes +Python libraries for accessing and working with data can be downloaded from the Github +repository and installed with the pip tool. Fig. 3 demonstrates how to load a dataset, randomly +sample from the pre-populated data in the dataset, call an external LLM API to generate novel +CoTs and answers, automatically evaluate the accuracy of generated answers, and finally save all +generated data to a JSON file. Fig. 4 depicts an excerpt of the resulting JSON file. + +Question +CoT +aqua1.00 +asdiv 0.02 +1.00 +asdiv +1.00 +commonsense_qa +1.00 +commonsense_qa +1.00 +0.8 +entailment_bank +1.00 +entailment_bank +1.00 +gsm8k0.020.04 +1.00 +gsm8k0.03 +1.00 +mawps0.02 +0.27 +0.05 +1.00 +mawps +1.00 +0.6 +med_qa +1.00 +med_qa +1.00 +medmc_qa +0.061.00 +medmc_qa +0.01 +1.00 +1.00 +0.55 +open_book_qa +open_book_ga +0.01 +1.00 +0.4 +pubmed_qa +0.030.04 +1.00 +pubmed_qa +1.00 +qed +1.00 +qed +1.00 +0.2 +strategy_qa +1.00 +strategy_qa +0.02 +0.011.00 +svamp 0.02 +0.19 +0.040.03 +1.00 +svamp +1.00 +worldtree +1.00 +0.01 +1.00 +worldtree +0.41 +0.01 +0.84 +1.00 +0 +aqua +asdiv +com +gsm8k +pubn +qed +svamp +worldtree +aqua +asdiy +gsm8k +ma +med +medr +oper +svamp +oper +bmed. +9ed +worldtree +tegy..qa +9a +eb" +9a +-9a +9a +9a +_qa +ga +9a +9a +9aThoughtSource: a central hub for large language model reasoning data | 12 +from cot import Collection +# Load a dataset +collection_worldtree = Collection(["worldtree"]) +# Randomly sample 10 rows of train split +collection_worldtree_10 = collection_worldtree.select(split="train", +number_samples=10) +# Create a config file for calling OpenAI API to generate new CoTs and answers. +config={ +"instruction_keys": ["qa-01"], # Determines which instructions are used +"cot_trigger_keys": ["kojima-01"], # Determines which cot triggers are used +"answer_extraction_keys": ["kojima-A-D"], # Determines which answer extraction +# prompts are used +"author" : "your_name", # Name of the person responsible for generation +"api_service": "openai", # Name of the API called ("openai", "huggingface_hub" +# or a mock for testing: "mock_api") +"engine": "text-davinci-002", # Name of the engine used +"temperature": 0, # Level of randomness in the generated output +"max_tokens": 512, # Maximum length of output generated by the model +"api_time_interval": 1.0, # Pause between two api calls in seconds +"verbose": False, # Determines whether the progress of the generation is printed +"warn": True, # Determines whether a warnings that external APIs will be called +# are printed +} +# Generate novel chains-of-thought and answer extractions +collection_worldtree_10.generate(config=config) +# Evaluate accuracy of model predictions +collection_worldtree_10.evaluate() +# Example output: {'accuracy': {'qa-01_kojima-01_kojima-A-D': 0.86}} +# Save all data (including evaluation data) to JSON file +collection_worldtree_10.dump("worldtree_10_evaluate.json") +Figure 3: Demonstration of the ThoughtSource API. Basic functionalities of the data loader, generator +and evaluator modules are demonstrated. + +ThoughtSource: a central hub for large language model reasoning data | 13 +{ +"id": "1242", +"ref_id": "", +"question": "Which is a characteristic of a sperm cell but not of an egg cell?", +"type": "multiplechoice", +"choices": [ +"round shape", +"presence of a tail", +"contains genetic information", +"involved in sexual reproduction" +], +"context": "", +"cot": [ +"A part of something means a characteristic of something.", +"A tail is not part of an egg cell.", +"A tail is a part of a sperm cell.", +"A part of something is present in that something." +], +"answer": [ +"presence of a tail" +], +"generated_cot": [ +{ +"id": "738b54ba-9a20-47e6-b8ff-7cb876103b92", +"fragments_version": "0.01", +"api_service": "openai", +"model": "{'name': 'text-davinci-002', 'temperature': 0, 'max_tokens': +512}", +"instruction": "qa-01", +"cot_trigger": "kojima-01", +"cot": "We know that both sperm and egg cells are involved in sexual +reproduction, so we can eliminate (D). \n\nWe also know that both +sperm and egg cells contain genetic information, so we can +eliminate (C). \n\nThat leaves us with (A) and (B). \n\nWe know +that sperm cells have a tail, but egg cells do not. Therefore, the +correct answer is (B).", +"answers": [ +{ +"id": "7f7cc26f-a3b3-4b59-9af7-35980514d0c3", +"answer_extraction": "kojima-A-D", +"answer": " B.", +"correct_answer": true +} +], +"author": "your_name", +"date": "2023/01/12 14:18:57", +"comment": "", +"annotation": [] +} +] +} +Figure 4: An excerpt of data generated by running the example code. Data for a single question from +Worldtree V2 are shown, including human-authored reference CoT, gold-standard answer, an +AI-generated CoT and extracted answer, as well as evaluation results. Some fields were omitted for +legibility. + +ThoughtSource: a central hub for large language model reasoning data | 14 +In a zero-shot setup, specific text fragments can be used to prompt question answering and CoT +reasoning in LLMs. ThoughtSource includes a curated list of text fragments that can be used to +generate novel CoTs (Fig. 5). Where possible, we also mapped individual CoTs in pre-existing +CoT datasets to the text fragments that were used in their creation. +"instructions": { +"qa-01": "Answer the following question through step-by-step reasoning.", +"qa-02": "Answer the following question through careful, concise step-by-step +reasoning.", +"qa-03": "Answer the following question through careful, concise step-by-step +reasoning. Avoid making up wrong statements. If the question does not +make sense or cannot be answered, write \"I cannot answer the +question\". +If you do not have a good answer, write \"I do not have a good answer\". +If you are uncertain, write \"I am uncertain about this\".", +[...] +}, +"cot_triggers": { +"kojima-01": "Answer: Let's think step by step.", +"kojima-02": "Answer: We should think about this step by step.", +"kojima-03": "Answer: First,", +"kojima-04": "Answer: Before we dive into the answer,", +[...] +"lievin-01": "Answer: Let's derive the differential diagnosis step by step.", +"lievin-02": "Answer: Let's use step by step inductive reasoning, given the +medical nature of the question.", +[...] +"lievin-26": "Answer: Let's follow a Bayesian step by step approach.", +"lievin-27": "Answer: Let's reflect on each option from the least likely to the +most likely.", +"lievin-28": "Answer: Let's use step by step Bayesian reasoning, given the +medical nature of the question." +}, +"answer_extractions":{ +"kojima-01": "Therefore, the answer is", +"kojima-02": "Therefore,", +"kojima-03": "The answer is", +"kojima-numerals": "Therefore, the answer (arabic numerals) is", +"kojima-yes-no": "Therefore, the answer (Yes or No) is", +"kojima-A-C": "Therefore, among A through C, the answer is", +"kojima-A-D": "Therefore, among A through D, the answer is", +[...] +} +Figure 5: An excerpt of the collection of prompt fragments. These fragments can be used to build +prompts for interacting with LLMs, allowing for empirical testing of how different prompts affect model +performance. +We provide two web-based interfaces for exploring and annotating ThoughtSource data, the +Dataset Viewer and the Annotator. The Dataset Viewer is a simple interface for exploring dataset +contents. The Annotator (Fig. 6) allows you to upload specific subsets of a dataset, provides +convenience functions for highlighting similarities between different generated CoTs and the +correctness of generated answers, and allows you to annotate individual CoTs interactively. The +annotator facilitates identifying strengths and weaknesses of different CoTs. Annotations can be +used for downstream model evaluation and further improving the capabilities of AI models +through fine-tuning / reinforcement learning. + +ThoughtSource: a central hub for large language model reasoning data | 15 +Figure 6: The ThoughtSource Annotator. The web-based interface allows for convenient inspection and +annotation of reasoning chains and answers. Text that is similar between CoTs can be automatically +highlighted based on an easily adjustable similarity threshold, facilitating a better understanding of +similarities and differences of different reasoning chains. +All tools and libraries, as well as more detailed demonstration notebooks, can be found on the +project Github page. +We plan to add more datasets and generated CoTs to the ThoughtSource repository, and we +welcome outside contributions. Novel CoTs for existing core datasets can be generated and +shared through the ThoughtSource APIs and JSON files. Completely new datasets can also be +added, as described in the Github repository's contribution guide. +Code Availability +All code, data and tools are openly available at https://github.com/OpenBioLink/ThoughtSource. +Our code and data are licensed under an MIT license, while data adapted from existing datasets +are available under the licenses of their respective sources. + + ThoughtSource Annotator +tfidf Visualisation threshold +Download current +Matthias Logout +Question 1242 (test) +A 28-year-old woman, gravida 1, para O, at 20 weeks' gestation comes to the physician with her husband for a prenatal visit. Her +pregnancy has been uncomplicated. They are planning to travel to Ethiopia next month to visit the husband's family +uterus that is consistent with a 20-week gestation. Which of the following drugs is most suitable for pre-exposure prophylaxis +against malaria? +A. Doxycycline +B. Primaquine +C. Mefloquine +D. Chloroquine +E. Proguanil +Let's think step by step. The first thing we need to +Let's think step by step like a medical expert. The +Let's differentiate using step by step reasoning like +do is figure out what kind of malaria is common in +first thing to consider is the safety of the +a medical expert: 1) Doxycycline is not suitable for +Ethiopia. According to this website, the most +medication for the mother and the developing +pre-exposure prophylaxis against malaria in this +common type of malaria in Ethiopia is Plasmodium +fetus. Doxycycline and mefloquine are not +case because the patient is pregnant. 2) +falciparum. The next thing we need to do is figure +recommended for use in pregnant women, so we +Mefloquine is not suitable for pre-exposure +out which of the four drugs listed is most effective +can eliminate those options. The next thing to +prophylaxis against malaria in this case because +against Plasmodium falciparum. According to this +consider is the efficacy of the medication. +the patient is pregnant. 3) Chloroquine is suitable +website, chloroguine and proguanil are not +Chloroquine is not effective in Ethiopia, so we can +for pre-exposure prophylaxis against malaria in +effective against Plasmodium falciparum. That +eliminate that option. That leaves us with +this case because the patient is not pregnant. 4) +leaves doxycycline and mefloquine. The next thing +proguanil, which is safe for use in pregnant +Proguanil is suitable for pre-exposure prophylaxis +we need to do is figure out which of the two +women and is effective in Ethiopia. +against malaria in this case because the patient is +remaining drugs has the fewest side effects + Answer: Proguanil +not pregnant. +According to this website, the most common side +☆ + Answer: Chloroquine +effects of doxycycline are nausea, vomiting, and +☆ + Incorrect reasoning +diarrhea. The most common side effects of + Insufficient knowledge + Incorrect reasoning +mefloquine are nausea, vomiting, diarrhea, and +headache. Based on the information above, the + Incorrect reading comprehension + Insufficient knowledge +most suitable drug for pre-exposure prophylaxis + Incorrect reading comprehension + Too verbose +against malaria in Ethiopia is mefloquine. + Too verbose +O Answer: Mefloquine + Incorrect reasoning + Insufficient knowledge + Incorrect reading comprehension + Too verboseThoughtSource: a central hub for large language model reasoning data | 16 +Acknowledgements +We thank primary dataset contributors that assisted with assembling the ThoughtSource +meta-dataset. +Author contributions +S.O. and K.H. wrote the code for accessing, converting, generating and analysing datasets, and +wrote parts of the manuscript and documentation. +V.L., C.E. and O.W. generated and analysed CoT data for medical datasets. +M.Ma. wrote the code of the annotator so�ware. +M.Mo. wrote a first prototype of code for accessing and converting datasets. +R.P. contributed to improving code and documentation quality. +M.S. conceived and supervised the project and wrote parts of the manuscript and documentation. +All authors have read and approved the final manuscript. +Competing interests +The authors declare that there are no conflicts of interest. +References +1. 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Are NLP Models really able to Solve Simple Math Word +Problems? in Proceedings of the 2021 Conference of the North American Chapter of the Association +for Computational Linguistics: Human Language Technologies 2080–2094 (Association for +Computational Linguistics, 2021). doi:10.18653/v1/2021.naacl-main.168. + diff --git a/9dFJT4oBgHgl3EQfoixI/content/tmp_files/load_file.txt b/9dFJT4oBgHgl3EQfoixI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e8baa5ad40234131b45635d91d65eabf61ba1e02 --- /dev/null +++ b/9dFJT4oBgHgl3EQfoixI/content/tmp_files/load_file.txt @@ -0,0 +1,795 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf,len=794 +page_content='ThoughtSource: A central hub for large language model reasoning data Simon Ott 1*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Konstantin Hebenstreit 1*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Valentin Liévin 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Christoffer Egeberg Hother 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Milad Moradi 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Maximilian Mayrhauser 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Robert Praas 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Ole Winther 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Matthias Samwald 1 1) Institute of Artificial Intelligence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Medical University of Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Austria 2) Section for Cognitive Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Technical University of Denmark,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Lyngby,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Denmark 3) School of Electrical Engineering and Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The Royal Institute of Technology (KTH),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Stockholm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Sweden 4) Department of Clinical Immunology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Copenhagen University Hospital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Copenhagen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Denmark equal contribution Corresponding author: Matthias Samwald (matthias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='samwald [at] meduniwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='at) Abstract Large language models (LLMs) such as GPT-3 and ChatGPT have recently demonstrated impressive results across a wide range of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' LLMs are still limited, however, in that they frequently fail at complex reasoning, their reasoning processes are opaque, they are prone to ‘hallucinate’ facts, and there are concerns about their underlying biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Letting models verbalize reasoning steps as natural language, a technique known as chain-of-thought prompting, has recently been proposed as a way to address some of these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Here we present the first release of ThoughtSource, a meta-dataset and so�ware library for chain-of-thought (CoT) reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The goal of ThoughtSource is to improve future artificial intelligence systems by facilitating qualitative understanding of CoTs, enabling empirical evaluations, and providing training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' This first release of ThoughtSource integrates six scientific/medical, three general-domain and five math word question answering datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Background & Summary The most recent generation of large language models (LLMs) has produced impressive results across a wide range of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Examples of such models include T0 1, GPT-3 2, InstructGPT 3 and ChatGPT (https://openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='com/blog/chatgpt/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' These models demonstrated remarkable ability to ThoughtSource: a central hub for large language model reasoning data | 2 generate text that is both realistic and coherent, as well as good performance on a broad spectrum of tasks, despite not explicitly being trained on them 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' However, despite this ability, LLMs are also limited in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' They o�en fail to produce accurate predictions due to their inability to accomplish complex reasoning, such as solving mathematical problems or question answering tasks requiring multi-hop reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Furthermore, they tend to be black boxes, making it difficult to understand how and why predictions are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' These limitations severely limit the application domains of LLMs and have the potential to cause harm, as lack of explainability and robustness can lead to critical failures and biases when these models are deployed in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' One recently proposed method for enabling complex reasoning and generating explanations with LLMs is to force models to explicitly verbalize reasoning steps as natural language, a technique known as chain-of-thought prompting 4,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' This method improved performance on a variety of tasks and sparked the active development of further refinements 6, such as decomposing problems and structuring reasoning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=', least-to-most prompting 7, ReAct 8, self-ask 9, maieutic prompting 10, successive prompting 11) and/or extending LLM capabilities by leveraging external services for tasks like information retrieval (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=', self-ask 9, IRCoT 12, DSP 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=" The terminology surrounding these rapidly evolving techniques is not settled, hence in this document, we refer to all approaches that result in a linear sequence of reasoning steps as 'chain-of-thought' (CoT)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Meta-datasets (datasets of datasets) that are easily accessible and standardized have proven useful for training and evaluating versatile LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Examples include SuperGLUE 14 for general-domain language model tasks, BigBIO 15 and BLURB 16 for biomedical tasks, or Pile 17 and ROOTS 18 as text corpora for LLM pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Datasets can be complemented by tools such as PromptSource, which was used to convert a large number of datasets into prompts fit for training and interrogating LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' PromptSource facilitated training the highly performant T0 model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Here we present the first release of ThoughtSource, a meta-dataset and so�ware library for chain-of-thought reasoning in LLMs (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='com/OpenBioLink/ThoughtSource).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The goals of ThoughtSource are to: — Facilitate qualitative understanding of CoTs generated by LLMs under various conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=', across tasks, models and prompts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' — Enable empirical and quantitative evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' — Provide a library of diverse CoT training data for improving performance, robustness, explainability and value-alignment of future LLM-based AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ThoughtSource: a central hub for large language model reasoning data | 3 Methods We selected NLP benchmarks for question answering and natural language inference for which pre-existing data for constructing CoTs was available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' For some of the datasets, one or multiple additional datasets were used as sources for additional CoTs, allowing for the comparison of different CoT generation methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We created data loader scripts compatible with the Hugging Face datasets library 19 for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Additionally, we collected metadata of attributes such as descriptions, websites and licenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We contacted dataset providers and encouraged them to choose an open source/open data license if licensing information was unavailable or unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We implemented two kinds of schemas: 1) source dataset schemas, which are unique to each dataset and provide data close to their original format;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' and 2) a standardized ThoughtSource schema, which maps all datasets into a common format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The ThoughtSource schema was created by extending the question answering schema of the BigBIO project 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We implemented tailored algorithms for converting each dataset because the collected datasets provide explanations in different ways, such as math expressions or structured graph-based explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Furthermore, we performed preprocessing such as capitalization and punctuation correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' To recover standard formatted text from pre-tokenized datasets, we reversed the tokenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' This preprocessing was performed only on data in the ThoughtSource schema, while data in the Source schemas was le� in their original formatting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' All code for running these conversions is available in our Github repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We developed a suite of Python libraries and tools for generating novel CoTs and answers by calling LLM APIs, as well as tools for evaluating, comparing and annotating datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We built upon the LangChain library (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='com/hwchase17/langchain/) for interfacing with a wide variety of external LLM APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Data Records This first release of ThoughtSource integrates six scientific/medical, three general-domain and five math word question answering datasets (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' For every dataset except for PubmedQA and MedQA we provide ‘reference CoTs’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We created these reference CoTs by converting rationales provided by original datasets into reasoning chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' These rationales, depending on the dataset, were created by human experts or obtained from crowdsourcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' For MedQA, MedMCQA and PubmedQA, we generated and integrated CoTs with the AI systems text-davinci-002 3 and code-davinci-002 20 (described in detail by co-authors Liévin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' in a separate manuscript 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Furthermore, we extended CommonsenseQA and StrategyQA with AI-generated CoTs created by few-shot 4 and zero-shot 5 prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Since current LLM models are still prone to errors, it should be noted that AI-generated CoTs may contain faulty reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ThoughtSource: a central hub for large language model reasoning data | 4 Table 1: Integrated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' For some core datasets, additional datasets were used as sources for additional CoTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Dataset License Scientific and medical question answering WorldTree V2 22 AI2 Mercury license EntailmentBank 23 CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='0 OpenBookQA 24 Apache License 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='0 MedQA (USMLE ) 25 Core dataset MIT CoT source: few-shot from Liévin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' 21 CC-BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='0 MedMCQA 26 Core dataset MIT CoT source: few-shot from Liévin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' 21 CC-BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='0 PubmedQA 27 Core dataset MIT CoT source: few-shot from Liévin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' 21 CC-BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='0 General-domain question answering CommonsenseQA 28 Core dataset MIT CoT source: ECQA ³ Community Data License Agreements Sharing license 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='0 CoT source: few-shot from Wei et al .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' 4, zero-shot from Kojima et al .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' 5 Unspecified StrategyQA 29 Core dataset MIT CoT source: few-shot from Wei et al .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' 4, zero-shot from Kojima et al .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' 5 Unspecified QED 30 CC BY-SA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='0 Math word problems AQUA-RAT 31 Apache 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='0 ASDiv 32 CC BY-NC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='0 GSM8K 33 MIT MAWPS 34 MIT SVAMP 35 MIT ThoughtSource: a central hub for large language model reasoning data | 5 Scientific/medical question answering datasets WorldTree V2 22 is one of the most detailed multi-hop science question answering datasets available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Finding the right multiple-choice answers requires a multi-hop inference combining between 1 and 16 facts (average: 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' It contains explanations created by experts in the form of multiple facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We concatenated these facts and applied a set of rules to improve style and grammaticality to yield reference CoTs that are close to natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' EntailmentBank 23 contains open-domain science exam questions and answers, along with systematic explanations that show how the correct answer is reached through a series of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' These steps are organized into a tree structure, known as an entailment tree, which starts with known facts and progresses through intermediate conclusions until the final answer is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' These entailment trees are also serialized into text-based proofs by traversing the trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We applied a set of rules to improve style and grammaticality in these proofs to yield reference CoTs that are close to natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' OpenBookQA 24 contains questions modeled a�er open-book exams of elementary-level science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' They require multi-step reasoning, commonsense knowledge, and a diverse application of core science facts to find the correct answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The dataset provides over 1,300 core science facts and a mapping to all of the questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' By design, questions in OpenBookQA are answered incorrectly by both retrieval-based and word co-occurrence algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The dataset contains a single-fact explanation of the correct answer for each question, which we adopted to create reference CoTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' MedQA 25 is a free-form multiple-choice OpenQA dataset containing questions from medical board exams in the US (USMLE), Mainland China and Taiwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We imported the English-language USMLE subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Reference CoTs are not provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' MedMCQA 26 is a multiple-choice question answering dataset containing real-world medical entrance exam questions from the All India Institute of Medical Sciences (AIIMS PG) and National Eligibility cum Entrance Test (NEET PG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Answer rationales authored by human experts were integrated as reference CoTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' PubmedQA 27 is a question answering dataset containing biomedical questions extracted from PubMed abstracts that can be answered with yes/no/maybe answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' In addition to the short answer, each question comes with a longer answer, which can be used as reference CoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' For MedQA, MedMCQA and PubmedQA we added CoTs generated with the AI systems text-davinci-002 3 and code-davinci-002 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ThoughtSource: a central hub for large language model reasoning data | 6 General-domain question answering datasets CommonsenseQA 28 is a collection of multiple-choice questions that test a wide range of general knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We created reference CoTs for the train and validation set derived from the crowd-sourced ECQA dataset³.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We also added AI-generated reasoning chains generated with few-shot 4 and zero-shot 5 prompting, which are available for the validation split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' StrategyQA 29 is a question answering dataset that tests the ability to reason through open-domain questions and provide Yes/No answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Each example includes a question, a decomposition of the question into reasoning steps, and evidence paragraphs from Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The dataset was created through a crowdsourcing process to gather creative and diverse questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Human-generated freetext reasoning chains are part of the train split of the original dataset and were used as CoTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The dataset also includes relevant paragraphs from Wikipedia, but these were not included in our CoTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We extended the StrategyQA dataset with AI-generated CoTs created through few-shot 4 and zero-shot 5 prompting, which are available for the train split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' QED 30 is a collection of expert-annotated structured explanations for answers to questions, built upon a subset of the Google Natural Questions dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Given a question and a passage from Wikipedia, QED uses linguistic information to represent explanations as a series of interpretable steps, such as referential equality, sentencehood, and entailment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Structured reasoning chains by experts are provided for all examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' To create reference CoTs, we extracted the sentence that entails the answer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' statements about referential equality in QED were converted to natural language and added as additional steps in the CoTs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' "The noun phrase […] in the sentence and the noun phrase […] in the question refer to the same thing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Math word problem datasets Algebra Question Answering with Rationales (AQUA-RAT) 31 is a large-scale multiple-choice dataset containing algebraic word problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Each problem consists of a question with five possible answers and a rationale, a step-by-step natural language explanation of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We used natural language explanations as reference CoTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Academia Sinica Diverse (ASDiv) math word problem (MWP) dataset 32 aims to provide more diverse language patterns and problem types than previous datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' It covers most of the math topics taught in elementary school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Each MWP is labeled with its grade level (for indicating difficulty), the needed math operation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' division) and includes a short explanation of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ASDiv contains explanations of answers in the form of nested math expressions using common operators such as addition, subtraction, division and multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We generated reference CoTs by converting these math expressions into natural language explanation chains using a rule-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ThoughtSource: a central hub for large language model reasoning data | 7 Grade School Math 8K (GSM8K) 33 contains grade school math word problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Despite their conceptual simplicity, these problems are more challenging to process than earlier datasets due to their linguistic diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The creators of GSM8K instructed crowd workers to write solutions to problems in free text format, which we used as reference CoTs in ThoughtSource, omitting any additional arithmetic specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Math Word Problems (MAWPS) 34 is an online platform that provides a collection of math word problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The problems have simple one- or two-line explanations for their solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' MAWPS includes datasets from various sources, offers tools for automatically creating datasets with specific characteristics as well as the possibility to tune lexical and template overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We converted explanatory math expressions to reference CoTs with an approach similar to the one used for ASDiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Simple Variations on Arithmetic Math Word Problems (SVAMP) 35 was created by applying carefully chosen variations to examples from existing datasets, such as ASDiv and MAWPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' These variations make it difficult for language models to solve the problems using simple heuristics, and instead require a deeper understanding and reasoning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We converted math expressions to reference CoTs with an approach similar to the one used for ASDiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Dataset schema Tables 2–5 provide descriptions and datatypes of the various fields in the ThoughtSource schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Any performed sample task leads to a generated CoT and answer to the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Annotations can be added programmatically or through an annotator tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Table 2: Fields of the ‘sample’ object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Field Description Datatype id Unique identifier of object string ref_id Identifier of external objects such as documents or other resources string question Question of task string type Type of the question answering task, currently one of [“multiplechoice”, “text”, “number”, “collection”] string choices Set of multiple options containing the gold answer list(string) context Additional context for answering the question string cot Reference CoT, o�en human-generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' list(string) answer Gold answer of task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Can contain multiple elements if type is collection list(string) generated_cot List of generated_cot objects list(generated_cot_object) ThoughtSource: a central hub for large language model reasoning data | 8 Table 3: Fields of the ‘generated_cot’ object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Field Description Datatype id Unique identifier of object string templates_version Version of the fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='json file string instruction Identifier of the cot trigger fragment stored in fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='json string cot_trigger Identifier of the cot trigger fragment stored in fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='json ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='string ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='cot_trigger_template ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='Template to specify structure of prompt text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='string ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='prompt_text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='Full text of prompt used for the CoT generation step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='string ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='answers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='List of generated answer objects ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='list(answer_object) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='cot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='Generated chain-of-thought ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='string ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='author ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='Name of the author ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='string ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='date ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='Date of the chain-of-thought generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='string ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='api_service ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='Identification of the used api service ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='string ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='Identification of the used language model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='string ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='comment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='Comment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='string ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='annotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='List of annotation objects ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='list(annotation_object) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='Table 4: Fields of the ‘answer’ object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Field Description Datatype id Unique identifier of object string answer_extraction Identifier of the answer extraction fragment stored in fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='json string cot_trigger_template Template to specify structure of prompt text string answer_extraction _text Full text of prompt used for the answer extraction step string answer Extracted answer string correct_answer True if the extracted answer is equal to the gold answer, else false bool Table 5: Fields of the ‘annotation’ object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Field Description Datatype author Name of the author string date Date of the creation of the annotation string key Specifies the label of the annotation string value Specifies the value of the annotation string ThoughtSource: a central hub for large language model reasoning data | 9 Table 6 shows the example counts, CoT counts and answer types of each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The majority of datasets in the current collection are of the multiple choice answer type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The medical dataset MedMCQA is the largest among all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Table 6: Statistics and answer types for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Note that generated CoTs are not available for all examples, and multiple CoTs might have been generated for any given example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' [Link: Notebook used for generating stats and graphs] Dataset ID Examples Reference CoTs Examples w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' generated CoTs Generated CoTs Answer type aqua 97,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='975 97,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='975 0 0 multiple choice asdiv 1218 1218 0 0 number commonsense_qa 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='102 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='962 1221 2437 multiple choice entailment_bank 1840 1840 0 0 text gsm8k 8792 8792 0 0 number mawps 1921 1921 0 0 number med_qa (USMLE) 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='723 0 1273 133,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='660 multiple choice medmc_qa 193,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='155 161,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='558 1000 104,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='987 multiple choice open_book_qa 5957 5957 0 0 multiple choice pubmed_qa 1000 0 500 2500 multiple choice qed 6175 6175 0 0 collection strategy_qa 2780 2290 2289 4532 bool svamp 1000 1000 0 0 number worldtree 4367 4365 0 0 multiple choice We analyzed the distribution of question and CoT field lengths (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' MedQA has the longest median question length, while PubMedQA has the longest median CoT length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Several datasets contain outlier CoTs with extremely long text lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Context fields were only filled for the PubmedQA and QED datasets, with mean context lengths of 116 and 56 tokens, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ThoughtSource: a central hub for large language model reasoning data | 10 Figure 1: Distribution of question and CoT field lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Technical Validation The datasets were reviewed by three team members and issues were tracked on the issue tracker of the associated GitHub repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' To characterize potential overlaps and relations between datasets, we calculated mutual n-gram overlap using n=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' To quantify the overlap between two sets of n-grams we use the Szymkiewicz–Simpson coefficient (overlap coefficient), which can be interpreted as the proportion of n-grams of the smaller dataset that are contained in the bigger dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='worldtree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='svamp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='工白 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='strategy_qa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='百T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='qed ' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='Number of tokens in CoTThoughtSource: a central hub for large language model reasoning data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='| 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='There is an overlap of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='0 between the set of questions in WorldTree v2 and EntailmentBank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The QA pairs in EntailmentBank were taken from the WorldTree v2 dataset 23, so all the questions in EntailmentBank are a subset of WorldTree v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Furthermore, there is significant overlap between the questions contained in ASDiv and SVAMP and those in ASDiv and MAWPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ASDiv and SVAMP have overlapped questions because a subset of examples from ASDiv was used as seed examples for the creation of SVAMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' For MAWPS and ASDiv, questions were crawled from web resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The overlap could be due to examples being crawled from the same web resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Besides overlaps in questions, we also identified overlaps in CoTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' WorldTree v2 provided an initial pool of atomic facts that the annotators could use to construct an explanation tree in EntailmentBank (in addition to creating their own facts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' This explains the high overlap of n-grams of CoTs in WorldTree v2 and EntailmentBank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Similarly, a subset of WorldTree v2 facts was used for the creation of explanations in OpenbookQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Figure 2: n-gram overlap in questions and CoTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Overlap is measured by mutual n-gram overlap using n=3, values <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='01 are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Usage Notes Python libraries for accessing and working with data can be downloaded from the Github repository and installed with the pip tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' 3 demonstrates how to load a dataset, randomly sample from the pre-populated data in the dataset, call an external LLM API to generate novel CoTs and answers, automatically evaluate the accuracy of generated answers, and finally save all generated data to a JSON file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' 4 depicts an excerpt of the resulting JSON file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Question CoT aqua1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='00 asdiv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='00 asdiv 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='00 commonsense_qa 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='00 commonsense_qa 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='8 entailment_bank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='00 entailment_bank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='00 gsm8k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='00 gsm8k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='00 mawps0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='00 mawps 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='00 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='00 svamp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='00 worldtree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='00 worldtree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='00 0 aqua asdiv com gsm8k pubn qed svamp worldtree aqua asdiy gsm8k ma med medr oper svamp oper bmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' 9ed worldtree tegy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='.qa 9a eb" 9a 9a 9a 9a _qa ga 9a 9a 9aThoughtSource: a central hub for large language model reasoning data | 12 from cot import Collection # Load a dataset collection_worldtree = Collection(["worldtree"]) # Randomly sample 10 rows of train split collection_worldtree_10 = collection_worldtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='select(split="train", number_samples=10) # Create a config file for calling OpenAI API to generate new CoTs and answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' config={ "instruction_keys": ["qa-01"],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' # Determines which instructions are used "cot_trigger_keys": ["kojima-01"],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' # Determines which cot triggers are used "answer_extraction_keys": ["kojima-A-D"],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' # Determines which answer extraction # prompts are used "author" : "your_name",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' # Name of the person responsible for generation "api_service": "openai",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' # Name of the API called ("openai",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' "huggingface_hub" # or a mock for testing: "mock_api") "engine": "text-davinci-002",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' # Name of the engine used "temperature": 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' # Level of randomness in the generated output "max_tokens": 512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' # Maximum length of output generated by the model "api_time_interval": 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='0, # Pause between two api calls in seconds "verbose": False, # Determines whether the progress of the generation is printed "warn": True, # Determines whether a warnings that external APIs will be called # are printed } # Generate novel chains-of-thought and answer extractions collection_worldtree_10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='generate(config=config) # Evaluate accuracy of model predictions collection_worldtree_10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content="evaluate() # Example output: {'accuracy': {'qa-01_kojima-01_kojima-A-D': 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='86}} # Save all data (including evaluation data) to JSON file collection_worldtree_10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='dump("worldtree_10_evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='json") Figure 3: Demonstration of the ThoughtSource API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Basic functionalities of the data loader, generator and evaluator modules are demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ThoughtSource: a central hub for large language model reasoning data | 13 { "id": "1242", "ref_id": "", "question": "Which is a characteristic of a sperm cell but not of an egg cell?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ", "type": "multiplechoice", "choices": [ "round shape", "presence of a tail", "contains genetic information", "involved in sexual reproduction" ], "context": "", "cot": [ "A part of something means a characteristic of something.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ", "A tail is not part of an egg cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ", "A tail is a part of a sperm cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ", "A part of something is present in that something.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='" ], "answer": [ "presence of a tail" ], "generated_cot": [ { "id": "738b54ba-9a20-47e6-b8ff-7cb876103b92", "fragments_version": "0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='01", "api_service": "openai", "model": "{\'name\': \'text-davinci-002\', \'temperature\': 0, \'max_tokens\': 512}", "instruction": "qa-01", "cot_trigger": "kojima-01", "cot": "We know that both sperm and egg cells are involved in sexual reproduction, so we can eliminate (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' \\n\\nWe also know that both sperm and egg cells contain genetic information, so we can eliminate (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' \\n\\nThat leaves us with (A) and (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' \\n\\nWe know that sperm cells have a tail, but egg cells do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Therefore, the correct answer is (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ", "answers": [ { "id": "7f7cc26f-a3b3-4b59-9af7-35980514d0c3", "answer_extraction": "kojima-A-D", "answer": " B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ", "correct_answer": true } ], "author": "your_name", "date": "2023/01/12 14:18:57", "comment": "", "annotation": [] } ] } Figure 4: An excerpt of data generated by running the example code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Data for a single question from Worldtree V2 are shown, including human-authored reference CoT, gold-standard answer, an AI-generated CoT and extracted answer, as well as evaluation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Some fields were omitted for legibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ThoughtSource: a central hub for large language model reasoning data | 14 In a zero-shot setup, specific text fragments can be used to prompt question answering and CoT reasoning in LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ThoughtSource includes a curated list of text fragments that can be used to generate novel CoTs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Where possible, we also mapped individual CoTs in pre-existing CoT datasets to the text fragments that were used in their creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' "instructions": { "qa-01": "Answer the following question through step-by-step reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ", "qa-02": "Answer the following question through careful, concise step-by-step reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ", "qa-03": "Answer the following question through careful, concise step-by-step reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Avoid making up wrong statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' If the question does not make sense or cannot be answered, write \\"I cannot answer the question\\".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' If you do not have a good answer, write \\"I do not have a good answer\\".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' If you are uncertain, write \\"I am uncertain about this\\".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ", [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='] }, "cot_triggers": { "kojima-01": "Answer: Let\'s think step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ", "kojima-02": "Answer: We should think about this step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ", "kojima-03": "Answer: First,", "kojima-04": "Answer: Before we dive into the answer,", [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='] "lievin-01": "Answer: Let\'s derive the differential diagnosis step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ", "lievin-02": "Answer: Let\'s use step by step inductive reasoning, given the medical nature of the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ", [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='] "lievin-26": "Answer: Let\'s follow a Bayesian step by step approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ", "lievin-27": "Answer: Let\'s reflect on each option from the least likely to the most likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ", "lievin-28": "Answer: Let\'s use step by step Bayesian reasoning, given the medical nature of the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='" }, "answer_extractions":{ "kojima-01": "Therefore, the answer is", "kojima-02": "Therefore,", "kojima-03": "The answer is", "kojima-numerals": "Therefore, the answer (arabic numerals) is", "kojima-yes-no": "Therefore, the answer (Yes or No) is", "kojima-A-C": "Therefore, among A through C, the answer is", "kojima-A-D": "Therefore, among A through D, the answer is", [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='] } Figure 5: An excerpt of the collection of prompt fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' These fragments can be used to build prompts for interacting with LLMs, allowing for empirical testing of how different prompts affect model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We provide two web-based interfaces for exploring and annotating ThoughtSource data, the Dataset Viewer and the Annotator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The Dataset Viewer is a simple interface for exploring dataset contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The Annotator (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' 6) allows you to upload specific subsets of a dataset, provides convenience functions for highlighting similarities between different generated CoTs and the correctness of generated answers, and allows you to annotate individual CoTs interactively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The annotator facilitates identifying strengths and weaknesses of different CoTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Annotations can be used for downstream model evaluation and further improving the capabilities of AI models through fine-tuning / reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' ThoughtSource: a central hub for large language model reasoning data | 15 Figure 6: The ThoughtSource Annotator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The web-based interface allows for convenient inspection and annotation of reasoning chains and answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Text that is similar between CoTs can be automatically highlighted based on an easily adjustable similarity threshold, facilitating a better understanding of similarities and differences of different reasoning chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' All tools and libraries, as well as more detailed demonstration notebooks, can be found on the project Github page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' We plan to add more datasets and generated CoTs to the ThoughtSource repository, and we welcome outside contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Novel CoTs for existing core datasets can be generated and shared through the ThoughtSource APIs and JSON files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=" Completely new datasets can also be added, as described in the Github repository's contribution guide." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Code Availability All code, data and tools are openly available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='com/OpenBioLink/ThoughtSource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Our code and data are licensed under an MIT license, while data adapted from existing datasets are available under the licenses of their respective sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=" ThoughtSource Annotator tfidf Visualisation threshold Download current Matthias Logout Question 1242 (test) A 28-year-old woman, gravida 1, para O, at 20 weeks' gestation comes to the physician with her husband for a prenatal visit." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Her pregnancy has been uncomplicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=" They are planning to travel to Ethiopia next month to visit the husband's family uterus that is consistent with a 20-week gestation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Which of the following drugs is most suitable for pre-exposure prophylaxis against malaria?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Doxycycline B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Primaquine C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Mefloquine D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Chloroquine E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=" Proguanil Let's think step by step." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=" The first thing we need to Let's think step by step like a medical expert." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=" The Let's differentiate using step by step reasoning like do is figure out what kind of malaria is common in first thing to consider is the safety of the a medical expert: 1) Doxycycline is not suitable for Ethiopia." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' According to this website, the most medication for the mother and the developing pre-exposure prophylaxis against malaria in this common type of malaria in Ethiopia is Plasmodium fetus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Doxycycline and mefloquine are not case because the patient is pregnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' 2) falciparum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The next thing we need to do is figure recommended for use in pregnant women, so we Mefloquine is not suitable for pre-exposure out which of the four drugs listed is most effective can eliminate those options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The next thing to prophylaxis against malaria in this case because against Plasmodium falciparum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' According to this consider is the efficacy of the medication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' the patient is pregnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' 3) Chloroquine is suitable website, chloroguine and proguanil are not Chloroquine is not effective in Ethiopia, so we can for pre-exposure prophylaxis against malaria in effective against Plasmodium falciparum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' That eliminate that option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' That leaves us with this case because the patient is not pregnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' 4) leaves doxycycline and mefloquine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The next thing proguanil, which is safe for use in pregnant Proguanil is suitable for pre-exposure prophylaxis we need to do is figure out which of the two women and is effective in Ethiopia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' against malaria in this case because the patient is remaining drugs has the fewest side effects Answer: Proguanil not pregnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' According to this website, the most common side ☆ Answer: Chloroquine effects of doxycycline are nausea, vomiting, and ☆ Incorrect reasoning diarrhea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' The most common side effects of Insufficient knowledge Incorrect reasoning mefloquine are nausea, vomiting, diarrhea, and headache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Based on the information above, the Incorrect reading comprehension Insufficient knowledge most suitable drug for pre-exposure prophylaxis Incorrect reading comprehension Too verbose against malaria in Ethiopia is mefloquine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Too verbose O Answer: Mefloquine Incorrect reasoning Insufficient knowledge Incorrect reading comprehension Too verboseThoughtSource: a central hub for large language model reasoning data | 16 Acknowledgements We thank primary dataset contributors that assisted with assembling the ThoughtSource meta-dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Author contributions S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' wrote the code for accessing, converting, generating and analysing datasets, and wrote parts of the manuscript and documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' generated and analysed CoT data for medical datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' wrote the code of the annotator so�ware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='Mo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' wrote a first prototype of code for accessing and converting datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' contributed to improving code and documentation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' conceived and supervised the project and wrote parts of the manuscript and documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' All authors have read and approved the final manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Competing interests The authors declare that there are no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Sanh, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content=' Multitask Prompted Training Enables Zero-Shot 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} +page_content='168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFJT4oBgHgl3EQfoixI/content/2301.11596v1.pdf'} diff --git a/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf b/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a1936ebbdaaeb36ef7cc39a14f11c7cf2bbd25ff --- /dev/null +++ b/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:28d5767900f39b9fb92a00b68b189f049b60f11b8b1ed6ca685ebb9521a7bb52 +size 851377 diff --git a/A9AyT4oBgHgl3EQfd_iI/vector_store/index.pkl b/A9AyT4oBgHgl3EQfd_iI/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..cb67f48afbe68b2ee6b89fee8e1b95cc84f702b9 --- /dev/null +++ b/A9AyT4oBgHgl3EQfd_iI/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f8c600f18fde33c8eeb696f7d32bf1c9179ccf368a7237e707bcb6f8ece52de4 +size 106867 diff --git a/ANAzT4oBgHgl3EQfF_sv/content/tmp_files/2301.01019v1.pdf.txt b/ANAzT4oBgHgl3EQfF_sv/content/tmp_files/2301.01019v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7425ae650be5fba124cb68fb8cc0970a33f833a8 --- /dev/null +++ b/ANAzT4oBgHgl3EQfF_sv/content/tmp_files/2301.01019v1.pdf.txt @@ -0,0 +1,1634 @@ +Correlation Loss: Enforcing Correlation between Classification and Localization +Fehmi Kahraman*,1, Kemal Oksuz*,1, Sinan Kalkan†,1,2, Emre Akbas†,1,2 +1Dept. of Computer Engineering. 2METU Center for Robotics and Artificial Intelligence (ROMER) +Middle East Technical University (METU), Ankara, Turkey +{fehmi.kahraman 01, kemal.oksuz, skalkan, eakbas}@metu.edu.tr +Abstract +Object detectors are conventionally trained by a weighted +sum of classification and localization losses. Recent studies +(e.g., predicting IoU with an auxiliary head, Generalized Fo- +cal Loss, Rank & Sort Loss) have shown that forcing these +two loss terms to interact with each other in non-conventional +ways creates a useful inductive bias and improves perfor- +mance. Inspired by these works, we focus on the correlation +between classification and localization and make two main +contributions: (i) We provide an analysis about the effects +of correlation between classification and localization tasks +in object detectors. We identify why correlation affects the +performance of various NMS-based and NMS-free detectors, +and we devise measures to evaluate the effect of correla- +tion and use them to analyze common detectors. (ii) Moti- +vated by our observations, e.g., that NMS-free detectors can +also benefit from correlation, we propose Correlation Loss, +a novel plug-in loss function that improves the performance +of various object detectors by directly optimizing correla- +tion coefficients: E.g., Correlation Loss on Sparse R-CNN, +an NMS-free method, yields 1.6 AP gain on COCO and 1.8 +AP gain on Cityscapes dataset. Our best model on Sparse +R-CNN reaches 51.0 AP without test-time augmentation on +COCO test-dev, reaching state-of-the-art. Code is available +at: https://github.com/fehmikahraman/CorrLoss. +1 +Introduction +Most object detectors optimize a weighted sum of classifi- +cation and localization losses during training. Results from +recent work suggest that performance improves when these +two loss functions are forced to interact with each other in +non-conventional ways as illustrated in Fig. 1. For example, +training an auxiliary (aux.) head to regress the localization +qualities of the positive examples, e.g. centerness, IoU or +mask-IoU, has proven useful (Jiang et al. 2018; Kim and Lee +2020; Tian et al. 2019; Zhang et al. 2020) (Fig. 1(b)). Other +methods remove such auxiliary heads and aim directly to +enforce correlation1 in the classification or localization task +during training; e.g., Average LRP Loss (Oksuz et al. 2020) +*These authors contributed equally. +†Equal contribution for senior authorship. +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +1In the rest of the paper, “correlation” will refer to the correla- +tion between classification scores and IoUs. +weighs the examples in the localization task by ranking them +with respect to (wrt.) their classification scores (Fig. 1(c)). +Using localization quality as an additional supervision sig- +nal for classification has been more commonly adopted (Fig. +1(d)) (Li et al. 2020; Liu et al. 2021; Oksuz et al. 2021a; +Zhang et al. 2021) in two main ways: (i) Score-based ap- +proaches aim to regress the localization qualities (Li et al. +2019, 2020; Zhang et al. 2021) in the classification score, +and (ii) ranking-based approaches enforce the classifier to +rank the confidence scores wrt. the localization qualities (Liu +et al. 2021; Oksuz et al. 2021a). +Improving correlation seems to have a positive effect on +performance of a variety of object detectors, as shown in +Fig. 2. However, the effect of correlation on object detectors +has not been thoroughly studied. We fill this gap in this pa- +per and first identify that correlation affects the performance +of object detectors at two levels: (i) Image-level correlation, +the correlation between the classification scores and local- +ization qualities (i.e., IoU for the rest of the paper) of the de- +tections in a single image before post-processing, which is +important to promote NMS performance, and (ii) Class-level +correlation, the correlation over the entire dataset for each +class after post-processing, which is related to the COCO- +style Average Precision (AP). Moreover, we quantitatively +define correlation at each level to enable analyses on how +well an object detector captures correlation (e.g., βcls in +Fig. 2(a)). Then, we provide an analysis on both levels of +correlation and draw important observations using common +models. Finally, to better exploit correlation, we introduce a +more direct mechanism to enforce correlation: Correlation +Loss, a simple plug-in and detector-independent loss term +(Fig. 1(e)), improving performance for a wide range of ob- +ject detectors including NMS-free detectors, aligning with +our analysis (Fig. 2(b)). Similar to the novel loss functions +(Li et al. 2020; Oksuz et al. 2021a; Zhang et al. 2021), our +Correlation Loss boosts the performance without an aux- +iliary head, but different from them, it is a simple plug-in +technique that can easily be incorporated into any object de- +tector, whether NMS-based or NMS-free. +Our main contributions are: (1) We identify how corre- +lation affects NMS-based and NMS-free detectors, and de- +sign quantitative measures to analyze a detector wrt. corre- +lation. (2) We analyze the effects of correlation at different +levels on various object detectors. (3) We propose Correla- +arXiv:2301.01019v1 [cs.CV] 3 Jan 2023 + +̂𝑠 ← ̂𝑠×%ℓ +Cls +Loc +Aux +ℒ!"# +ℒ"$! +ℒ%&' +(b) Auxiliary Head +(d) Novel Cls. Loss +(a) No correlation +Cls +Loc +ℒ!"# +ℒ"$! +̂𝑠 +%𝐵 +̂𝑠 +%𝐵 +%ℓ +Cls +Loc +ℒ!"# +ℒ"$! +̂𝑠 +%𝐵 +(c) Novel Loc. Loss +Cls +Loc +ℒ!"# +ℒ"$! +̂𝑠 +%𝐵 +Cls +Loc +ℒ!"# +ℒ"$! +̂𝑠 +%𝐵 +ℒ!$(( +(e) Correlation Loss (Ours) +Legend +̂𝑠 : Classification Scores +)𝐵 : Box Coordinates +)ℓ : Localization Quality +(e.g. centerness) +ℒ : A Loss Function +Figure 1: Different ways of handling the classification and localization tasks from the perspective of correlation. (a) Conven- +tional case of optimizing the two tasks independently (e.g., Chen et al. 2020; Sun et al. 2021b). (b) An additional auxiliary head +predicts centerness (Zhang et al. 2020) or IoU (Jiang et al. 2018; Kim and Lee 2020), which introduces additional learnable +parameters. (c) Novel loss functions replace the standard localization loss (Oksuz et al. 2020) or (d) novel classification loss +(Li et al. 2020; Oksuz et al. 2021a) by more complicated ones to leverage correlation. (e) Our Correlation Loss explicitly opti- +mizes a correlation coefficient. It is a simple, plug-in loss function which does not introduce additional parameters and has the +flexibility to supervise classification or localisation head as well as both. Black and colored arrows respectively denote the loss +functions (i.e., during training) & the network outputs (i.e., during inference). +38.5 +39.0 +39.5 +40.0 APC +40 +41 +42 +43 +44 +45 +46 +47 +cls +FL +Aux. +QFL +RS +FL +Aux. +QFL +RS +w/o Corr.Loss +w. Corr.Loss (Ours) +(a) Detection vs. Correlation +ATSS +Sparse RCNN +YOLACT +25 +30 +35 +40 +45 +50 +APC ++1.1 AP ++2.8% gain ++1.6 AP ++4.2% gain ++0.7 AP ++2.4% gain +Baseline +Ours +(b) Effect of our Corr. Loss +Figure 2: (a) Detection performance, measured by COCO- +style AP (APC) vs. correlation quality, measured by class- +level correlation (βcls - see Section 3.1 for details). The +methods proposed to improve the correlation between classi- +fication and localization tasks also improve APC. Compare +using aux. head, QFL, RS Loss with the baseline ATSS only +using Focal Loss (FL – all in red dots) to see the positive cor- +relation between APC and βcls. Our Correlation Loss as a +plug-in loss function explicitly optimizes a correlation coef- +ficient and improves the detection performance (APC) over +different settings of ATSS (i.e. using FL, aux. head, QFL, RS +Loss) consistently owing to increasing βcls, validating our +hypothesis (compare green stars with red dots). (b) Our Cor- +relation Loss is simple-to-use and improves various meth- +ods (i) NMS-based ATSS (w/o aux. head) by 1.1APC, (ii) +NMS-free Sparse R-CNN by 1.6APC and (iii) YOLACT, +an instance segmentation method by 0.7APC. +tion Loss as a plug-in loss function to optimize correlation +explicitly. Thanks to its simplicity, our loss function can be +easily incorporated into a diverse set of object detectors and +improves the performance of e.g., Sparse R-CNN up to 1.6 +AP and 2.0AP75, suggesting, for the first time, that NMS- +free detectors can also benefit from correlation. Our best +model yields 51.0 AP, reaching state-of-the art. +2 +Background and Related Work +Object Detection Pipeline. We group object detectors wrt. +their usage of NMS (Fig. 3 presents overview & notation): +1. NMS-based Detectors. To detect all objects with dif- +ferent scales, locations and aspect ratios; most methods +(He et al. 2017; Kong et al. 2020; Law and Deng 2018; +Lin et al. 2020; Ren et al. 2017; Tian et al. 2019; Zhang +et al. 2020) employ a large number of object hypotheses +(e.g., anchors, points), which are labeled as positive (a.k.a. +foreground) or negative (a.k.a. background) during training, +based on whether/how they match GT boxes (Zhang et al. +2020, 2019). In this setting, there is no restriction for an ob- +ject to be predicted by multiple object hypotheses, causing +duplicates. Accordingly, during inference, NMS picks the +detection with the largest confidence score among the detec- +tions that overlap more than a predetermined IoU threshold +to avoid duplicate detections. +2. NMS-free Detectors. An emerging research direction is +to remove the need for doing NMS, simplifying the detec- +tion pipeline (Carion et al. 2020; Dai et al. 2021; Roh et al. +2022; Sun et al. 2021b,a; Zhu et al. 2021). This is achieved +by ensuring a one-to-one matching between the GTs and de- +tections, which supervises the detector to avoid duplicates in +the first place. +Methods Enforcing Correlation. One common way to +ensure correlation is to use an additional auxiliary head, su- +pervised by the localization quality of a detection such as +centerness (Tian et al. 2019; Zhang et al. 2020), IoU (Jiang +et al. 2018), mask IoU (Huang et al. 2019) or uncertainty +(He et al. 2019), during training. During inference, the pre- +dictions of the auxiliary head are then combined with those +of the classifier to improve detection performance. Recent +methods show that the auxiliary head can be removed, and +either (i) the regressor can prioritize the positive examples +(Oksuz et al. 2020) or (ii) the classifier can be supervised to +prioritize detections with confidence scores. The latter is en- +sured either by regressing the IoUs by the classifier (Li et al. +2020; Zhang et al. 2021) or by training the classifier to rank +confidence scores (Liu et al. 2021; Oksuz et al. 2021a) wrt. +IoUs. Unlike these methods, TOOD (Feng et al. 2021) takes +correlation into account mainly while designing the model, +particularly the detection head, i.e., not the loss function. +Correlation Coefficients. Correlation coefficients mea- +sure the strength and direction of the “relation” between two + +Ƹ𝑠𝑝𝑟𝑒 +𝐼 +(score) +෠𝐵𝑝𝑟𝑒 +𝐼 +(box) +Image, I +Post-processing +Object +Detector +Remove +background +NMS +Top-k +For each class c +in image I +For each image I +Collect +Ƹ𝑠𝑝𝑜𝑠𝑡 +𝑐 +& ෠𝐵𝑝𝑜𝑠𝑡 +𝑐 +over all +images +Ƹ𝑠𝑝𝑜𝑠𝑡 +𝐼,𝑐 (score) +෠𝐵𝑝𝑜𝑠𝑡 +𝐼,𝑐 +(box) +Figure 3: Object detection pipeline and notation. Given an input image, I, NMS-based detectors yield raw detections before +post-processing, each of which has a predicted bounding box (BB) and an array of confidence scores over GT classes. We +denote the confidence scores and the predicted BBs pertaining to the positive detections, i.e., the detections matching with GT +objects during training, by ˆsI +pre and ˆBI +pre, respectively. To obtain final detections, raw detections are post-processed in three +steps: (i) Detections with low confidence scores, i.e., background, are removed, (ii) duplicates are eliminated by NMS, and +(iii) top-k scoring detections are kept. As for these final detections, we denote the confidence scores and BBs of true positive +detections for class c in a single image I by ˆsI,c +post and ˆBI,c +post respectively, and over the entire dataset by ˆsc +post and ˆBc +post. As for +NMS-free detectors; NMS, dashed gray box in post-processing, is excluded, hence post-processing is lighter. +sets, X = {x1, ..., xN} and Y = {y1, ..., yN}. Different re- +lations are evaluated by different correlation coefficients: (i) +Pearson correlation coefficient, denoted by α(·, ·), measures +the linear relationship between the sets, (ii) Spearman corre- +lation coefficient, β(·, ·), corresponds to the ranking relation- +ship and (iii) Concordance correlation coefficient, γ(·, ·), is +more strict, measuring the similarity of the values and max- +imized when xi = yi for all i ∈ 1, ..., N. All correlation +coefficients have a range of [−1, +1] where positive/neg- +ative correlation corresponds to increasing/decreasing rela- +tion, while 0 implies no correlation between X and Y . +Comparative Summary. In this paper, we comprehen- +sively identify and analyze the effect of explicitly correlat- +ing classification and localization in object detectors. Unlike +other methods that also enforce correlation, some of which +are tested only on a single architecture (Huang et al. 2019; +Jiang et al. 2018; Tian et al. 2019), we propose a simple solu- +tion by directly optimizing the correlation coefficient, which +is auxiliary-head free and easily applicable to all object de- +tectors, whether NMS-based or NMS-free. Also, ours is the +first to work on NMS-free detectors in this context. +3 +Effects of Correlation on Object Detectors +This section presents why maximizing correlation is impor- +tant for object detectors, introduces measures to evaluate ob- +ject detectors wrt. correlation and provides an analysis on +methods designed for improving correlation. +3.1 +How Correlation Affects Object Detectors +Detectors are affected by correlation at two levels (Fig. 4): +Image-level Correlation. This level of correlation corre- +sponds to the correlation between the classification scores +and IoUs of the detections in a single image before post- +processing, and accordingly, we measure it with the Spear- +man correlation coefficient2, β(·, ·), averaged over images. +2While analyzing object detectors in terms of correlation, we +employ Spearman correlation coefficient, β(·, ·), to measure the +relation between the ranks of the values (i.e., scores and IoUs) in- +stead of the values themselves, and aim to isolate the correlation +quality from the localization and classification performances. +Denoting the set of images to be evaluated by I and IoUs +between the BBs of the positive detections ( ˆBI +pre, Fig. 3) +and their associated GTs by IoUI +pre, image-level correlation +is measured as follows: +βimg = 1 +|I| +� +I∈I +β(IoUI +pre, ˆsI +pre). +(1) +Maximizing image-level correlation is important for +NMS-based detectors since NMS aims to suppress dupli- +cates, i.e., to keep only a single detection for each GT when +there is more than one. More particularly among overlap- +ping detections (e.g., dark and light green detections in the +detector output image in Fig. 4(a)), NMS picks the one with +the larger score, and hence, if there is positive correlation +between the confidence scores and IoUs of those overlap- +ping detections, then the one with the best IoU (e.g., dark +green detection in Fig. 4(a)) will survive and detection per- +formance will increase. +Class-level Correlation. This level of correlation indi- +cates the correlation between the classification scores and +IoUs of the detections obtained after post-processing for +each class. Since class-level correlation is related to COCO- +style AP, APC, we average β(·, ·) over classes to be consis- +tent with the computation of APC: +βcls = 1 +|C| +� +c∈C +β(IoUc +post, ˆsc +post), +(2) +where C is the set of classes in the dataset and IoUc +post is the +set IoUs of BBs of true positives for class c ( ˆBc +post, Fig. 3). +Class-level correlation affects the performance of all de- +tectors since it is directly related to APC, the performance +measure itself. To be more specific, APC for a single class +is defined as the average of APs computed over 10 differ- +ent IoU thresholds, IoU ∈ {0.50, 0.55, ..., 0.95}, validating +the true positives. For a specific threshold IoU, the detec- +tions are first sorted with respect to the classification scores, +and then precision and recall pairs are calculated on each +detection. Using these pairs, a precision-recall (PR) curve is +obtained, and the area under the PR curve corresponds to + +Positively correlated +NMS +Solid BBs +: Ground truths +Dashed BBs +: Detections before +post-processing ( Ƹ𝑠𝑝𝑟𝑒 +𝐼 +, ෠𝐵𝑝𝑟𝑒 +𝐼 +in Fig. 3) +Positively correlated - High APC +(b) Class-level Correlation for better AP +AP Calculation +𝐼𝑜𝑈 +0.80 +N/A +0.60 +N/A +0.50 +Ƹ𝑠 +0.80 +0.70 +0.60 +0.55 +0.50 +𝐼𝑜𝑈 +0.80 +N/A +0.60 +N/A +0.50 +Ƹ𝑠 +0.50 +0.55 +0.60 +0.70 +0.80 +Negatively correlated - Low APC +Precision +Pos. Detections +APIoU +P50 +1.00 +0.67 +0.60 +0.76 +P75 +1.00 +0.00 +0.00 +0.33 +Precision +Pos. Detections +APIoU +P50 +1.00 +0.67 +0.60 +0.76 +P75 +0.20 +0.00 +0.00 +0.07 +Negatively correlated +Solid BBs +: Ground truths +Dashed BBs +: Detections after +post-processing ( Ƹ𝑠𝑝𝑜𝑠𝑡 +𝐼,𝑐 , ෠𝐵𝑝𝑜𝑠𝑡 +𝐼,𝑐 in Fig. 3) +(a) Image-level Correlation for better NMS +High +IoU +Low +IoU +Figure 4: How correlation affects detection performance. (a) Image-level correlation: Given detections before post-processing, +NMS benefits from image-level correlation, thereby yielding detections with better IoU. Compare IoUs of detections in “posi- +tively correlated” (i.e., when the dark-colored ones have larger score) and “negatively correlated” (i.e., when the light-colored +ones have larger score) outputs after NMS. (b) Class-level correlation: Given detections after post-processing, APs with larger +IoUs and COCO-style AP benefit from positive class-level correlation (compare APIoU columns in “positively correlated” and +“negatively correlated” outputs after AP Calculation to see lower AP75 for the “negatively correlated” output in the red cell). +PIoU: Precision computed on a detection using the threshold IoU, True positives are color-coded in tables and input, white +cells: false positives, and hence their IoU is not available, N/A. +the single AP value, APIoU. When the correlation between +classification and localization is maximized among true pos- +itives, larger precision values are obtained on the same de- +tections in larger IoU values (e.g. P75 of orange detection is +1.00 and 0.20 with positive and negative correlation respec- +tively in Fig. 4(b)). +3.2 +Analyses of Object Detectors wrt. Correlation +Dataset and Implementation Details. Unless otherwise +specified; we (i) employ the widely-used COCO dataset (Lin +et al. 2014) by training the models on trainval35K (115K +images), testing on minival (5k images), comparing with +SOTA on test-dev (20k images), (ii) build upon the mmde- +tection framework (Chen et al. 2019), (iii) rely on AP-based +measures and also use Optimal LRP (oLRP) (Oksuz et al. +2021b), βimg (Eq. 1) and βcls (Eq. 2) to provide more in- +sights, (iv) keep the standard configuration of the models, +(v) use a ResNet-50 backbone with FPN (Lin et al. 2017), +(vi) train models on 4 GPUs (A100 or V100 type GPUs) +with 4 images on each GPU (16 batch size). +Analysis Setup. We conduct experiments to analyze the +effects of the image-level (βimg – Table 1) and class-level +(βcls – Table 2) correlations. For both analyses, we com- +pare three sets of methods, all of which are incorporated +into the common ATSS baseline (Zhang et al. 2020) (see +Sec. 2 for a discussion of these methods): (i) AP Loss and +Focal Loss as methods not enforcing correlation, (ii) using +an auxiliary head to enforce correlation, and (iii) Quality Fo- +cal Loss (QFL), aLRP Loss and Rank & Sort Loss as recent +loss functions enforcing correlation. In our class-level anal- +ysis, we also employ NMS-free methods to demonstrate the +effects of correlation on that approach. +We compare the methods based on (i) their AP-based per- +formance, (ii) our proposed measures for correlation (Eqs. +1 and 2), and finally (iii) lower/upper bounds, AP+1 +C /AP−1 +C , +obtained by modifying the ranking of the confidence scores +pertaining to the GT classes of the positive detections to +minimize/maximize Eq. 1 in Table 1 and Eq. 2 in Table 2. +More particularly, in Table 1, given ˆsI +pre and ˆBI +pre (Fig. 3), +we collect the GT class probabilities of positive detections +and change their ranking in ˆsI +pre within an image follow- +ing the ranking order of IoUs (computed using ˆBI +pre), and in +Table 2, we do the same operation class-wise for true posi- +tives given ˆsc +post and ˆBc +post (Fig. 3). To decouple other types +of errors as much as possible; in Table 1, we do not modify +the scores of the negative detections, the predicted BBs and +the scores of non-GT classes of the positive detections, and +in Table 2, we do not modify the scores of the false positives +and the predicted BBs of the true positives. Note that achiev- +ing the upper bound in (iii) for image-level correlation also +corresponds to perfectly minimizing RS Loss. +Observations. We observe in Tables 1 and 2 that: +(1) Our proposed measures in Eqs. 1 and 2 can measure +the improvements in correlation consistently. In Tables 1 and +2, (i) aLRP Loss and RS Loss are proposed to improve AP +Loss and (ii) aux. head and QFL are proposed to improve Fo- +cal Loss. In both tables, the proposed methods are shown to +improve their baselines in terms of βimg and βcls, suggest- +ing that our measures can consistently evaluate image-level +and class-level correlations respectively. +(2) NMS-free detectors can also potentially benefit from +correlation. All detectors, including NMS-free ones, can ex- +ploit class-level correlation (compare APC and AP+1 +C to see +∼ 10 points gap in Table 2). Still, existing methods do not +enforce this correlation on NMS-free detectors. +(3) Existing methods enforcing correlation have still a +large room for improvement. Considering that βimg +∈ +[27.2%, 33.8%] (Table 1) and βcls ∈ [37.5%, 47.0%] (Table +2), there is still room for improvement wrt. correlation. + +Performance +Modify ranking of scores +Method +APC AP50 AP75 βimg +AP−1 +C +AP−1 +50 +AP−1 +75 +AP+1 +C +AP+1 +50 +AP+1 +75 +Not Enforcing Correlation +ATSS w. AP Loss (Chen et al. 2020) +38.1 +58.2 +41.0 +27.2 +24.9 +53.2 +19.2 +57.0 +72.4 +62.2 +ATSS w. Focal Loss (Lin et al. 2020) +38.7 +57.6 +41.5 +27.3 +25.6 +51.8 +21.1 +55.8 +70.6 +60.5 +Using Aux. Head +ATSS w. ctr. head (Zhang et al. 2020) +39.3 +57.5 +42.6 +28.7 +16.8 +32.4 +15.3 +49.8 +64.8 +54.2 +Using Novel Loss +ATSS w. aLRP Loss (Oksuz et al. 2020) +37.7 +57.4 +39.9 +33.8 +22.7 +48.8 +17.5 +54.2 +70.4 +58.7 +ATSS w. QFL (Li et al. 2020) +39.7 +58.1 +42.7 +33.2 +25.7 +51.1 +21.9 +55.8 +70.9 +60.6 +ATSS w. RS Loss (Oksuz et al. 2021a) +39.9 +58.9 +42.6 +30.9 +26.2 +53.9 +21.3 +57.1 +71.8 +62.1 +Table 1: Evaluation of NMS-based detectors in terms of image-level correlation. See Eq. 1 for βimg. AP+1 +IoU and AP−1 +IoU refer to +the upper & lower bound APs (see analysis setup for details). The values are in %. Our βimg captures correlation consistently, +e.g. that (i) Focal Loss is improved by ctr. head and QFL and (ii) AP Loss is improved by aLRP Loss and RS Loss wrt. βimg. +Also, there is still room for improvement for object detectors wrt. βimg with a range between 27.2% and 33.8%. +Performance +Modify ranking of scores +Method +APC AP50 AP75 +βcls +AP−1 +C +AP−1 +50 +AP−1 +75 +AP+1 +C +AP+1 +50 +AP+1 +75 +Not Enforcing Correlation +- NMS-free Detectors +Sparse R-CNN (Sun et al. 2021b) +37.7 +55.8 +40.5 +37.5 +30.1 +55.8 +28.9 +48.6 +55.8 +52.7 +DETR (Carion et al. 2020) +40.1 +60.6 +42.0 +47.0 +32.9 +60.6 +30.6 +51.9 +60.6 +55.8 +- NMS-based Detectors +ATSS w. AP Loss (Chen et al. 2020) +38.1 +58.2 +41.0 +39.4 +30.0 +58.2 +26.6 +48.5 +58.2 +54.0 +ATSS w. Focal Loss (Lin et al. 2020) +38.7 +57.6 +41.5 +40.3 +30.2 +57.6 +27.3 +48.7 +57.6 +53.6 +Using Aux. Head +ATSS w. ctr. head (Zhang et al. 2020) +39.3 +57.4 +42.5 +42.5 +30.2 +57.4 +27.6 +48.7 +57.4 +53.5 +Using Novel Loss +ATSS w. aLRP Loss (Oksuz et al. 2020) +37.7 +57.4 +39.9 +42.0 +29.1 +57.4 +25.0 +47.8 +57.4 +52.7 +ATSS w. QFL (Li et al. 2020) +39.7 +58.1 +42.7 +45.7 +30.6 +58.1 +27.7 +49.1 +58.1 +53.9 +ATSS w. RS Loss (Oksuz et al. 2021a) +39.9 +58.9 +42.6 +43.2 +31.1 +58.9 +28.1 +49.8 +58.9 +54.8 +Table 2: Evaluation of detectors wrt. class-level correlation. See Eq. 2 for βcls. AP+1 +IoU & AP−1 +IoU denote upper & lower bound +APs (analysis setup for details). Values are in %. NMS-free detectors can also benefit from class-level correlation (compare +AP+1 +C with APC for Sparse R-CNN), and as in βimg (c.f. Table 1 and its caption), βcls measures the correlation consistently. +AP+1 +50 = AP−1 +50 = AP50 since only modifying TPs validated from IoU=0.50 does not effect AP50 (see Fig. 4(b) for an example). +(4) While significantly important, improving correlation +may not always imply performance improvement. For exam- +ple, aLRP Loss in Table 1 has the largest correlation but the +lowest APC. Such a situation can arise, for example, when a +method does not have good localization performance. In the +extreme case, assume a detector yields perfect βimg, image- +level ranking correlation, but the IoUs of all positive exam- +ples are less than 0.50 implying no TP at all. Hence, boost- +ing the correlation, while simultaneously preserving a good +performance in each branch, is critical. +4 +Correlation Loss: A Novel Loss Function +for Object Detection +Correlation (Corr.) Loss is a simple plug-in loss function to +improve correlation of classification and localization tasks. +Correlation Loss is unique in that it can be easily incorpo- +rated into any object detector, whether NMS-based or NMS- +free (see Observation (2) - Sec. 3.2), and improves perfor- +mance without affecting the model size, inference time and +with negligible effect on training time (Sec. 5.4). Further- +more, from a fundamental perspective, Corr. Loss can su- +pervise both of the classification and localisation heads for a +better correlation while existing methods generally focus on +a single head such as classification (Fig. 1). +Definition. Given an object detector with loss function +LOD, our Correlation Loss (Lcorr) is simply added using a +weighting hyper-parameter λcorr: +LOD + λcorrLcorr. +(3) +Lcorr is the Correlation Loss defined as: +Lcorr = 1 − ρ( ˆ +IoU,ˆs), +(4) +where ρ(·, ·) is a correlation coefficient; ˆs and +ˆ +IoU are the +confidence scores of the GT class and IoUs of the predicted +BBs pertaining to the positive examples in the batch. +Practical Usage. To avoid promoting trivial cases with +high correlation but low performance (Observation (4) - +Sec. 3.2), similar to QFL (Li et al. 2020) and RS Loss +(Oksuz et al. 2021a), we only use the gradients of Lcorr +wrt. classification score, i.e., we backpropagate the gradi- +ents through only the classifier. We mainly adopt two dif- +ferent correlation coefficients for ρ(·, ·) and obtain two ver- +sions of Correlation Loss: (i) Concordance Loss, defined as +the Correlation Loss when Concordance correlation coeffi- +cient is optimized (ρ(·, ·) = γ(·, ·)), which aims to match + +Method +APC ↑AP50 ↑AP75 ↑ oLRP ↓ +NMS-based +Retina Net (Lin et al. 2020) +36.5 +55.4 +39.1 +70.7 +w. Conc.Corr (Ours) +37.0 +55.7 +39.7 +70.2 +w. Spear.Corr (Ours) +37.5 +55.4 +40.5 +69.7 +Fovea Box (Kong et al. 2020) +36.4 +56.5 +38.6 +70.2 +w. Conc.Corr (Ours) +37.1 +56.4 +39.6 +69.7 +w. Spear.Corr (Ours) +37.0 +55.6 +39.3 +70.0 +ATSS (Zhang et al. 2020) +38.7 +57.6 +41.5 +69.0 +w. Conc.Corr (Ours) +39.8 +57.9 +43.2 +68.2 +w. Spear.Corr (Ours) +39.3 +56.6 +42.5 +68.7 +PAA (Kim and Lee 2020) +39.9 +57.3 +43.4 +68.6 +w. Conc.Corr (Ours) +40.7 +58.8 +44.3 +67.7 +w. Spear.Corr (Ours) +40.4 +58.0 +43.7 +67.8 +NMS-free +Sparse R-CNN (Sun et al. 2021b) 37.7 +55.8 +40.5 +69.5 +w. Conc.Corr (Ours) +38.9 +57.2 +41.8 +68.1 +w. Spear.Corr (Ours) +39.3 +56.7 +42.5 +68.3 +Table 3: Comparison on detectors not considering correla- +tion. Accordingly, we remove aux. heads from ATSS (Zhang +et al. 2020) and PAA (Kim and Lee 2020) for fair compari- +son (see Table 6 for comparison with aux. heads and novel +loss functions). We use ResNet-50 and train the models for +12 epochs. Simply incorporating our Corr. Loss provides +(i) ∼ 1APC improvement for NMS-based detectors consis- +tently and (ii) ∼ 1.5APC on the NMS-free Sparse R-CNN. +the confidence scores with IoUs. (ii) Spearman Loss as Cor- +relation Loss when Spearman correlation coefficient is op- +timized (ρ(·, ·) = β(·, ·)), thereby enforcing the ranking +of the classification scores considering IoUs. To tackle the +non-differentiability of ranking operation while computing +Spearman Loss, we leverage the differentiable sorting oper- +ation from Blondel et al. (Blondel et al. 2020). When apply- +ing our Correlation Loss to NMS-free methods, which use +an iterative multi-stage loss function, we incorporate Lcorr +to every stage. +5 +Experimental Evaluation +We evaluate Corr. Loss on (i) the COCO dataset with five +different object detectors of various types (Sparse R-CNN as +NMS-free, FoveaBox as anchor-free, RetinaNet as anchor- +based, ATSS and PAA using auxiliary head), and one in- +stance segmentation method, YOLACT; and (ii) an addi- +tional dataset (Cityscapes) for the method with the largest +gain, i.e., Sparse R-CNN. +5.1 +Comparison with Methods Not Considering +Correlation +We train these five object detectors and the instance segmen- +tation method (Tables 3 and 5) with and without our Corr. +Loss (as Concordance Loss or Spearman Loss). +NMS-based Detectors. Table 3 suggests ∼ 1.0APC gain +on NMS-based detectors: (i) Spearman Loss (λcorr = 0.1) +improves RetinaNet by 1.0APC and oLRP, (ii) Concor- +dance Loss (λcorr = 0.2) enhances anchor-free FoveaBox +by 0.7APC, and (iii) Concordance Loss (λcorr = 0.3) im- +proves ATSS and PAA by ∼ 1APC and ∼ 1oLRP. +NMS-free Detectors. Our results in Table 3 suggest that +Sparse R-CNN, an NMS-free method, can also benefit from +Method +AP +AP50 +AP75 +Sparse R-CNN +39.0 +63.1 +37.6 +w. Spear.Corr (Ours) +40.8 +64.4 +40.8 +Table 4: Results on Cityscapes dataset. +Method +APmask +C +APmask +50 +APmask +75 +YOLACT (Bolya et al. 2019) +28.3 +47.8 +28.8 +w. Conc.Corr (Ours) +28.8 +48.3 +29.6 +w. Spear.Corr (Ours) +29.0 +48.3 +30.0 +Table 5: Comparison with YOLACT. +our Corr. Loss: (i) Both Concordance (λcorr = 0.3) and +Spearman Losses (λcorr += 0.2) improve baseline; (ii) +Spearman Loss improves APC significantly by up to 1.6; +(iii) as hypothesized, the gains are owing to APs with larger +IoUs, e.g., AP75 improves by up to 2.0, and (iv) gains persist +in a stronger setting of Sparse R-CNN (Appendix). +Cityscapes dataset. To see the effect of Corr. Loss over +different scenarios, we train Sparse R-CNN with Spear- +man Loss (the model that has the best gain over baseline +in Table 3), on the Cityscapes dataset (Cordts et al. 2016) +(λcorr = 0.6), a dataset for autonomous driving object de- +tection. Table 4 presents that (i) Spearman Loss also im- +proves baseline Sparse R-CNN on Cityscapes by 1.8 AP and +(ii) our gain mainly originates from APs with larger IoUs, +i.e. AP75 improves by more than 3 points, from 37.6 to 40.8. +Instance Segmentation. We train YOLACT (Bolya et al. +2019) as an instance segmentation method with Corr. Loss +and observed 0.7 mask AP gain using Spearman Loss +(λcorr = 0.5 - Table 5), implying 1.7% relative gain. +5.2 +Comparison with Methods Enforcing +Correlation +Table 6 compares Corr. Loss. with using an aux. head +(Zhang et al. 2020), QFL (Li et al. 2020) and RS Loss (Ok- +suz et al. 2021a) on the common ATSS baseline (Zhang et al. +2020) wrt. detection and correlation: +Detection Performance. Reaching 39.8APC without an +aux. head, Concordance Loss (Table 6) outperforms using an +aux. head, which introduces additional learnable parameters +(39.8 vs 39.3APC), and reaches on-par performance with +the recently proposed, relatively complicated loss functions, +Aux.QFLRS LossOurs APC AP50 AP75 oLRP ↓ βimg ↑βcls ↑ +38.7 57.6 41.5 +68.9 +27.3 +40.3 +✓ +39.3 57.5 42.6 +68.6 +28.7 +42.5 +✓ +39.7 58.1 42.7 +68.0 +33.2 +45.7 +✓ +39.9 58.9 42.6 +67.9 +30.9 +43.2 +✓ +39.8 57.6 43.1 +68.2 +31.6 +45.2 +✓ +✓ +40.0 58.0 43.3 +68.0 +31.1 +44.8 +✓ +✓ +39.9 58.2 43.2 +67.7 +34.6 +45.6 +✓ +✓ +40.2 58.6 43.5 +67.9 +33.6 +46.1 +Table 6: Comparison with methods enforcing correlation. +Corr. Loss (i) reaches similar results with existing methods +on ATSS, (ii) is complementary to those methods thanks to +its simple design and (iii) once combined with RS Loss, out- +performs compared methods. + +Method +Backbone +Epochs APC AP50 AP75 APS APM APL +Venue +NMS-based +ATSS (Zhang et al. 2020) +ResNet-101-DCN +24 +46.3 64.7 +50.4 27.7 49.8 58.4 +CVPR 2020 +GFLv2 (Li et al. 2019) +ResNet-101-DCN +24 +48.3 66.5 +52.8 28.8 51.9 60.7 +CVPR 2021 +aLRP Loss (Oksuz et al. 2020) +ResNeXt-101-DCN +100 +48.9 69.3 +52.5 30.8 51.5 62.1 NeurIPS 2020 +VFNet (Zhang et al. 2021) +ResNet-101-DCN +24 +49.2 67.5 +53.7 29.7 52.6 62.4 +CVPR 2021 +DW (Li et al. 2022) +ResNet-101-DCN +24 +49.3 67.6 +53.3 29.2 52.2 63.5 +CVPR 2022 +TOOD (Feng et al. 2021) +ResNet-101-DCN +24 +49.6 67.4 +54.1 30.5 52.7 62.4 +ICCV 2021 +RS-Mask R-CNN+ (Oksuz et al. 2021a) ResNeXt-101-DCN +36 +50.2 70.3 54.8 31.5 53.5 63.9 +ICCV 2021 +NMS-free +TSP R-CNN (Sun et al. 2021c) +ResNet-101-DCN +96 +47.4 66.7 +51.9 29.0 49.7 59.1 +ICCV 2021 +Sparse R-CNN (Sun et al. 2021b) +ResNeXt-101-DCN +36 +48.9 68.3 +53.4 29.9 50.9 62.4 +CVPR 2021 +Dynamic DETR (Dai et al. 2021) +ResNeXt-101-DCN +36 +49.3 68.4 +53.6 30.3 51.6 62.5 +ICCV 2021 +Deformable DETR (Zhu et al. 2021) +ResNeXt-101-DCN +50 +50.1 69.7 +54.6 30.6 52.8 64.7 +ICLR 2021 +Ours +Corr-Sparse R-CNN +ResNet-101-DCN +36 +49.6 67.8 +54.1 29.2 52.3 64.9 +Corr-Sparse R-CNN +ResNeXt-101-DCN +36 +51.0 69.2 55.7 31.1 53.7 66.3 +Table 7: SOTA comparison on COCO test-dev. Our Corr-Sparse R-CNN (i) performs on-par or better compared to recent NMS- +based methods, all of which also enforce correlation, and (ii) outperforms NMS-free methods by a notable margin. Results are +obtained from papers. +QFL (Li et al. 2020) and RS Loss (Oksuz et al. 2021a). Be- +sides, owing to its simple usage, Concordance Loss is com- +plementary to existing methods: It yields 40.0APC with an +aux. head (+0.7 APC) and 40.2APC with RS Loss (+0.3 +APC) without introducing additional learnable parameters. +Correlation Analysis. To provide insight, we report βimg +(Eq. 1) and βcls (Eq. 2) in Table 6: Our Concordance Loss (i) +improves baseline correlation significantly, enhancing βimg +(from 27.3% to 31.6%) and βcls (from 40.3% to 45.2%) +both by ∼ 5%, and (ii) results in better correlation than all +methods wrt. βimg and βcls once combined with QFL and +RS Loss respectively. This set of results confirms that Con- +cordance Loss improves correlation between classification +and localization tasks in both image-level and class-level. +5.3 +Comparison with SOTA +Here, we prefer Sparse R-CNN owing to its competitive de- +tection performance and our large gains. We train our “Corr- +Sparse R-CNN” for 36 epochs with DCNv2 (Zhu et al. 2019) +and multiscale training by randomly resizing the shorter side +within [480, 960] similar to common practice (Oksuz et al. +2021a; Zhang et al. 2021; Sun et al. 2021b). Table 7 presents +the results on COCO test-dev (Lin et al. 2014): +NMS-based Methods. On the common ResNet-101- +DCN backbone and with similar data augmentation, our +Corr-Sparse R-CNN yields 49.6APC at 13.7 fps (on a V100 +GPU) outperforming recent NMS-based methods, all of +which also enforce correlation, e.g., (i) RS-R-CNN (Ok- +suz et al. 2021a) by 1.8APC, (ii) GFLv2 (Li et al. 2019) +by more than 1APC, and (iii) VFNet (Zhang et al. 2021) in +terms of not only APC but also efficiency (with 12.6 fps on +a V100 GPU). On ResNeXt-101-DCN, our Corr-Sparse R- +CNN provides 51.0APC at 6.8 fps, surpassing all methods +including RS-Mask R-CNN+ (50.2APC at 6.4 fps), addi- +tionally using masks and Carafe FPN (Wang et al. 2019). +NMS-free Methods. Our Corr-Sparse R-CNN outper- +forms (i) TSP R-CNN (Sun et al. 2021c) by more than +2APC on ResNet-101-DCN with significantly less training, +(ii) Sparse R-CNN (Sun et al. 2021b) by ∼ 2APC and De- +formable DETR (Zhu et al. 2021), a recent strong NMS-free +method, by ∼ 1APC on ResNeXt-101-DCN. +5.4 +Ablation & Hyper-parameter Analyses +Optimizing Different Correlation Coefficients. Spearman +Loss yields better localization performance, i.e. the lowest +localization error wrt. oLRPLoc in all experiments while +it rarely yields the best oLRPFP or oLRPFN, implying its +contribution to classification to be weaker than Concordance +Loss (see Appendix for components of oLRP). We also +tried Pearson Correlation Coefficient on ATSS and Sparse +R-CNN but it performed worse compared to either using +Spearman or Concordance (Appendix). +Backpropagating Through Different Heads. On Sparse +R-CNN, we observed that the performance degrades when +we backpropagate either only localization head (37.5 AP) or +both heads (38.9 AP). Hence, we preferred backpropagating +the gradients only through the classification head (39.3 AP). +Effect on Training Time. Using Spearman or Concor- +dance Loss to train Sparse R-CNN, computing the loss for 6 +times each iteration, increases iteration time 0.50 sec to 0.51 +sec on V100 GPUs, suggesting a negligible overhead. +Sensitivity to λcorr. We found it sufficient to search over +{0.1, 0.2, 0.3.0.4, 0.5, 0.6} to tune λcorr. Appendix presents +empirical results for grid search. +5.5 +Additional Material +This paper is accompanied by an Appendix containing (i) the +effect of Corr.Loss on Sparse R-CNN using its stronger set- +ting, (ii) components of oLRP for detectors in Table 3, (iii) +results when Pearson Correlation Coefficient is optimized, +(iv) our grid search to tune λcorr. +6 +Conclusion +In this paper, we defined measures to evaluate object detec- +tors wrt. correlation, provided analyses on several methods +and proposed Correlation Loss as an auxiliary loss function +to enforce correlation for object detectors. Our extensive +experiments on six detectors show that Correlation Loss. +consistently improves the detection and correlation perfor- +mances, and reaches SOTA results. + +Acknowledgments +This work was supported by the Scientific and Technolog- +ical Research Council of Turkey (T ¨UB˙ITAK) (under grant +120E494). 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(Sun et al. 2021b) +Method +Dataset +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +ATSS +COCO +38.7 38.8 39.3 39.8 39.7 39.7 39.6 +YOLACT +COCO +28.3 28.6 28.8 28.8 29.0 28.8 28.6 +Sparse R-CNN +COCO +37.7 38.7 39.3 39.1 39.0 38.1 38.0 +Sparse R-CNN Cityscapes 39.0 39.0 38.3 39.9 40.0 40.0 40.8 +Table A.8: Grid search to tune λcorr on different models. +We present the results for concordance correlation coeffi- +cient for ATSS and YOLACT, and spearman correlation co- +efficient for Sparse R-CNN models. 0.0 corresponds to not +including our Correlation Loss. +Method +AP +AP50 +AP75 +Sparse R-CNN (Sun et al. 2021b) +45.0 +64.1 +48.9 +w. Conc.Corr (Ours) +45.5 +64.4 +49.7 +w. Spear.Corr (Ours) +46.1 +64.0 +50.4 +Table A.9: Comparison with stronger Sparse R-CNN. +(Table A.9), we train Sparse R-CNN with 36 epochs train- +ing, 300 proposals, multi-scale training and random crop- +ping. Table A.9 presents that the improvement of our Spear- +man Loss on this strong baseline is ∼ 1 AP points. +Using Pearson Correlation Coefficient. We tried opti- +mizing pearson correlation coefficient as well and observed +that while it has similar performance with concordance cor- +relation coefficient on ATSS and spearman correlation coef- +ficient on Sparse R-CNN, it does not outperform the other +two in both of the cases (Table A.10). Considering the sim- +ilarities of spearman and concordance correlation coeffi- +cients in terms of scoring the relation of the values, we +preferred concordance correlation coefficient over spearman +correlation coefficient due to the fact that concordance corre- +lation coefficient enforces the scores to be equal to the IoUs +imposing a tighter constraint than pearson correlation coef- +ficient. +The components of oLRP. Table A.11 shows the the +components of oLRP for different detectors corresponding +to Table 3 in the paper. As discussed in the paper, Spearman +Loss yields better localization performance, i.e. the lowest +localization error wrt. oLRPLoc in all experiments while +it rarely yields the best oLRPFP or oLRPFN, implying its +contribution to classification to be weaker than Concordance +Loss. +Method +APC +AP50 +AP75 +ATSS w/o aux head +38.7 +57.6 +41.5 +w. Pearson Corr +39.4 +56.6 +42.7 +w. Conc.Corr +39.8 +57.9 +43.2 +w. Spear.Corr +39.3 +56.6 +42.5 +Sparse-RCNN +37.7 +55.9 +40.5 +w. Pearson Corr +39.3 +56.6 +42.2 +w. Conc.Corr +38.9 +57.2 +41.8 +w. Spear.Corr +39.3 +56.7 +42.5 +Table A.10: Effect of using Pearson correlation coefficient. + +Method +APC ↑ +AP50 ↑ +AP75 ↑ +oLRP ↓ +oLRPLoc ↓ +oLRPFP ↓ +oLRPFN ↓ +Retina Net (Lin et al. 2020) +36.5 +55.4 +39.1 +70.7 +16.8 +32.0 +48.1 +w. Conc.Corr (Ours) +37.0 +55.7 +39.7 +70.2 +16.3 +30.8 +49.3 +w. Spear.Corr (Ours) +37.5 +55.4 +40.5 +69.7 +16.0 +31.3 +48.4 +Fovea Box (Kong et al. 2020) +36.4 +56.5 +38.6 +70.2 +17.0 +30.2 +47.2 +w. Conc.Corr (Ours) +37.1 +56.4 +39.6 +69.7 +16.6 +28.6 +48.1 +w. Spear.Corr (Ours) +37.0 +55.6 +39.3 +70.0 +16.3 +31.0 +47.9 +ATSS (Zhang et al. 2020) +38.7 +57.6 +41.5 +69.0 +16.0 +29.1 +47.0 +w. Conc.Corr (Ours) +39.8 +57.9 +43.2 +68.2 +15.4 +29.1 +46.9 +w. Spear.Corr (Ours) +39.3 +56.6 +42.5 +68.7 +15.2 +31.2 +46.7 +PAA (Kim and Lee 2020) +39.9 +57.3 +43.4 +68.6 +15.0 +30.4 +47.0 +w. Conc.Corr (Ours) +40.7 +58.8 +44.3 +67.7 +15.2 +28.5 +46.3 +w. Spear.Corr (Ours) +40.4 +58.0 +43.7 +67.8 +14.9 +29.5 +46.6 +Sparse R-CNN (Sun et al. 2021b) +37.7 +55.8 +40.5 +69.5 +16.0 +28.7 +48.6 +w. Conc.Corr (Ours) +38.9 +57.2 +41.8 +68.1 +15.7 +27.7 +47.2 +w. Spear.Corr (Ours) +39.3 +56.7 +42.5 +68.3 +15.3 +27.1 +48.4 +Table A.11: Components of oLRP for Table 3 in the paper. + diff --git a/ANAzT4oBgHgl3EQfF_sv/content/tmp_files/load_file.txt b/ANAzT4oBgHgl3EQfF_sv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f462d751d6d664aa93ed44385c077698b1532f4b --- /dev/null +++ b/ANAzT4oBgHgl3EQfF_sv/content/tmp_files/load_file.txt @@ -0,0 +1,1803 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf,len=1802 +page_content='Correlation Loss: Enforcing Correlation between Classification and Localization Fehmi Kahraman*,1, Kemal Oksuz*,1, Sinan Kalkan†,1,2, Emre Akbas†,1,2 1Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' of Computer Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2METU Center for Robotics and Artificial Intelligence (ROMER) Middle East Technical University (METU), Ankara, Turkey {fehmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='kahraman 01, kemal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='oksuz, skalkan, eakbas}@metu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='tr Abstract Object detectors are conventionally trained by a weighted sum of classification and localization losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Recent studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', predicting IoU with an auxiliary head, Generalized Fo- cal Loss, Rank & Sort Loss) have shown that forcing these two loss terms to interact with each other in non-conventional ways creates a useful inductive bias and improves perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Inspired by these works, we focus on the correlation between classification and localization and make two main contributions: (i) We provide an analysis about the effects of correlation between classification and localization tasks in object detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' We identify why correlation affects the performance of various NMS-based and NMS-free detectors, and we devise measures to evaluate the effect of correla- tion and use them to analyze common detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (ii) Moti- vated by our observations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', that NMS-free detectors can also benefit from correlation, we propose Correlation Loss, a novel plug-in loss function that improves the performance of various object detectors by directly optimizing correla- tion coefficients: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', Correlation Loss on Sparse R-CNN, an NMS-free method, yields 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 AP gain on COCO and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 AP gain on Cityscapes dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Our best model on Sparse R-CNN reaches 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 AP without test-time augmentation on COCO test-dev, reaching state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Code is available at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='com/fehmikahraman/CorrLoss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 1 Introduction Most object detectors optimize a weighted sum of classifi- cation and localization losses during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Results from recent work suggest that performance improves when these two loss functions are forced to interact with each other in non-conventional ways as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' For example, training an auxiliary (aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=') head to regress the localization qualities of the positive examples, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' centerness, IoU or mask-IoU, has proven useful (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Kim and Lee 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Other methods remove such auxiliary heads and aim directly to enforce correlation1 in the classification or localization task during training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', Average LRP Loss (Oksuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' †Equal contribution for senior authorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 1In the rest of the paper, “correlation” will refer to the correla- tion between classification scores and IoUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' weighs the examples in the localization task by ranking them with respect to (wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=') their classification scores (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Using localization quality as an additional supervision sig- nal for classification has been more commonly adopted (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 1(d)) (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Oksuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021) in two main ways: (i) Score-based ap- proaches aim to regress the localization qualities (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021) in the classification score, and (ii) ranking-based approaches enforce the classifier to rank the confidence scores wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' the localization qualities (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Oksuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Improving correlation seems to have a positive effect on performance of a variety of object detectors, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' However, the effect of correlation on object detectors has not been thoroughly studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' We fill this gap in this pa- per and first identify that correlation affects the performance of object detectors at two levels: (i) Image-level correlation, the correlation between the classification scores and local- ization qualities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', IoU for the rest of the paper) of the de- tections in a single image before post-processing, which is important to promote NMS performance, and (ii) Class-level correlation, the correlation over the entire dataset for each class after post-processing, which is related to the COCO- style Average Precision (AP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Moreover, we quantitatively define correlation at each level to enable analyses on how well an object detector captures correlation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', βcls in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Then, we provide an analysis on both levels of correlation and draw important observations using common models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Finally, to better exploit correlation, we introduce a more direct mechanism to enforce correlation: Correlation Loss, a simple plug-in and detector-independent loss term (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 1(e)), improving performance for a wide range of ob- ject detectors including NMS-free detectors, aligning with our analysis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Similar to the novel loss functions (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Oksuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021), our Correlation Loss boosts the performance without an aux- iliary head, but different from them, it is a simple plug-in technique that can easily be incorporated into any object de- tector, whether NMS-based or NMS-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Our main contributions are: (1) We identify how corre- lation affects NMS-based and NMS-free detectors, and de- sign quantitative measures to analyze a detector wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' corre- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (2) We analyze the effects of correlation at different levels on various object detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (3) We propose Correla- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='01019v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='CV] 3 Jan 2023 ̂𝑠 ← ̂𝑠×%ℓ Cls Loc Aux ℒ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' "# ℒ"$!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=" ℒ%&' (b) Auxiliary Head (d) Novel Cls." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Loss (a) No correlation Cls Loc ℒ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' "# ℒ"$!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' ̂𝑠 %𝐵 ̂𝑠 %𝐵 %ℓ Cls Loc ℒ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' "# ℒ"$!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' ̂𝑠 %𝐵 (c) Novel Loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Loss Cls Loc ℒ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' "# ℒ"$!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' ̂𝑠 %𝐵 Cls Loc ℒ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' "# ℒ"$!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' ̂𝑠 %𝐵 ℒ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='$(( (e) Correlation Loss (Ours) Legend ̂𝑠 : Classification Scores )𝐵 : Box Coordinates )ℓ : Localization Quality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' centerness) ℒ : A Loss Function Figure 1: Different ways of handling the classification and localization tasks from the perspective of correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (a) Conven- tional case of optimizing the two tasks independently (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (b) An additional auxiliary head predicts centerness (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) or IoU (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Kim and Lee 2020), which introduces additional learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (c) Novel loss functions replace the standard localization loss (Oksuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) or (d) novel classification loss (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Oksuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021a) by more complicated ones to leverage correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (e) Our Correlation Loss explicitly opti- mizes a correlation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' It is a simple, plug-in loss function which does not introduce additional parameters and has the flexibility to supervise classification or localisation head as well as both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Black and colored arrows respectively denote the loss functions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', during training) & the network outputs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', during inference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 APC 40 41 42 43 44 45 46 47 cls FL Aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' QFL RS FL Aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' QFL RS w/o Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Loss w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Loss (Ours) (a) Detection vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Correlation ATSS Sparse RCNN YOLACT 25 30 35 40 45 50 APC +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 AP +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8% gain +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 AP +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2% gain +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 AP +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4% gain Baseline Ours (b) Effect of our Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Loss Figure 2: (a) Detection performance, measured by COCO- style AP (APC) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' correlation quality, measured by class- level correlation (βcls - see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' The methods proposed to improve the correlation between classi- fication and localization tasks also improve APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Compare using aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' head, QFL, RS Loss with the baseline ATSS only using Focal Loss (FL – all in red dots) to see the positive cor- relation between APC and βcls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Our Correlation Loss as a plug-in loss function explicitly optimizes a correlation coef- ficient and improves the detection performance (APC) over different settings of ATSS (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' using FL, aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' head, QFL, RS Loss) consistently owing to increasing βcls, validating our hypothesis (compare green stars with red dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (b) Our Cor- relation Loss is simple-to-use and improves various meth- ods (i) NMS-based ATSS (w/o aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' head) by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1APC, (ii) NMS-free Sparse R-CNN by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6APC and (iii) YOLACT, an instance segmentation method by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' tion Loss as a plug-in loss function to optimize correlation explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Thanks to its simplicity, our loss function can be easily incorporated into a diverse set of object detectors and improves the performance of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', Sparse R-CNN up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 AP and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0AP75, suggesting, for the first time, that NMS- free detectors can also benefit from correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Our best model yields 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 AP, reaching state-of-the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2 Background and Related Work Object Detection Pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' We group object detectors wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' their usage of NMS (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 3 presents overview & notation): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' NMS-based Detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' To detect all objects with dif- ferent scales, locations and aspect ratios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' most methods (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Law and Deng 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) employ a large number of object hypotheses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', anchors, points), which are labeled as positive (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' foreground) or negative (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' background) during training, based on whether/how they match GT boxes (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' In this setting, there is no restriction for an ob- ject to be predicted by multiple object hypotheses, causing duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Accordingly, during inference, NMS picks the detection with the largest confidence score among the detec- tions that overlap more than a predetermined IoU threshold to avoid duplicate detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' NMS-free Detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' An emerging research direction is to remove the need for doing NMS, simplifying the detec- tion pipeline (Carion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Roh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' This is achieved by ensuring a one-to-one matching between the GTs and de- tections, which supervises the detector to avoid duplicates in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Methods Enforcing Correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' One common way to ensure correlation is to use an additional auxiliary head, su- pervised by the localization quality of a detection such as centerness (Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020), IoU (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2018), mask IoU (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2019) or uncertainty (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2019), during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' During inference, the pre- dictions of the auxiliary head are then combined with those of the classifier to improve detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Recent methods show that the auxiliary head can be removed, and either (i) the regressor can prioritize the positive examples (Oksuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) or (ii) the classifier can be supervised to prioritize detections with confidence scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' The latter is en- sured either by regressing the IoUs by the classifier (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021) or by training the classifier to rank confidence scores (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Oksuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021a) wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' IoUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Unlike these methods, TOOD (Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021) takes correlation into account mainly while designing the model, particularly the detection head, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', not the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Correlation Coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Correlation coefficients mea- sure the strength and direction of the “relation” between two Ƹ𝑠𝑝𝑟𝑒 𝐼 (score) \u0de0𝐵𝑝𝑟𝑒 𝐼 (box) Image, I Post-processing Object Detector Remove background NMS Top-k For each class c in image I For each image I Collect Ƹ𝑠𝑝𝑜𝑠𝑡 𝑐 & \u0de0𝐵𝑝𝑜𝑠𝑡 𝑐 over all images Ƹ𝑠𝑝𝑜𝑠𝑡 𝐼,𝑐 (score) \u0de0𝐵𝑝𝑜𝑠𝑡 𝐼,𝑐 (box) Figure 3: Object detection pipeline and notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Given an input image, I, NMS-based detectors yield raw detections before post-processing, each of which has a predicted bounding box (BB) and an array of confidence scores over GT classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' We denote the confidence scores and the predicted BBs pertaining to the positive detections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', the detections matching with GT objects during training, by ˆsI pre and ˆBI pre, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' To obtain final detections, raw detections are post-processed in three steps: (i) Detections with low confidence scores, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', background, are removed, (ii) duplicates are eliminated by NMS, and (iii) top-k scoring detections are kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' As for these final detections, we denote the confidence scores and BBs of true positive detections for class c in a single image I by ˆsI,c post and ˆBI,c post respectively, and over the entire dataset by ˆsc post and ˆBc post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' As for NMS-free detectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' NMS, dashed gray box in post-processing, is excluded, hence post-processing is lighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' sets, X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', xN} and Y = {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', yN}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Different re- lations are evaluated by different correlation coefficients: (i) Pearson correlation coefficient, denoted by α(·, ·), measures the linear relationship between the sets, (ii) Spearman corre- lation coefficient, β(·, ·), corresponds to the ranking relation- ship and (iii) Concordance correlation coefficient, γ(·, ·), is more strict, measuring the similarity of the values and max- imized when xi = yi for all i ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' All correlation coefficients have a range of [−1, +1] where positive/neg- ative correlation corresponds to increasing/decreasing rela- tion, while 0 implies no correlation between X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Comparative Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' In this paper, we comprehen- sively identify and analyze the effect of explicitly correlat- ing classification and localization in object detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Unlike other methods that also enforce correlation, some of which are tested only on a single architecture (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2019), we propose a simple solu- tion by directly optimizing the correlation coefficient, which is auxiliary-head free and easily applicable to all object de- tectors, whether NMS-based or NMS-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Also, ours is the first to work on NMS-free detectors in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 3 Effects of Correlation on Object Detectors This section presents why maximizing correlation is impor- tant for object detectors, introduces measures to evaluate ob- ject detectors wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' correlation and provides an analysis on methods designed for improving correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 How Correlation Affects Object Detectors Detectors are affected by correlation at two levels (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 4): Image-level Correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' This level of correlation corre- sponds to the correlation between the classification scores and IoUs of the detections in a single image before post- processing, and accordingly, we measure it with the Spear- man correlation coefficient2, β(·, ·), averaged over images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2While analyzing object detectors in terms of correlation, we employ Spearman correlation coefficient, β(·, ·), to measure the relation between the ranks of the values (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', scores and IoUs) in- stead of the values themselves, and aim to isolate the correlation quality from the localization and classification performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Denoting the set of images to be evaluated by I and IoUs between the BBs of the positive detections ( ˆBI pre, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 3) and their associated GTs by IoUI pre, image-level correlation is measured as follows: βimg = 1 |I| � I∈I β(IoUI pre, ˆsI pre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (1) Maximizing image-level correlation is important for NMS-based detectors since NMS aims to suppress dupli- cates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', to keep only a single detection for each GT when there is more than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' More particularly among overlap- ping detections (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', dark and light green detections in the detector output image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 4(a)), NMS picks the one with the larger score, and hence, if there is positive correlation between the confidence scores and IoUs of those overlap- ping detections, then the one with the best IoU (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', dark green detection in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 4(a)) will survive and detection per- formance will increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Class-level Correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' This level of correlation indi- cates the correlation between the classification scores and IoUs of the detections obtained after post-processing for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Since class-level correlation is related to COCO- style AP, APC, we average β(·, ·) over classes to be consis- tent with the computation of APC: βcls = 1 |C| � c∈C β(IoUc post, ˆsc post), (2) where C is the set of classes in the dataset and IoUc post is the set IoUs of BBs of true positives for class c ( ˆBc post, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Class-level correlation affects the performance of all de- tectors since it is directly related to APC, the performance measure itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' To be more specific, APC for a single class is defined as the average of APs computed over 10 differ- ent IoU thresholds, IoU ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='55, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='95}, validating the true positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' For a specific threshold IoU, the detec- tions are first sorted with respect to the classification scores, and then precision and recall pairs are calculated on each detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Using these pairs, a precision-recall (PR) curve is obtained, and the area under the PR curve corresponds to Positively correlated NMS Solid BBs : Ground truths Dashed BBs : Detections before post-processing ( Ƹ𝑠𝑝𝑟𝑒 𝐼 , \u0de0𝐵𝑝𝑟𝑒 𝐼 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 3) Positively correlated - High APC (b) Class-level Correlation for better AP AP Calculation 𝐼𝑜𝑈 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='80 N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='60 N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='50 Ƹ𝑠 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='50 𝐼𝑜𝑈 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='80 N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='60 N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='50 Ƹ𝑠 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='80 Negatively correlated - Low APC Precision Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Detections APIoU P50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='76 P75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='33 Precision Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Detections APIoU P50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='76 P75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='07 Negatively correlated Solid BBs : Ground truths Dashed BBs : Detections after post-processing ( Ƹ𝑠𝑝𝑜𝑠𝑡 𝐼,𝑐 , \u0de0𝐵𝑝𝑜𝑠𝑡 𝐼,𝑐 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 3) (a) Image-level Correlation for better NMS High IoU Low IoU Figure 4: How correlation affects detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (a) Image-level correlation: Given detections before post-processing, NMS benefits from image-level correlation, thereby yielding detections with better IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Compare IoUs of detections in “posi- tively correlated” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', when the dark-colored ones have larger score) and “negatively correlated” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', when the light-colored ones have larger score) outputs after NMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (b) Class-level correlation: Given detections after post-processing, APs with larger IoUs and COCO-style AP benefit from positive class-level correlation (compare APIoU columns in “positively correlated” and “negatively correlated” outputs after AP Calculation to see lower AP75 for the “negatively correlated” output in the red cell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' PIoU: Precision computed on a detection using the threshold IoU, True positives are color-coded in tables and input, white cells: false positives, and hence their IoU is not available, N/A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' the single AP value, APIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' When the correlation between classification and localization is maximized among true pos- itives, larger precision values are obtained on the same de- tections in larger IoU values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' P75 of orange detection is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='00 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='20 with positive and negative correlation respec- tively in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 4(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 Analyses of Object Detectors wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Correlation Dataset and Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Unless otherwise specified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' we (i) employ the widely-used COCO dataset (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2014) by training the models on trainval35K (115K images), testing on minival (5k images), comparing with SOTA on test-dev (20k images), (ii) build upon the mmde- tection framework (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2019), (iii) rely on AP-based measures and also use Optimal LRP (oLRP) (Oksuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021b), βimg (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 1) and βcls (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2) to provide more in- sights, (iv) keep the standard configuration of the models, (v) use a ResNet-50 backbone with FPN (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2017), (vi) train models on 4 GPUs (A100 or V100 type GPUs) with 4 images on each GPU (16 batch size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Analysis Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' We conduct experiments to analyze the effects of the image-level (βimg – Table 1) and class-level (βcls – Table 2) correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' For both analyses, we com- pare three sets of methods, all of which are incorporated into the common ATSS baseline (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2 for a discussion of these methods): (i) AP Loss and Focal Loss as methods not enforcing correlation, (ii) using an auxiliary head to enforce correlation, and (iii) Quality Fo- cal Loss (QFL), aLRP Loss and Rank & Sort Loss as recent loss functions enforcing correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' In our class-level anal- ysis, we also employ NMS-free methods to demonstrate the effects of correlation on that approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' We compare the methods based on (i) their AP-based per- formance, (ii) our proposed measures for correlation (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 1 and 2), and finally (iii) lower/upper bounds, AP+1 C /AP−1 C , obtained by modifying the ranking of the confidence scores pertaining to the GT classes of the positive detections to minimize/maximize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 1 in Table 1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2 in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' More particularly, in Table 1, given ˆsI pre and ˆBI pre (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 3), we collect the GT class probabilities of positive detections and change their ranking in ˆsI pre within an image follow- ing the ranking order of IoUs (computed using ˆBI pre), and in Table 2, we do the same operation class-wise for true posi- tives given ˆsc post and ˆBc post (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' To decouple other types of errors as much as possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' in Table 1, we do not modify the scores of the negative detections, the predicted BBs and the scores of non-GT classes of the positive detections, and in Table 2, we do not modify the scores of the false positives and the predicted BBs of the true positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Note that achiev- ing the upper bound in (iii) for image-level correlation also corresponds to perfectly minimizing RS Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' We observe in Tables 1 and 2 that: (1) Our proposed measures in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 1 and 2 can measure the improvements in correlation consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' In Tables 1 and 2, (i) aLRP Loss and RS Loss are proposed to improve AP Loss and (ii) aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' head and QFL are proposed to improve Fo- cal Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' In both tables, the proposed methods are shown to improve their baselines in terms of βimg and βcls, suggest- ing that our measures can consistently evaluate image-level and class-level correlations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (2) NMS-free detectors can also potentially benefit from correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' All detectors, including NMS-free ones, can ex- ploit class-level correlation (compare APC and AP+1 C to see ∼ 10 points gap in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Still, existing methods do not enforce this correlation on NMS-free detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (3) Existing methods enforcing correlation have still a large room for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Considering that βimg ∈ [27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2%, 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8%] (Table 1) and βcls ∈ [37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5%, 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0%] (Table 2), there is still room for improvement wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Performance Modify ranking of scores Method APC AP50 AP75 βimg AP−1 C AP−1 50 AP−1 75 AP+1 C AP+1 50 AP+1 75 Not Enforcing Correlation ATSS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' AP Loss (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 ATSS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Focal Loss (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 Using Aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Head ATSS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' ctr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' head (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 Using Novel Loss ATSS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' aLRP Loss (Oksuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 ATSS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' QFL (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 ATSS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' RS Loss (Oksuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021a) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 Table 1: Evaluation of NMS-based detectors in terms of image-level correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' See Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 1 for βimg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' AP+1 IoU and AP−1 IoU refer to the upper & lower bound APs (see analysis setup for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' The values are in %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Our βimg captures correlation consistently, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' that (i) Focal Loss is improved by ctr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' head and QFL and (ii) AP Loss is improved by aLRP Loss and RS Loss wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' βimg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Also, there is still room for improvement for object detectors wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' βimg with a range between 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2% and 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Performance Modify ranking of scores Method APC AP50 AP75 βcls AP−1 C AP−1 50 AP−1 75 AP+1 C AP+1 50 AP+1 75 Not Enforcing Correlation NMS-free Detectors Sparse R-CNN (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021b) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 DETR (Carion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 NMS-based Detectors ATSS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' AP Loss (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 ATSS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Focal Loss (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 Using Aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Head ATSS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' ctr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' head (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 Using Novel Loss ATSS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' aLRP Loss (Oksuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 ATSS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' QFL (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 ATSS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' RS Loss (Oksuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021a) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 Table 2: Evaluation of detectors wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' class-level correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' See Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2 for βcls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' AP+1 IoU & AP−1 IoU denote upper & lower bound APs (analysis setup for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Values are in %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' NMS-free detectors can also benefit from class-level correlation (compare AP+1 C with APC for Sparse R-CNN), and as in βimg (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Table 1 and its caption), βcls measures the correlation consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' AP+1 50 = AP−1 50 = AP50 since only modifying TPs validated from IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='50 does not effect AP50 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 4(b) for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (4) While significantly important, improving correlation may not always imply performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' For exam- ple, aLRP Loss in Table 1 has the largest correlation but the lowest APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Such a situation can arise, for example, when a method does not have good localization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' In the extreme case, assume a detector yields perfect βimg, image- level ranking correlation, but the IoUs of all positive exam- ples are less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='50 implying no TP at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Hence, boost- ing the correlation, while simultaneously preserving a good performance in each branch, is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 4 Correlation Loss: A Novel Loss Function for Object Detection Correlation (Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=') Loss is a simple plug-in loss function to improve correlation of classification and localization tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Correlation Loss is unique in that it can be easily incorpo- rated into any object detector, whether NMS-based or NMS- free (see Observation (2) - Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2), and improves perfor- mance without affecting the model size, inference time and with negligible effect on training time (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Further- more, from a fundamental perspective, Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Loss can su- pervise both of the classification and localisation heads for a better correlation while existing methods generally focus on a single head such as classification (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Given an object detector with loss function LOD, our Correlation Loss (Lcorr) is simply added using a weighting hyper-parameter λcorr: LOD + λcorrLcorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (3) Lcorr is the Correlation Loss defined as: Lcorr = 1 − ρ( ˆ IoU,ˆs), (4) where ρ(·, ·) is a correlation coefficient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' ˆs and ˆ IoU are the confidence scores of the GT class and IoUs of the predicted BBs pertaining to the positive examples in the batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Practical Usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' To avoid promoting trivial cases with high correlation but low performance (Observation (4) - Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2), similar to QFL (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) and RS Loss (Oksuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021a), we only use the gradients of Lcorr wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' classification score, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', we backpropagate the gradi- ents through only the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' We mainly adopt two dif- ferent correlation coefficients for ρ(·, ·) and obtain two ver- sions of Correlation Loss: (i) Concordance Loss, defined as the Correlation Loss when Concordance correlation coeffi- cient is optimized (ρ(·, ·) = γ(·, ·)), which aims to match Method APC ↑AP50 ↑AP75 ↑ oLRP ↓ NMS-based Retina Net (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Spear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 Fovea Box (Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Spear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 ATSS (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Spear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 PAA (Kim and Lee 2020) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Spear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 NMS-free Sparse R-CNN (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021b) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Spear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 Table 3: Comparison on detectors not considering correla- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Accordingly, we remove aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' heads from ATSS (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) and PAA (Kim and Lee 2020) for fair compari- son (see Table 6 for comparison with aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' heads and novel loss functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' We use ResNet-50 and train the models for 12 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Simply incorporating our Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Loss provides (i) ∼ 1APC improvement for NMS-based detectors consis- tently and (ii) ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5APC on the NMS-free Sparse R-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' the confidence scores with IoUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (ii) Spearman Loss as Cor- relation Loss when Spearman correlation coefficient is op- timized (ρ(·, ·) = β(·, ·)), thereby enforcing the ranking of the classification scores considering IoUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' To tackle the non-differentiability of ranking operation while computing Spearman Loss, we leverage the differentiable sorting oper- ation from Blondel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (Blondel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' When apply- ing our Correlation Loss to NMS-free methods, which use an iterative multi-stage loss function, we incorporate Lcorr to every stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 5 Experimental Evaluation We evaluate Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Loss on (i) the COCO dataset with five different object detectors of various types (Sparse R-CNN as NMS-free, FoveaBox as anchor-free, RetinaNet as anchor- based, ATSS and PAA using auxiliary head), and one in- stance segmentation method, YOLACT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' and (ii) an addi- tional dataset (Cityscapes) for the method with the largest gain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', Sparse R-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 Comparison with Methods Not Considering Correlation We train these five object detectors and the instance segmen- tation method (Tables 3 and 5) with and without our Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Loss (as Concordance Loss or Spearman Loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' NMS-based Detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Table 3 suggests ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0APC gain on NMS-based detectors: (i) Spearman Loss (λcorr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1) improves RetinaNet by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0APC and oLRP, (ii) Concor- dance Loss (λcorr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2) enhances anchor-free FoveaBox by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7APC, and (iii) Concordance Loss (λcorr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3) im- proves ATSS and PAA by ∼ 1APC and ∼ 1oLRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' NMS-free Detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Our results in Table 3 suggest that Sparse R-CNN, an NMS-free method, can also benefit from Method AP AP50 AP75 Sparse R-CNN 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Spear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 Table 4: Results on Cityscapes dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Method APmask C APmask 50 APmask 75 YOLACT (Bolya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2019) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Spear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 Table 5: Comparison with YOLACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' our Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Loss: (i) Both Concordance (λcorr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3) and Spearman Losses (λcorr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2) improve baseline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (ii) Spearman Loss improves APC significantly by up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (iii) as hypothesized, the gains are owing to APs with larger IoUs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', AP75 improves by up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0, and (iv) gains persist in a stronger setting of Sparse R-CNN (Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Cityscapes dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' To see the effect of Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Loss over different scenarios, we train Sparse R-CNN with Spear- man Loss (the model that has the best gain over baseline in Table 3), on the Cityscapes dataset (Cordts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2016) (λcorr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6), a dataset for autonomous driving object de- tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Table 4 presents that (i) Spearman Loss also im- proves baseline Sparse R-CNN on Cityscapes by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 AP and (ii) our gain mainly originates from APs with larger IoUs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' AP75 improves by more than 3 points, from 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 to 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Instance Segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' We train YOLACT (Bolya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2019) as an instance segmentation method with Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Loss and observed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 mask AP gain using Spearman Loss (λcorr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 - Table 5), implying 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7% relative gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 Comparison with Methods Enforcing Correlation Table 6 compares Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' with using an aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' head (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020), QFL (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) and RS Loss (Ok- suz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021a) on the common ATSS baseline (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' detection and correlation: Detection Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Reaching 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8APC without an aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' head, Concordance Loss (Table 6) outperforms using an aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' head, which introduces additional learnable parameters (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 vs 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3APC), and reaches on-par performance with the recently proposed, relatively complicated loss functions, Aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='QFLRS LossOurs APC AP50 AP75 oLRP ↓ βimg ↑βcls ↑ 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 27.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 Table 6: Comparison with methods enforcing correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Loss (i) reaches similar results with existing methods on ATSS, (ii) is complementary to those methods thanks to its simple design and (iii) once combined with RS Loss, out- performs compared methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Method Backbone Epochs APC AP50 AP75 APS APM APL Venue NMS-based ATSS (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) ResNet-101-DCN 24 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 CVPR 2020 GFLv2 (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2019) ResNet-101-DCN 24 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 CVPR 2021 aLRP Loss (Oksuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) ResNeXt-101-DCN 100 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 62.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 Corr-Sparse R-CNN ResNeXt-101-DCN 36 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 Table 7: SOTA comparison on COCO test-dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Our Corr-Sparse R-CNN (i) performs on-par or better compared to recent NMS- based methods, all of which also enforce correlation, and (ii) outperforms NMS-free methods by a notable margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Results are obtained from papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' QFL (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) and RS Loss (Oksuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Be- sides, owing to its simple usage, Concordance Loss is com- plementary to existing methods: It yields 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0APC with an aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' head (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 APC) and 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2APC with RS Loss (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 APC) without introducing additional learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Correlation Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' To provide insight, we report βimg (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 1) and βcls (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2) in Table 6: Our Concordance Loss (i) improves baseline correlation significantly, enhancing βimg (from 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3% to 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6%) and βcls (from 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3% to 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2%) both by ∼ 5%, and (ii) results in better correlation than all methods wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' βimg and βcls once combined with QFL and RS Loss respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' This set of results confirms that Con- cordance Loss improves correlation between classification and localization tasks in both image-level and class-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 Comparison with SOTA Here, we prefer Sparse R-CNN owing to its competitive de- tection performance and our large gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' We train our “Corr- Sparse R-CNN” for 36 epochs with DCNv2 (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2019) and multiscale training by randomly resizing the shorter side within [480, 960] similar to common practice (Oksuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Table 7 presents the results on COCO test-dev (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2014): NMS-based Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' On the common ResNet-101- DCN backbone and with similar data augmentation, our Corr-Sparse R-CNN yields 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6APC at 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 fps (on a V100 GPU) outperforming recent NMS-based methods, all of which also enforce correlation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=', (i) RS-R-CNN (Ok- suz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021a) by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8APC, (ii) GFLv2 (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2019) by more than 1APC, and (iii) VFNet (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021) in terms of not only APC but also efficiency (with 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 fps on a V100 GPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' On ResNeXt-101-DCN, our Corr-Sparse R- CNN provides 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0APC at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 fps, surpassing all methods including RS-Mask R-CNN+ (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2APC at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 fps), addi- tionally using masks and Carafe FPN (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' NMS-free Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Our Corr-Sparse R-CNN outper- forms (i) TSP R-CNN (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021c) by more than 2APC on ResNet-101-DCN with significantly less training, (ii) Sparse R-CNN (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021b) by ∼ 2APC and De- formable DETR (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021), a recent strong NMS-free method, by ∼ 1APC on ResNeXt-101-DCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 Ablation & Hyper-parameter Analyses Optimizing Different Correlation Coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Spearman Loss yields better localization performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' the lowest localization error wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' oLRPLoc in all experiments while it rarely yields the best oLRPFP or oLRPFN, implying its contribution to classification to be weaker than Concordance Loss (see Appendix for components of oLRP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' We also tried Pearson Correlation Coefficient on ATSS and Sparse R-CNN but it performed worse compared to either using Spearman or Concordance (Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Backpropagating Through Different Heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' On Sparse R-CNN, we observed that the performance degrades when we backpropagate either only localization head (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 AP) or both heads (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 AP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Hence, we preferred backpropagating the gradients only through the classification head (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 AP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Effect on Training Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Using Spearman or Concor- dance Loss to train Sparse R-CNN, computing the loss for 6 times each iteration, increases iteration time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='50 sec to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='51 sec on V100 GPUs, suggesting a negligible overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Sensitivity to λcorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' We found it sufficient to search over {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6} to tune λcorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Appendix presents empirical results for grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 Additional Material This paper is accompanied by an Appendix containing (i) the effect of Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Loss on Sparse R-CNN using its stronger set- ting, (ii) components of oLRP for detectors in Table 3, (iii) results when Pearson Correlation Coefficient is optimized, (iv) our grid search to tune λcorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 6 Conclusion In this paper, we defined measures to evaluate object detec- tors wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' correlation, provided analyses on several methods and proposed Correlation Loss as an auxiliary loss function to enforce correlation for object detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Our extensive experiments on six detectors show that Correlation Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' consistently improves the detection and correlation perfor- mances, and reaches SOTA results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Acknowledgments This work was supported by the Scientific and Technolog- ical Research Council of Turkey (T ¨UB˙ITAK) (under grant 120E494).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' We also gratefully acknowledge the computa- tional resources kindly provided by T ¨UB˙ITAK ULAKBIM High Performance and Grid Computing Center (TRUBA) and METU Robotics and Artificial Intelligence Center (ROMER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Akbas is supported by the “Young Scientist Awards Program (BAGEP)” of Science Academy, Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' References Blondel, M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Wan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Ji, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' and Ye, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' FreeAn- chor: Learning to Match Anchors for Visual Object Detec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems (NeurIPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Hu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Lin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' and Dai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Deformable Con- vNets V2: More Deformable, Better Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' In IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Su, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Lu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' and Dai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Deformable {DETR}: Deformable Transformers for End- to-End Object Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' In International Conference on Learning Representations (ICLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' APPENDIX Sensitivity to λcorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' In Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8, we see that (i) λcorr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 provides the best performance overall, (ii) the perfor- mance is not very sensitive to λcorr and (iii) a grid search over {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6} is sufficient (outside of this range, performance drops).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' The effect of Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Loss on Sparse R-CNN using its stronger setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Following Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021b) Method Dataset 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 ATSS COCO 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 YOLACT COCO 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 Sparse R-CNN COCO 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 Sparse R-CNN Cityscapes 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8: Grid search to tune λcorr on different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' We present the results for concordance correlation coeffi- cient for ATSS and YOLACT, and spearman correlation co- efficient for Sparse R-CNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 corresponds to not including our Correlation Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Method AP AP50 AP75 Sparse R-CNN (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021b) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Spear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9: Comparison with stronger Sparse R-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' (Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9), we train Sparse R-CNN with 36 epochs train- ing, 300 proposals, multi-scale training and random crop- ping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 presents that the improvement of our Spear- man Loss on this strong baseline is ∼ 1 AP points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Using Pearson Correlation Coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' We tried opti- mizing pearson correlation coefficient as well and observed that while it has similar performance with concordance cor- relation coefficient on ATSS and spearman correlation coef- ficient on Sparse R-CNN, it does not outperform the other two in both of the cases (Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Considering the sim- ilarities of spearman and concordance correlation coeffi- cients in terms of scoring the relation of the values, we preferred concordance correlation coefficient over spearman correlation coefficient due to the fact that concordance corre- lation coefficient enforces the scores to be equal to the IoUs imposing a tighter constraint than pearson correlation coef- ficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' The components of oLRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='11 shows the the components of oLRP for different detectors corresponding to Table 3 in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' As discussed in the paper, Spearman Loss yields better localization performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' the lowest localization error wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' oLRPLoc in all experiments while it rarely yields the best oLRPFP or oLRPFN, implying its contribution to classification to be weaker than Concordance Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Method APC AP50 AP75 ATSS w/o aux head 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Pearson Corr 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Spear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 Sparse-RCNN 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Pearson Corr 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Spear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='10: Effect of using Pearson correlation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Method APC ↑ AP50 ↑ AP75 ↑ oLRP ↓ oLRPLoc ↓ oLRPFP ↓ oLRPFN ↓ Retina Net (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Spear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 Fovea Box (Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Spear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 ATSS (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2020) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Spear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 PAA (Kim and Lee 2020) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Spear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 Sparse R-CNN (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' 2021b) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='6 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Conc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='8 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='2 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content=' Spear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='Corr (Ours) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='5 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='1 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='4 Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} +page_content='11: Components of oLRP for Table 3 in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfF_sv/content/2301.01019v1.pdf'} diff --git a/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf b/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf new file mode 100644 index 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a/FNFQT4oBgHgl3EQfRTbc/content/tmp_files/2301.13286v1.pdf.txt b/FNFQT4oBgHgl3EQfRTbc/content/tmp_files/2301.13286v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0580a5edbe413757bbef22095630611f058e2dee --- /dev/null +++ b/FNFQT4oBgHgl3EQfRTbc/content/tmp_files/2301.13286v1.pdf.txt @@ -0,0 +1,2283 @@ +arXiv:2301.13286v1 [math.RT] 30 Jan 2023 +THE SEMI-INFINITE COHOMOLOGY OF WEYL MODULES +WITH TWO SINGULAR POINTS +GIORGIA FORTUNA, DAVIDE LOMBARDO, +ANDREA MAFFEI, VALERIO MELANI +Abstract. In their study of spherical representations of an affine Lie algebra +at the critical level and of unramified opers, Frenkel and Gaitsgory introduced +what they called the Weyl module Vλ corresponding to a dominant weight +λ. This object plays an important role in the theory. In [4], we introduced a +possible analogue Vλ,µ +2 +of the Weyl module in the setting of opers with two +singular points, and in the case of sl(2) we proved that it has the ‘correct’ +endomorphism ring. In this paper, we compute the semi-infinite cohomology +of Vλ,µ +2 +and we show that it does not share some of the properties of the semi- +infinite cohomology of the Weyl module of Frenkel and Gaitsgory. For this +reason, we introduce a new module ˜Vλ,µ +2 +which, in the case of sl(2), enjoys all +the expected properties of a Weyl module. +1. Introduction +Let g be a complex simple Lie algebra and let ˆg be its affinization. Choose a +Borel subalgebra and a maximal toral subalgebra, and let G be a simply connected +algebraic group with Lie algebra equal to g. As a particular case of a more general +conjecture, Frenkel and Gaitsgory proved in [6] that the semi-infinite cohomology +gives an isomorphism between the category ˆgcrit-modJG of spherical representations +of ˆg at the critical level (that is, representations of ˆg at the critical level with a +compatible action of JG = G(C[[t]])) and the category of quasi-coherent sheaves +on the space of unramified opers Opunr +1 +over gL, the Langlands dual of g. +As +they explain, the space of unramified opers is the disjoint union of its connected +components Opλ,unr +1 +, and the category of spherical representations is the product of +certain subcategories ˆgcrit-modJG,λ, where in both cases λ ranges over all dominant +weights of G. The equivalence given by semi-infinite cohomology specialises to an +equivalence between ˆgcrit-modJG,λ and the category of quasi-coherent sheaves over +Opλ,unr +1 +. The space Opλ,unr +1 +is a non-reduced indscheme, and its reduced version, +denoted by Opλ +1, is an affine scheme. +In this paper we will denote by Zλ +1 its +coordinate ring. +In this theory, an important role is played by the Weyl module Vλ +1. This module +enjoys the following fundamental properties: +Endˆg(Vλ +1) ≃ Zλ +1 +and +Ψ0(Vλ +1) ≃ Zλ +1 , +where Ψn is the n-th semi-infinite cohomology group. Moreover the semi-infinite +cohomology groups Ψn(Vλ +1) are trivial for n ̸= 0. +Dennis Gaitsgory suggested to Giorgia Fortuna to study the space of unramified +opers and spherical representations in a more general context, see [3]; in fact, the +definition of unramified opers as well as the definition of spherical representations +can be generalized in the presence of more than one singularity, raising the question +on whether or not certain statements remain true and what happens when these +singularities collide. +1 + +2 +FORTUNA, LOMBARDO, MAFFEI, MELANI +In [4] we took some steps in this direction, by studying the case of sl(2). In +particular, we introduced a version of the Weyl module Vλ,µ +2 +of critical level of the +affine Lie algebra with two singularities ˆg2. Thinking of t as a coordinate near the +first singularity and s as a coordinate near the second singularity, this is the version +of the affine Lie algebra over the ring A = C[[a]], where a = (t−s). As an A module +is equal to K2 ⊗C ⊕AC2 where K2 = C[[a, t]][1/t(t− a)] and C2 is a central element +(see [4], Section 3.3 for the complete definition). +We also introduced reduced scheme over A of unramified opers Opλ,µ +2 +which gener- +alize the schemes Opλ +1. Both objects depend on two integral dominant weights λ, +µ of G, and we proved that +Endˆg2(Vλ,µ +2 +) ≃ Zλ,µ +2 +, +where Zλ,µ +2 +is the coordinate ring of Opλ,µ +2 +. +In this article we study the semi-infinite cohomology of Vλ,µ +2 +in order to un- +derstand what relation it has with the ring Zλ,µ +2 +in order to understand how the +equivalence Ψ0(Vλ +1) ≃ Zλ +1 generalizes. +This is done in Section 4, where we compute the cohomology of Vλ,µ +2 +; in Section 5 +we study the action of Z2, the center of a completion ˆU2 of the enveloping algebra +of ˆg2 at the critical level on this module (see Section 2.2). +In particular, we prove that the specialisation at a = 0 and the localization at +a ̸= 0 of the semi-infinite cohomology of Vλ,µ +2 +are isomorphic to the specialisation +and localization of Zλ,µ +2 +, respectively. However, in contrast to our intuition, we also +show the following result which says that Ψ0(Vλ,µ +2 +) doesn’t exactly generalize the +equivalence Ψ0(Vλ +1) ≃ Zλ +1 as expected: +Theorem A (Theorem 4.9 and Proposition 5.3). We have Ψn(Vλ,µ +2 +) = 0 for n ̸= 0. +Moreover, Ψ0(Vλ,µ +2 +) is not isomorphic to Zλ,µ +2 +as a Z2-module. +For this computation, we rely on the formalism introduced by Casarin in [1], +which makes it possible to use vertex algebras also in the context of opers with +two singularities. Once this formalism is in place, for the computation of the semi- +infinite cohomology we can follow closely the approach taken by Frenkel and Ben +Zvi in [5, Chapter 15] for the case of one singularity. +In the last section, we restrict our attention to the Lie algebra sl(2) and introduce +a submodule �Vλ,µ +2 +of Vλ,µ +2 +, which is generated by the highest weight vector. We +prove that this module is the correct one to consider, in the sense that it has the +expected cohomology groups and endomorphism ring, as the following result shows. +Theorem B (Proposition 6.3, Theorem 6.5 and Proposition 6.6). If g = sl(2) then +we have Ψn(�Vλ,µ +2 +) = 0 for n ̸= 0. Moreover, we have +Endˆg2(�Vλ,µ +2 +) ≃ Zλ,µ +2 +and +Ψ0(�Vλ,µ +2 +) ≃ Zλ,µ +2 +. +We now briefly explain the connection between these results and Conjecture 3.6.1 +in Fortuna’s Thesis [3]. As a particular case the conjecture predicts an equivalence +between quasi-coherent sheaves over the space of unramified opers with two singu- +larities and the category of spherical representations over ˆg2: that is the space of +smooth representations of ˆg2 with a compatible action of J2G = G(C[[a, t]]). +The conjecture stated in [3] predicts an equivalence of similar categories not only +in the presence of two singularities but in the presence of n-possible singularities. +In particular for any finite set with n elements I we can define the space of opers +on the formal disc with n-singularities OpI and the subspace of unramified opers +Opunr +I +(see Section 3.5 in [3]). These are spaces over the product of n-copies of +the formal disc. These are easily seen to be factorization spaces, which means that + +SEMI-INFINITE COHOMOLOGY OF WEYL MODULES +3 +this spaces specialise nicely when restricted along or outside the diagonals of this +product (see Section 3.1.5 in [3]). There are not substantial differences between +the treatment we do here or in [4] of Op2 and the general case. The only minor +difference is that we fix a singularity to be 0. These spaces are indschemes, and +so we can define the categories QCoh(OpI), and QCoh(Opunr +I +) of quasi-coherent +sheaves on OpI and Opur +I +(see Section 3.5.3 in [3] for the actual definition), and the +nice factorization properties which make them factorization categories (see Section +3.1.2 in [3]). +Similarly, for a finite set I we can define a Lie algebra ˆgI and study its smooth +representations at the critical level. The objects constructed in this way live also +on the product of n copies of the formal disc, and they also have nice factorization +properties, in particular the collection of (completions of the) enveloping algebras +specialized at the critical level ˆUI of the algebras ˆgI, is what is called a factorization +algebra (see Section 3.1.3 in [3]). As a conseguence the collection of the categories of +smooth representations at the critical of the Lie algebras ˆgI, denoted by ˆgI,crit-mod +and their subcategories of spherical representations ˆgI,crit-modJG can be organized +also in a factorization category. The semi-infinite cohomology can be defined also +in this generality and defines a functor +ΨI : ˆgI,crit-mod −→ D(QCoh(OpI)) +compatible with the factorization properties. While in Fortuna’s thesis all these +constructions are obtained somehow for free using the language of chiral algebras +(see Section 3.1.6 in [3]), in this paper we use the language of vertex algebras and +the formalism introduced by Casarin [1]. Let us notice that, from this point of +view, there are no differences in treating the case with two singular points and the +case with an arbitrary finite number of singular points. For example, the proof of +Theorem A above can be repeated verbatim in the case of n singular points. More +generally we believe that all the technical difficulties in the study of this problem +already appear in the case of two singularities. +It is easy to see from the factorization properties and the analogous statement +for the case of one singularty by Frenkel and Gaitsgory (see [7]) that the semi- +infinite cohomology of a ˆgI-spherical module is supported on Opunr +I +. Hence semi- +infinite cohomology restricts to a functor ΨI : ˆgI,crit-modJG −→ D(QCoh(Opunr +I +)). +Conjecture 3.6.1 in [3] states that this functor is exact and +Ψ0 +I : ˆgI,crit-modJG −→ QCoh(Opunr +I +) +is an equivalence of categories. In Appendix A.2 of [3], one possible strategy to +prove this conjecture is sketched, using the factorization structure and the result +proved in the case of one singularity to deduce the general case. In particular, +thanks to Proposition A.2.2 and Proposition A.2.3 of [3] and Theorems A and B +above, a more careful study of the modules Vλ,µ +2 +or �Vλ,µ +2 +might help in finding a +proof of [3, Conjecture 3.6.1], in the case of g = sl(2). +The paper is organized as follows. In the first section we recall some definitions +from [4]. In Section 3 we recall the formalism introduced by Casarin [1] and we +use it to define semi-infinite cohomology and prove some of its basic properties. In +Sections 3 and 4 we compute the semi-infinite cohomology of Vλ,µ +2 +and in Section +5 we compute the semi-infinite cohomology of �Vλ,µ +2 +. +We thank Luca Casarin for many useful discussions and in particular for explain- +ing to us the formalism introduced in [1]. It seems to us that Casarin’s approach +provides a natural framework to treat questions concerning opers with several singu- +larities, making the theory much more transparent than it was in [4]. In particular, + +4 +FORTUNA, LOMBARDO, MAFFEI, MELANI +the results of [1] allowed us to streamline several arguments and calculations which +would have been quite hard to carry out using the direct approach of [4]. +2. Basic constructions +In this section we recall some basic constructions from [4], to which we refer +for further details, and we introduce the notion of semi-infinite cohomology in the +context of affine Lie algebras with more than one singular point. +2.1. Rings. We follow [4, Section 1], to which the reader is referred for more details. +We introduce the rings +A = C[[a]], +Q = C((a)), +R2 = C[[t, s]], +K2 = C[[t, s]][1/ts], +where a = t − s. Recall that we have expansion maps (given by suitable natural +inclusions) and a specialisation map (which sends a to 0 and t, s to t, see Section +1.1 in [4]) +Et : K2[a−1] −→ Q((t)), +Es : K2[a−1] −→ Q((s)), +Sp : K2 −→ C((t)). +We also write E = Et × Es : K2[a−1] −→ Q((t)) × Q((s)). Recall from [4, Section +1.1] that Sp induces an isomorphism K2/(a) ≃ C((t)). These rings have natural +topologies: with respect to these, the image of E is dense, and E(R2[a−1]) is dense +in Q((t)) × Q((s)). +These rings are also equipped with residue maps +Res2 : K2 → A +Res1 : C((t)) → C, +Rest : Q((t)) → Q, +Ress : Q((s)) → Q, +which behave nicely with respect to specialisation and expansion (see [4, Section +1.2]). Finally, we recall Lemma 1.10 in [4]. +Lemma 2.1 ([4], Lemma 1.10). Let M, N be two A-modules and ϕ : M −→ N be +a morphism of A-modules. Then +a) if M is flat and ϕa : M[a−1] −→ N[a−1] is injective, then ϕ is injective. +b) if N is flat, ϕa : M[a−1] −→ N[a−1] is surjective, and ϕ : M/aM −→ N/aN is +injective, then ϕ is surjective. +In particular, if M and N are flat, ϕa : M[a−1] −→ N[a−1] is an isomorphism, +and ϕ : M/aM −→ N/aN is injective, then ϕ is an isomorphism. +2.2. Affine Lie algebras and completion of the enveloping algebra. We +follow [4, Section 3]. Let g be a finite-dimensional Lie algebra over the complex +numbers and denote by κ the Killing form of g. Recall from [4, Sections 3.1 and 3.3] +that for each of the rings of the previous section we introduce an affine Lie algebra: +ˆg1 is the usual affine Lie algebra (we take for convenience the version defined by +Laurent polynomial and not Laurent series), ˆgt and ˆgs are also versions of the the +usual affine Lie algebra, while ˆg2 is an A-Lie algebra having as underlying A-module +the space +ˆg2 = C[t, s][1/ts] ⊗C g ⊕ A C2. +We also introduce the Lie algebra ˆgt,s = ˆgt ⊕ ˆgs/(Ct − Cs) (see [4, Section 3.3]). +For each of these Lie algebras, we introduce the corresponding universal envel- +oping algebra, which we suitably complete and then specialize at the critical level +by imposing that the central element acts as −1/2 (see Sections 3.1 and 3.3 in [4]). +In particular +ˆU2 = lim +←− +n +U(ˆg2) +(C2 = −1/2, tnsnC[t, s] ⊗ g)left.id. +Recall from [4, Section 3.4] that the expansion maps and the specialisation maps +induce morphisms at the level of Lie algebras. +In particular, the specialisation +map Sp : ˆU2 −→ ˆU1 induces an isomorphism between ˆU2/a ˆU2 and ˆU1, while the + +SEMI-INFINITE COHOMOLOGY OF WEYL MODULES +5 +expansion map induces a morphism E : ˆU2[a−1] −→ ˆUt,s which is injective and has +dense image. +Moreover, the natural inclusions ˆgt ֒→ ˆgt,s and ˆgs ֒→ ˆgt,s induce a morphism +ˆUt ⊗ ˆUs −→ ˆUt,s +which is also injective and with dense image (see [4, Section 3.3]). +2.3. Weyl modules. We follow [4, Section 6]. We choose a Borel subalgebra and +a maximal toral subalgebra of g, which we denote by b and t respectively. This +data induces a choice of weights, integral weights and dominant weights. For every +integral dominant weight λ, [7] introduced the Weyl module Vλ +1 over the affine Lie +algebra ˆg1. The representation V = V0 +1, which has a structure of vertex algebra, will +play a particularly important role for us. This vertex algebra enjoys the following +universal property. +Lemma 2.2. Let U be a vertex algebra such that there exists a linear map x �→ ux +from g to U such that +(ux)(0)(uy) = u[x,y] +(ux)(1)(uy) = −1 +2κ(x, y)|0⟩U +(ux)(n)(uy) = 0 +for all n ⩾ 2. There exists a unique morphism of vertex algebras α : V → U such +that α(xt−1|0⟩V) = ux for all x ∈ g. +Weyl modules Vλ +t and Vλ +s can also be defined for the Lie algebras ˆgt and ˆgs, +without any significant change from [7]. In [4], we introduced a generalization of +these modules. Given two dominant weights λ, µ, we consider the irreducible repres- +entations V λ and V µ of the Lie algebra g having highest weights λ, µ, respectively. +In [4, Definition 6.2], given two dominant integral weights λ, µ we introduced the +module +Vλ,µ +2 += Indˆg2 +ˆg+ +2 +� +A ⊗C V λ ⊗C V µ� +, +where ˆg+ +2 = C[t, s] ⊗ g ⊕ A C2 acts on A ⊗C V λ ⊗C V µ as +f(t, s)x · (p(a) ⊗ u ⊗ v) = f(0, −a)p(a) ⊗ xu ⊗ v + f(a, 0)p(a) ⊗ u ⊗ xv, +while C2 acts as −1/2. In [4] we called this object the Weyl module of weights +(λ, µ), although, as we will see, it does not have the the same properties as its +1-singularity analogue. +We also define +Wλ,µ +1 += Indˆg1 +ˆg+ +1 +� +V λ ⊗C V µ� +, +where ˆg+ +1 = C[t] ⊗ g ⊕ C C1 acts on V λ ⊗C V µ as f(t)x · (u ⊗ v) = f(0)x · (u ⊗ v) +and C1 acts as −1/2. +The specialisation and expansion maps are defined also for Weyl modules, and +induce the following isomorphisms [4, Lemma 6.3]: +Vλ,µ +2 +aVλ,µ +2 +≃ Wλ,µ +1 +, +Vλ,µ +2 +[a−1] ≃ Vλ +t ⊗Q Vµ +s . +(2.1) +2.4. Clifford algebra. We now define the Clifford algebra with two singularities, +generalizing the construction of the classical case (see for example [5, Chapter 15]). +Let n+ be the nilpotent radical of b and set +X2 = K2 ⊗C n+ ⊕ K2 ⊗C n∗ ++. +We equip X2 with the unique A-bilinear form such that K2 ⊗C n+ and K2 ⊗C n∗ ++ +are isotropic subspaces and +(f ⊗ x; g ⊗ ϕ) = Res2(fg) ϕ(x) + +6 +FORTUNA, LOMBARDO, MAFFEI, MELANI +for all f, g ∈ K2, x ∈ n+ and ϕ ∈ n∗ ++. We denote by Cℓ2 the associated Clifford +algebra over A. +There are obvious variants of the same construction where we replace K2 with the +ring C[t±1] or one of the rings Q[t±1], Q[s±1], Q[t±1] × Q[s±1]. We obtain Clifford +algebras that we denote by Cℓ1, Cℓt, Cℓs, Cℓt,s. The algebra CℓU in [5, Section 15.1.1] +is a completion of Cℓ1. +These Clifford algebras have a natural grading called the charge and denoted by +ch. It can be defined as follows: the elements of the base ring have charge 0, while +for ψ ∈ n and ψ∗ ∈ n∗ we have +ch ψ = −1, +ch ψ∗ = 1. +(2.2) +The relations defining each Clifford algebra are homogeneous, hence the charge +induces a well-defined grading on the Clifford algebra. +We now introduce completions of the tensor product ˆU2 ⊗A Cℓ2. We define +ˆU2 ˆ⊗ACℓ2 = lim +←− +n +ˆU2 ⊗A Cℓ2 +� +(ts)nR2g ⊗ 1, 1 ⊗ (ts)nR2n+, 1 ⊗ (ts)nR2n∗ ++ +� +left ideal +and we notice that, as in the case of the algebra ˆU2, this A-module has a nat- +ural structure of A-algebra. We introduce the completed Clifford algebras ˆU1 ˆ⊗Cℓ1, +ˆUt ˆ⊗QCℓt, ˆUs ˆ⊗QCℓs, and ˆUt,s ˆ⊗QCℓt,s. The specialisation and expansion map de- +termine morphisms +Sp : ˆU2 ˆ⊗ACℓ2 −→ ˆU1 ˆ⊗Cℓ1 +and +E : ( ˆU2 ˆ⊗ACℓ2)[a−1] −→ ˆUt,s ˆ⊗QCℓt,s. +Arguing exacly as in [4, Lemmas 3.7 and 3.9] we see that E is injective with dense +image, while the specialisation map induces an isomorphism ˆU2 ˆ⊗ACℓ2/a( ˆU2 ˆ⊗ACℓ2) ≃ +ˆU1 ˆ⊗Cℓ1. Finally, we have an injective map I : ˆUt ˆ⊗QCℓt → ˆUt,s ˆ⊗QCℓt,s induced by +the natural inclusion Kt → Kt,s = Kt × Ks given by f �→ (f, 0). Similarly, we have +an injective map J : ˆUs ˆ⊗QCℓs → ˆUt,s ˆ⊗QCℓt,s. As in Section 3.3 of [4], the product +of these maps I ⊗ J : ( ˆUt ˆ⊗QCℓt) ⊗Q ( ˆUs ˆ⊗QCℓs) → ˆUt,s ˆ⊗QCℓt,s is injective with +dense image. +2.5. Fock module. We now describe the “fermionic” Fock spaces corresponding to +the Clifford algebras defined in the previous section. As above, for the construction +in the case of one singularity we refer to [5, Section 15.1.4]: here we mimic this +definition in the case of two singularities. We define Cℓ+ +2 as the A-subalgebra of Cℓ2 +generated by R2 ⊗ n+ and R2 ⊗ n∗ ++ and we define the Fock module +Λ2 = Cℓ2 ⊗Cℓ+ +2 A |0⟩Λ2 +where R2 ⊗ n+ and R2 ⊗ n∗ ++ acts trivially on |0⟩Λ2. The charge (see equation (2.2)) +induces a grading on the Fock space by setting +ch |0⟩Λ2 = 0. +We denote by Λn +2 the subspace of elements of degree n. Similar constructions can be +given for all the other Clifford algebras Cℓ1, Cℓt, Cℓs, and Cℓt,s, giving Fock modules +Λ1, Λt, Λs, and Λt,s. +Specialisation and expansion, induce maps also at the level of the Fock spaces. +Arguing as in [4, Section 6] (where we considered the module Vλ,µ +2 +), it is easy to +prove the following Lemma: +Lemma 2.3. +a) The specialisation map Sp : Λ• +2 −→ Λ• +1 is homogeneous of degree zero and +induces an isomorphism Λ• +2/aΛ• +2 ≃ Λ• +1. +b) We have a homogeneous isomorphism of degree zero Λ• +t,s ≃ Λ• +t ⊗Q Λ• +s. + +SEMI-INFINITE COHOMOLOGY OF WEYL MODULES +7 +c) The expansion map E : Λ• +2[a−1] −→ Λ• +t ⊗Q Λ• +s is a homogeneous isomorphism +of degree zero. +Recall also that the Fock space Λ = Λ1 has a natural structure of vertex super- +algebra with the following universal property. +Lemma 2.4. Let U be a vertex superalgebra such that there exists a linear map +x �→ ux from n∗ ++ ⊕ n∗ ++ to the space of odd elements of U such that +(1) for all ϕ, ψ ∈ n and for all ϕ∗, ψ∗ ∈ n∗ ++ +(uψ)(n)(uϕ) = (uψ∗)(n)(uϕ∗) = (uψ)(m)(uψ∗) = (uψ∗)(m)(uψ) = 0 +for all n ⩾ 0 and for all m ⩾ 1; +(2) (uψ)(0)(uψ∗) = (uψ∗)(0)(uψ) = ⟨ψ, ψ∗⟩|0⟩U for all ψ ∈ n and ψ∗ ∈ n∗ ++. +Then there exists a unique morhism of vertex superalgebras α : Λ → U such that +α(ψt−1|0⟩Λ) = uψ and α(ψ∗t−1|0⟩Λ) = uψ∗. +2.6. Bases. For each of the objects introduced above – base rings, enveloping al- +gebras, Clifford algebras, and Fock spaces – it is not hard to construct explicit +bases (or topological bases). We give the details in the case of two singularities. +The construction of a basis depends on the choice of a basis of C[t, s][1/ts] as an +A-module. Following [4], Section 1.1 and Equation (4.1) we introduce the following +bases, indexed by 1 +2Z: for n ∈ Z we define +� +zn = tnsn +zn+ 1 +2 = tn+1sn +� +wn = tnsn +wn+ 1 +2 = tnsn+1 +The elements zm for m ∈ +1 +2Z form a basis of C[t, s][1/ts] as an A-module, and +the elements wn are the dual basis with respect to the residue bilinear form: more +precisely, one has +Res2(znw−m− 1 +2 +) = δn,m. +This specific choice of basis is not particularly important, and several others would +be possible. However, some properties need to be satisfied for our approach to +work. In particolar with our choice, the elements zm (or wm) with m ⩾ 0 form an +A-basis of C[t, s]. +Since K2 is an A-free module, we deduce that the enveloping algebras of g2 and +Cℓ2 are A-free modules. Moreover, as R2 is a direct summand of K2, we also deduce +that Vλ,µ +2 +and Λ2 are also A-free modules. Explicit bases of these modules, as well +as an explicit topological basis of the algebra ˆU2 ˆ⊗ACℓ2, can be obtained using the +Poincar´e-Birkhoff-Witt theorem and its analogue for Clifford algebras. +3. Vertex algebras and semi-infinite cohomology +In this section, we recall some results obtained by Casarin [1] which allow us to +use the formalism of vertex algebras also in the context of several singularities. In +particular, using this formalism we develop a notion of semi-infinite cohomology for +ˆU2-modules. +3.1. Distributions and vertex algebra morphisms. Let R be a complete topo- +logical associative A-algebra. Following [1, Definition 3.0.4], we denote by FA(K2, R) +the space of continuous A-linear morphisms from K2 to R and call it the space of +2-fields. +We refer to [1] for the definitions of mutually local 2-fields (Definition +3.1.1), of the n-products X(n)Y of two 2-fields (Definitions 3.1.2 and 3.1.7) and +of the derivative ∂(X) of a 2-field (before Definition 3.0.2). The definition in [1] +applies also to the other ring we are considering: K1, Kt, Ks, Kt,s. + +8 +FORTUNA, LOMBARDO, MAFFEI, MELANI +In particular to define n products it is necessary to choose what in [1] is called +a global coordinate (see definition ). We choose always t as a global coordinate. +More explicitly for the rings K2, K1, Kt and Ks we choose t = s + a as a global +coordinate, and for the ring Kt,s = Kt × K2 we choose (t, t) = (t, s + a). +We also use some foundational results proved in this context in [1]. In particular, +the following result will be crucial for us. +Theorem 3.1 ([1], Theorem 3.2.3). Let F be a C-linear subspace of FA(K2, R) of +mutually local 2-fields closed under derivation and n-products. Let +1 be a field such +that +1(f) is central for every f ∈ K2, that ∂ +1 = 0 and such that +1(n)X = δn,−1X for +all X ∈ F. Then the vector dpace F + C1, endowed with n-products and derivation +T = ∂, is a C-vertex algebra with +1 as vacuum vector. +It is straightforward to generalize the constructions and results in [1] to the case +of superalgebras R. +We are interested in the case where R is the superalgebra ˆU2 ˆ⊗ACℓ2. For x ∈ g, +ψ ∈ n+ and ψ∗ ∈ n∗ ++ we define the 2-fields +x(2)(g) = (x⊗g)⊗1Cℓ2, +ψ[2](g) = 1 ˆU2⊗(ψ⊗g), +(ψ∗)[2](g) = 1 ˆU2⊗(ψ∗⊗g) (3.1) +for all g ∈ K2. The first of these fields has even parity with respect to the superal- +gebra structure, while the second and third ones are odd. These fields are mutually +local. We consider the minimal C-linear subspace F(2) of ˆU2 ˆ⊗ACℓ2 closed under +n-products and derivation and containing the fields (3.1). Moreover, we define +12(f) = Res2(f) +� +1 ˆU2 ⊗ 1Cℓ2 +� +. +It is easy to check that this data satisfies the hypothesis of Theorem 3.1. Therefore, +V(2) = F(2) + C12 has a structure of vertex superalgebra, and by the universal +properties of the vertex algebra V (Lemma 2.2) and of the vertex superalgebra Λ• +(Lemma 2.4) it follows that there exists a morphism of vertex superalgebras +Φ(2) : V ⊗C Λ• −→ V(2). +(3.2) +This homomorphism will allow us to easily introduce many elements in V(2), hence +also in ˆU2 ˆ⊗ACℓ2. +Similar constructions apply if the algebra ˆU2 ˆ⊗ACℓ2 is replaced by the algebras +ˆU1 ˆ⊗Cℓ1, ˆUt ˆ⊗QCℓt, etc. Hence, we construct the fields x(1), ψ[1], x(t), ψ[t], the vertex +superalgebras V(1), V(t), and homomorphisms of vertex algebras Φ(1) : V⊗C Λ• −→ +V(1), Φ(t) : V ⊗C Λ• −→ V(t), etc. +Notice that we have a specialisation morphism SpF : FA(K2, ˆU2 ˆ⊗Cℓ2) −→ +FC(K1, ˆU2 ˆ⊗Cℓ1) and an expansion map EF : FA(K2, ˆU2 ˆ⊗ACℓ2) −→ FQ(Kt,s, ˆU2 ˆ⊗QCℓt,s), +determined by the conditions +� +SpF(X) +� +(Sp(f)) = Sp(ϕ(f)) +and +� +EF(X) +� +(E(f)) = E(ϕ(f)). +These maps commute with n-products and derivations and satisfy SpF(12) = +11 +and EF(12) = +1t,s. Moreover, by construction they satisfy +SpF(x(2)) = x(1) +and +EF(x(2)) = x(t,s) +for x ∈ g. Similar relations hold for ψ[2] and (ψ∗)[2]. This implies in particular that +the homomorphisms SpF and EF restrict to homomorphisms of vertex algebras +Sp : V(2) −→ V(1) and E : V(2) −→ V(t,s) such that +Sp ◦Φ(2) = Φ(1) +E ◦ Φ(2) = Φ(t,s). +We can also describe the morphism Φ(2) through the morphisms Φ(t) and Φ(s). +Recall from the end of Section 2.4 the maps I, J from ˆUt ˆ⊗QCℓt and ˆUs ˆ⊗QCℓs to +ˆUt,s ˆ⊗QCℓt,s. These maps induce maps at the level of fields IF : FQ(Kt, ˆUt ˆ⊗QCℓt) → + +SEMI-INFINITE COHOMOLOGY OF WEYL MODULES +9 +FQ(Kt,s, ˆUt,s ˆ⊗QCℓt,s) and JF : FQ(Ks, ˆUs ˆ⊗QCℓs) → FQ(Kt,s, ˆUt,s ˆ⊗QCℓt,s), given +by +IF(X)(f, g) = I(X(f)) +and +JF(X)(f, g) = J(X(g)) +for all (f, g) ∈ Kt×Ks = Kt,s. The maps IF and JF preserve n-products, commute +with derivations, and satisfy IF(1t) + JF(1s) = +1t,s. Moreover we notice that I(u) +and J(v) commute for all u ∈ ˆUt ˆ⊗QCℓt and v ∈ ˆUs ˆ⊗QCℓs. By the discussion in [1, +Section 7.2], this implies +IF ◦ Φ(t) + JF ◦ Φ(s) = Φ(t,s). +This is the only statement where it is relevant the choice of the global coordinate +we have done in Section 3.1. +3.2. Semi-infinite cohomology. We now define a notion of semi-infinite cohomo- +logy for ˆU2-modules, in analogy with the analogous notion for ˆU1-modules described +for example in [5, Chapter 15]. To this end, we introduce some notation for elements +in the vertex superalgebra V ⊗ Λ•. As in the the case of ˆU1, to describe these ele- +ments we choose a basis Ja of g compatible with the decomposition g = n−⊕t⊕n+, +where n+ is the nilpotent radical of b and n− is the radical of the opposite nilpotent +borel subalgebra. We denote by cb,d +e +the structure coefficients of the Lie bracket +with respect to this basis. We denote by Φ ⊔ Γ the indexing set of the basis Ja, so +that, if α ∈ Φ, then Jα = eα = f−α is a root vector of weight α and, if α ∈ Γ, then +Jα ∈ t. We also denote by ψ∗ +α for α ∈ Φ+ the basis of n∗ ++ dual to the basis eα of +n+. +With each element in n+ ⊗ · · · ⊗ n+ ⊗ n∗ ++ ⊗ · · · ⊗ n∗ ++ we associate an element in +the vertex superalgebra Λ as follows: +N(ψ1 ⊗ . . . ψℓ ⊗ ψ∗ +1 ⊗ · · · ⊗ ψ∗ +m) = (ψ1t−1) · · · (ψℓt−1) · (ψ∗ +1t−1) · · · (ψ∗ +mt−1) · |0⟩Λ. +Similarly, with an element in g ⊗ n∗ ++ we associate an element in the vertex +superalgebra V ⊗ Λ∗ by setting +M(x ⊗ ψ∗) = (xt−1) · |0⟩V ⊗ (ψ∗t−1) · |0⟩Λ. +Following [5, Chapter 15] we define +q =M(I) − 1 +2|0⟩V ⊗ N(B) = +� +α∈Φ+ +(eαt−1) · |0⟩V ⊗ (ψ∗ +αt−1) · |0⟩Λ +− 1 +2 +� +α,β∈Φ+ +cα,β +α+β |0⟩V ⊗ (eα+βt−1) · (ψ∗ +αt−1) · (ψ∗ +βt−1) · |0⟩Λ, +where I ∈ g ⊗ n∗ ++ represents the inclusion of n+ in g and B ∈ n+ ⊗ n∗ ++ ⊗ n∗ ++ is the +Lie bracket. We now define the boundary operator d(2) +std ∈ ˆU2 ˆ⊗ACℓ2 as follows: +d(2) +std := +� +Φ(2)(q) +� +(1). +The boundary operator that we will use to define the semi-infinite cohomology +is a deformation of d(2) +std. Let ψ∗ +pr = � +α simple ψ∗ +α ∈ n∗ ++, and define +χ(2) = 1 ˆU2 ⊗ ψ∗ +pr = Φ(2)(N(ψ∗ +pr))(1) ∈ ˆU2 ⊗A Cℓ2. +Similar constructions yield χ(s), χ(t), χ(s), and χ(s,t). Finally set +d(2) = d(2) +std + χ(2). +As we will check in Section 3.3, this is an element that squares to zero, and therefore, +it can be used to define the semi-infinite cohomology of a ˆU2-module. + +10 +FORTUNA, LOMBARDO, MAFFEI, MELANI +Similarly we can define d(1) +std, χ(1), d(1), d(t) +std, χ(t), d(t), and so on, as elements of +the corresponding superalgebras. By the discussion at the end of Section 3.1 we +have +Sp(d(2)) = d(1), +E(d(2)) = d(t,s), +and +I(d(t)) + J(d(s)) = d(t,s). +Definition 3.2. Let M be an ˆU2 module. Consider the ˆU2 ˆ⊗ACℓ2-graded module +M ⊗AΛ• +2, where the grading is given by charge on Λ• +2. The element d(2) acts on this +module as a boundary operator of degree one. Define Ψn(M) as the corresponding +cohomology of degree n. +Similar constructions apply to modules over the algebras ˆU1, ˆUt, ˆUs or ˆUt,s. +Let Z2 be the center of the algebra ˆU2, and similarly introduce the center Z1 of +ˆU1 and the centers Zt and Zs of ˆUt and ˆUs. If M is an ˆU2-module, the action of Z2 +on M ⊗A Λ• +2 commutes with the differential d(2) and preserves the charge, hence +induces an action of Z2 on the semi-infinite cohomology groups of M. A similar +action is defined in the case of ˆU1-modules or ˆUt-modules. +Recall that a module M over a topological algebra R is said to be smooth if the +action of R on M is continuous with respect to the discrete topology on M. Notice +that, if M is a smooth ˆU2-module, then, since the map E has dense image, the +action of ˆU2 on M extends to a smooth action of ˆUt,s on M[a−1]. Similarly, if Mt +is a smooth ˆUt-module and Ms is a smooth ˆUs-module, then there is an induced +action of ˆUt,s on Mt ⊗Q Ms. In the next section we will use the following properties +of the semi-infinite cohomology. +Lemma 3.3. +a) Given a short exact sequence of ˆU2-modules, there is an induced long exact +sequence in semi-infinite cohomology. +b) Let M be an ˆU1-module. The semi-infinite cohomology of M as an ˆU1-module +is isomorphic to the semi-infinite cohomology of M considered as an ˆU2- +module through the map Sp. +c) Let M be an ˆUt,s-module. The semi-infinite cohomology of M as an ˆUt,s- +module is isomorphic to the semi-infinite cohomology of M considered as an +ˆU2-module through the map E. In particular, this applies to the case where +M = N[a−1] is the localization of a smooth ˆU2-module N. +d) Let Mt be a smooth ˆUt-module, Ms be a smooth ˆUs-module, and let M := +Mt ⊗Q Ms, regarded as a ˆUt,s-module. +The complex computing the semi- +infinite cohomology of M is the total complex associated with the double com- +plex given by the tensor product of the complex computing the semi-infinite +cohomology of Mt and that of Ms. In particular, being the base ring Q a +field, if Mt and Ms have non zero semi-infinite cohomology only in degree +zero, then M considered as an ˆUt,s-module has semi-infinite cohomology only +in degree zero and the cohomology in degree zero is isomorphic to the product +of the tensor product of Ψ0(Mt) and Ψ0(Ms). +Proof. Part a) follows from the fact that Λ2 is a free module over A. +Part b) follows from the fact that, since a ∈ A acts trivially on M, by Lemma +2.3 a) we have +M ⊗A Λ• +2 ≃ M ⊗C +Λ• +2 +aΛ• +2 +≃ M ⊗C Λ• +1 +and moreover, by construction, d(1) = Sp(d(2)). +Part c) follows from the fact that, since the action of a on M is invertible, by +Lemma 2.3 c) we have +M ⊗A Λ• +2 = M ⊗A Λ• +2[a−1] = M ⊗A Λ• +t,s + +SEMI-INFINITE COHOMOLOGY OF WEYL MODULES +11 +and, moreover, by construction, d(t,s) = E(d(2)). +Finally, from Lemma 2.3 c) we have +(Mt ⊗Q Λ• +t) ⊗Q (Ms ⊗Q Λ• +s) ≃ M ⊗Q Λ• +t,s. +Part d) then follows from the equality d(t,s) = I(d(t)) + J(d(s)). +□ +3.3. Commutation relations. For their computation of the semi-infinite cohomo- +logy of V, Frenkel and Ben Zvi (see [5] Chapter 15) relied on the choice of a clever +basis of V ⊗ Λ. For all x ∈ g, they define +ˆx = xt−1 · |0⟩V ⊗ |0⟩Λ + N(αx), +where αx ∈ n∗ ++⊗n∗ ++ represents the linear map n+ → n+ obtained as the composition +of adx : n+ −→ n+, the natural projection π : g −→ g/b−, and the inverse of the +isomorphism n+ ∼= g/b− induced by π. Using the map Φ(2) from Equation (3.2) we +define +ˆx(2) = Φ(2)(ˆx). +To compute the semi-infinite cohomology of Vλ,µ +2 +we will need some information +about the commutation relations among the elements ˆx(2), ψ[2], and (ψ∗)[2], and +the boundary operators. These are easy to compute because all these objects are +constructed through the map Φ(2). Let us make this remark precise. Given an +element x in V⊗Λ, denote by x(z) the corresponding field in the vertex superalgebra +and by x(2) : K2 −→ ˆU2 ˆ⊗Cℓ2 the 2-field Φ(2)(x). +For any choice of elements +x, y ∈ V ⊗ Λ, the commutator of the corresponding fields is given by +[x(z), y(w)] = +� +n⩾0 +1 +n!(x(n)y)(w)∂n +wδ(z − w). +We have a similar Operator Product Expansion formula for 2-fields (see [1], Pro- +position 3.1.4) +[x(2)(f), y(2)(g)] = +� +n⩾0 +1 +n! +� +(x(2))(n)(y(2)) +� +(g ∂nf), +where the product (x(2))(n)(y(2))) is the product of 2-fields defined in [1]. However, +since Φ(2) is a map of vertex algebras we get (x(2))(n)(y(2)) = (x(n)y)(2). Hence, if +we know the commutator of x(z), y(w), we immediately deduce that of x(2) and +y(2). +Similar considerations apply when we want to compute [x(2)(1), y(2)(g)] assuming +we know the commutator of x(0) and y(w). In this case, the usual OPE formula +gives [x(0), y(w)] = (x(0)y)(w), while the OPE formula for 2-fields gives +[x(2)(1), y(2)(g)] = +� +(x(2))(0)(y(2)) +� +(g). +Using again the fact that Φ(2) is a map of vertex algebras, we get +[x(2)(1), y(2)] = Φ(2) +�� +[x(0), y(w)](|0⟩V ⊗ |0⟩Λ) +� +|w=0 +� +. +These formulas are enough to determine all commutation relations among the ele- +ments ˆx(2), ψ[2], (ψ∗)[2] and the boundary operators from those obtained by Frenkel +and Ben Zvi in [5, Chapter 15], without the need of any further computation. We +summarise these results in Proposition 3.4 below, which (in light of the above) +follows from Sections 15.2.4 and 15.2.9 of [5]. +In the statement, we denote by +epr, hpr, fpr the sl(2)-triple such that fpr = � +α simple λαfα, κ(fpr, eα) = 1 for all +simple root α and hpr ∈ t. + +12 +FORTUNA, LOMBARDO, MAFFEI, MELANI +Proposition 3.4. for all x ∈ g, y ∈ b, z ∈ n+, w ∈ b−, ψ ∈ n+ and ψ∗ ∈ n∗ ++ we +have: +a) +(d(2) +std)2 = 0, +[d(2) +std, χ(2)]+ = 0, +b) +(χ(2))2 = 0, +(d(2))2 = 0, +c) [χ(2), ψ[2]]+ = ⟨ψ∗ +pr, ψ⟩ +1, +[χ(2), (ψ∗)[2]]+ = 0, +d) +[χ(2), ˆz(2)] = 0, +[χ(2), ˆw(2)] = +� +α ∈Φ+ +κ([fpr, z], eα)ψ∗ +α, +e) [d(2) +std, ψ[2]]+ = ˆψ(2), +[d(2) +std, (ψ∗)[2]]+ = −1 +2Φ +� +1 ˆU2 ⊗ N(ψ∗ ◦ B) +� +, +f) [d(2) +std, ˆy(2)]+ = 0 +where in the second formula of e) the element ψ∗ ◦ B ∈ n∗ ++ ⊗ n∗ ++ represents the +composition of the bracket with the map ψ∗. Moreover, if we choose a basis Ja as +at the beginnin of Section 3.2, for all γ ∈ Φ+ we have +[d(2) +std, ˆf (2) +γ ]+ = +� +α∈Φ+,a∈Φ−⊔Γ +cα,−γ +a +( ˆJa)(2) +(−1)(ψ∗ +α)[2] +− 1 +2 κ(e−γ, fγ) ∂(ψ∗ +−γ)[2] − +� +α,β∈Φ+, a∈Φ⊔Γ +cα,a +β cβ,−γ +a +∂(ψ∗ +α)[2] +By specialisation and localization we obtain that similar formulas hold also in +the case of our various other superalgebras ˆUt ˆ⊗Cℓt, ˆUt,s ⊗ Cℓt,s, . . . +4. The semi-infinite cohomology of Vλ,µ +2 +In this section we compute the semi-infinite cohomology of Vλ,µ +2 +. We denote by +C• +2 = C• +2(λ, µ) the complex Vλ,µ +2 +⊗A Λ• +2 and similarly we introduce the complexes +C• +t = C• +t (λ) = Vλ +t ⊗Q Λ• +t and C• +s = C• +s (µ) = Vµ +s ⊗Q Λ• +s. We further introduce +the complexes C• +1(ν) = Vν +1 ⊗C Λ• +1 and C• +1(λ, µ) = Wλ,µ +1 +⊗C Λ• +1. Hence, we have +C• +1(λ, µ) ≃ ⊕C• +1(ν), where the sum ranges over the irreducible factors of V λ ⊗ V µ +counted with multiplicity. +We denote by Op1 the indscheme of opers on the punctured disc and, for every +integral dominant weight ν, we write Opν +1 for the associated connected component +of the space of unramified opers without monodromy, equipped with its reduced +structure (see, for example, [7] for a more complete definition). We also denote by +vν a highest weight vector in the g-module V ν. Feigin and Frenkel [2] constructed +an isomorphism F1 : Funct(Op1) −→ Z1 between the space of functions over Op1 +and the center Z1 of ˆU1. Recall the following result, which combines Theorem 1, +Theorem 2 and the proof of Proposition 1 in [7]. +Theorem 4.1 (Frenkel and Gaitsgory [7]). The action of Z1 on Vν +1 and the Feigin- +Frenkel isomorphism induce an isomorphism +G1 : Funct(Opν +1) −→ Endˆg1(Vν +1). +Moreover, the element vν ⊗|0⟩Λ is a cocycle in C• +1(ν) and the map z �→ [z ·vν ⊗|0⟩Λ +from Z1 to Ψ0(Vν +1) induces isomorphisms of Z1-modules +Funct(Opν +1) ≃ Endˆg1(Vν +1) ≃ Ψ0(Vν +1). +Finally, Ψn(Vν +1) vanishes for all n ̸= 0. +The result of Frenkel and Gaitsgory generalises easily to the case of the modules +Vλ +t and Vµ +s . + +SEMI-INFINITE COHOMOLOGY OF WEYL MODULES +13 +By Lemma 2.1 and Lemma 2.3, as in the proof of Lemma 3.3, by the compatibility +of boundary operators we get homomorphisms of complexes Sp : C• +2 → C• +1(λ, µ) +and E : C• +2 → C• +t (λ) ⊗Q C• +s (µ). These induce isomorphisms +C• +2[a−1] ≃ C• +t (λ) ⊗Q C• +s (µ) +and +C• +2 +aC• +2 +≃ C• +1(λ, µ). +(4.1) +From these isomorphisms and Theorem 4.1 it follows easily that Ψn(Vλ,µ +2 +) is zero +for n ̸= 0, 1, and we could also get information on the cohomology in degrees zero +and one. +However, it is easier to compute these cohomology groups directly by adapting +the strategy employed by Frenkel and Ben Zvi in [5, Chapter 15]. In order to do +this, we now introduce certain subcomplexes of C• +2. We denote by 1V0,0 +2 +the element +1 ∈ A ⊗C C ⊗C C ⊂ V0,0 +2 . +Definition 4.2. We denote by E• +2 the subcomplex of C• +2(0, 0) spanned by elements +of the form +ˆx(2) +1 (g1) · · · ˆx(2) +a (ga) · 1V0,0 +2 +⊗ ψ(2) +1 (ℓ1) · · · ψ(2) +b (ℓb) · |0⟩Λ2 +(4.2) +where xi, ψi ∈ n+ and g1, . . . , ga, ℓ1, . . . ℓb ∈ K2. By the commutation relations of +Section 3.3 we see that E• +2 is a subcomplex of C• +2(0, 0). +We define also analogous complexes E• +t , E• +s and E• +1. These complexes were de- +noted by C′ in [5] and by C0 in [7]. By construction, these subcomplexes are compat- +ible with specialisation and localization, and there are isomorphisms E• +2/aE• +2 ≃ E• +1 +and E• +2[a−1] ≃ E• +t ⊗Q E• +s. +Definition 4.3. We denote by D• +2 = D• +2(λ, µ) the subcomplex of C• +2(λ, µ) spanned +by elements of the form +ˆy(2) +1 (h1) · · · ˆy(2) +c (hc) · w ⊗ (ψ∗ +1)(2)(k1) · · · (ψ∗ +d)(2)(kd) · |0⟩Λ2 +(4.3) +where w ∈ V λ ⊗ V µ, yi ∈ b− = n− + t, ψ∗ +i ∈ n∗ ++ and h1, . . . , hc, k1, . . . , kd ∈ K2. +By the commutation relations of Section 3.3 we see that E• +2 is a subcomplex of +C• +2(λ, µ). +We define also analogous complexes D• +t (λ), D• +s(µ) and D• +1(ν). These complexes +were denoted by C0 in [5] and by C′ in [7]. Finally, we denote by D• +1(λ, µ) the +analogous subcomplex of C• +1(λ, µ). By construction, these subcomplexes are com- +patible with specialisation and localization, and there are isomorphisms D• +2/aD• +2 ≃ +D• +1(λ, µ) and D• +2[a−1] ≃ D• +t (λ) ⊗Q D• +s(µ). +There is an isomorphism of complexes E• +2 ⊗ D• +2 −→ C• +2 defined by +� +x · 1V0,0 +2 +⊗ ψ · |0⟩Λ2 +� +⊗ +� +y · w ⊗ ψ∗ · |0⟩Λ2 +� +�−→ x · y · w ⊗ ψ · ψ∗ · |0⟩Λ2, +where x = ˆx(2) +1 (g1) · · · ˆx(2) +a (ga) and ψ = ψ(2) +1 (ℓ1) · · · ψ(2) +b (ℓb) are as in Equation (4.2), +y = ˆy(2) +1 (h1) · · · ˆy(2) +c (hc) and ψ∗ = (ψ∗)(2)(k1) · · · (ψ∗)(2)(kd) are as in Equation +(4.3), and w is an element of V λ ⊗ V µ. +We now compute the cohomology of the complex E• +2. We will need the following +result by Frenkel and Ben Zvi. +Lemma 4.4 ([5, Section 15.2.6]). Hn(E• +1) = 0 for n ̸= 0 and Ψ0(E• +1) = C[|0⟩V ⊗ +|0⟩Λ]. +This result generalizes easily to the case of E• +t and E• +s. Localizing and special- +izing, we deduce the following lemma. +Lemma 4.5. Hn(E• +2) = 0 for n ̸= 0 and H0(E• +2) = A[1V0,0 +2 +⊗ |0⟩Λ2]. + +14 +FORTUNA, LOMBARDO, MAFFEI, MELANI +Proof. By definition, the complex E• +2 is concentrated in non-positive degrees. Hence, +the long exact sequence induced by +0 +� E• +2 +a· +� E• +2 +� E• +1 +� 0 +implies that Hn(E• +2) is torsion free for every n, and that the specialisation of +H0(E• +2) is isomorphic to H0(E• +1). Since semi-infinite cohomology commutes with +localization (Lemma 3.3), using Lemma 2.1 and Lemma 4.4 we get the desired +result. +□ +We now compute the cohomology of D• +2. The strategy is similar, but the ar- +gument is less straightforward since we do not have an explicit representative for +H0(D• +1). Following the strategy in [5], we introduce the following bigraded struc- +ture on D• +2. Recall that the height ht(α) of a root α is equal to the sum of the +coefficients of α when written as a sum of simple roots. Let also epr, hpr, fpr be an +sl(2)-triple such that fpr = � +α simple fα and hpr belongs to t. +Definition 4.6. We define a bidegree, with values in 1 +2Z × 1 +2Z and denoted by +bideg, as follows. On elements of ˆg2, we set +bideg(x ⊗ g) = (−n, n) +if x ∈ g is such that [hpr, x] = 2 n x and g ∈ K2. We set also the bidegree of the +central element C2 ∈ ˆg2 to be (0, 0). This induces a bidegree on U(ˆg2). On the +space X2 = K2 ⊗ n+ ⊕ K2 ⊗ n∗ ++ (see Section 2.4) we define +bideg eα ⊗ g = (− ht(α), −1 + ht(α)) +bideg ψ∗ +α ⊗ g = (ht(α), 1 − ht(α)) +for α a positive root and g any element of K2. This induces a bidegree on the +Clifford algebra Cℓ2. Moreover, if W is any finite-dimensional representation of g, +then we set +bideg w = (−n, n) +if w ∈ W is such that hpr · w = 2 n w. These choices induces a bidegree on the +module C• +2(λ, µ), and the element ˆx(2)(g) is homogeneous of bidegree (−n, n) if +[hpr, x] = 2 n x. Finally, notice that if an element has bidegree (p, q), then it has +charge p + q. In particular, we introduce the submodule Dp,q +2 +of elements of Dp+q +2 +of bidegree (p, q). +We notice also that bideg d(2) +std = (0, 1) and that bideg χ(2) = (1, 0). In particu- +lar, D•,• +2 +is a double complex and D• +2 is the associated total complex. Following +Frenkel and Ben Zvi [5, Chapter 15], the cohomology of the rows of this double +complex is easy to describe. Let a be the centralizer of fpr in g. Recall from [5, +Lemma 15.1.3 and Section 15.2.9] that the space spanned by monomials of the form +(ˆp1)n1 · · · (ˆpk)nk ·|0⟩V ⊗|0⟩Λ with pi ∈ a generates a commutative vertex subalgebra +F1 of V ⊗ Λ• isomorphic to S•(a ⊗ t−1C[t−1]). As in Section 3.3, it follows that for +x, y ∈ a the fields ˆx(2) and ˆy(2) commute. +We define F2(λ, µ) as the span of elements of the form +ˆx(2) +1 (g1) · · · ˆx(2) +k (gk) · (v ⊗ |0⟩Λ2) ∈ Vλ,µ +2 +⊗A Λ• +2 +with x1, . . . , xk ∈ a and v ∈ V λ ⊗ V µ. +Notice that all these elements have +charge equal to zero, and that the space F2(λ, µ) splits as a direct sum F2(λ, µ) = +� +q F −q,q +2 +(λ, µ) according to the bidegree introduced above. Moreover, by Propos- +ition 3.4 d), these elements are annihilated by the action of χ(2). +Similarly we construct subspaces F −q,q +1 +(ν) ⊂ Vν +1 ⊗C Λ• +1, F −q,q +t +(λ) ⊂ Vλ +t ⊗Q Λ• +t , +F −q,q +s +(µ) ⊂ Vλ +s ⊗Q Λ• +s, and F −q,q +1 +(λ, µ) ⊂ Wλ,µ +1 +⊗C Λ• +1, In particular, F −q,q +1 +(λ, µ) = + +SEMI-INFINITE COHOMOLOGY OF WEYL MODULES +15 +� +ν F −q,q +1 +(ν) where the sum is over all irreducible factors of V λ ⊗C V µ. By con- +struction, the specialisation and localization maps induce isomorphisms +F −q,q +2 +(λ, µ) +aF −q,q +2 +(λ, µ) +≃ F −q,q +1 +(λ, µ) +and +F −q,q +2 +(λ, µ)[a−1] ≃ +� +b+c=q +F −b,b +t +(λ) ⊗Q F −c,c +s +(µ). +Recall the following result on the cohomology of D•,q +1 +with respect to the bound- +ary χ(1). +Lemma 4.7 ([5, Lemma 15.2.10] and [7]). Let 2pν = ⟨ν, hpr⟩. +a) Dp,q +1 (ν) = 0 for q > pν and for p < −q. In particular, Dp,q +1 += 0 for q > pλ+µ +and for p < −q; +b) Hn(D•,q +1 (ν)) = 0 for n ̸= −q. In particular, Hn(D•,q +1 (λ, µ)) = 0 for n ̸= −q; +c) The map v �→ [v] from F −q,q +1 +(ν) to H−q(D•,q +1 (ν)) is an isomorphism. +Finally, it follows from c) that the map v �→ [v] from F −q,q +1 +(λ, µ) to H−q(D•,q +1 (λ, µ)) +is also an isomorphism. +Similar results hold for the complexes D•,q +t +(λ) and D•,q +s (µ). From this result we +deduce the cohomology of the complex D•,q +q +with respect to the boundary operator +χ(2). +Lemma 4.8. Let 2p0 = ⟨λ + µ, hpr⟩ as above. +a) Dp,q +2 += 0 for q > p0 and for p < −q; +b) Hn(D•,q +2 ) = 0 for n ̸= −q; +c) The map v �→ [v] from F −q,q +2 +(λ, µ) to H−q(D•,q +2 (λ, µ)) is an isomorphism of +A-modules. +Proof. Part a) is clear for the definition of Dp,q +2 += 0. For parts b) and c), we start +by studying the localization of the cohomology groups of D•,q +2 . Equivalently, we aim +to compute the cohomology of the localization of the row D•,q +2 . This localization +can be rewritten as +� +b+c=q +D•,b +t (λ) ⊗ D•,c +s (µ). +In particular, it follows from Lemma 4.7 that its cohomology is concentrated in +degree −q, and that its cohomology in this degree is given by +� +b+c=q +F −b,b +t +(λ) ⊗ F −c,c +s +(µ), +which is the localization of F −q,q +2 +(λ, µ). +Since specialisation is compatible with +bideg, we have an isomorphism D•,q +2 /aD•,q +2 +≃ D•,q +1 (λ, µ). Using Lemma 4.7, the +associated long exact sequence shows that Hn(D•,q +2 ) is torsion-free for n ̸= −q + 1, +and that the map +ι : H−q(D•,q +2 )/aH−q(D•,q +2 ) → H−q(D•,q +1 (λ, µ)) +is injective. +We now prove c). Notice that both F −q,q +2 +(λ, µ) and H−q(D•,q +2 (λ, µ)) are torsion- +free. We have already shown that the localization of the natural maps between them +is an isomorphism. To study its specialisation, we compose it with the injection ι. +This composition is the isomorphism of the last remark of Lemma 4.7. We conclude +by applying Lemma 2.1. +In order to prove b), it is enough to notice that from the above discussion we +know that, for n ̸= −q, the module Hn(D•,q +2 ) = 0 is torsion-free, and that its +localization is trivial. +□ + +16 +FORTUNA, LOMBARDO, MAFFEI, MELANI +Let now be ϕ(q) +i +be an A-basis of F −q,q +2 +(λ, µ). Since the cohomology in degree +−q of the complex D•,q+1 +2 +is zero, there exists an element ϕ(q) +i,1 ∈ D−q−1,q+1 +2 +such +that χ(2)(ϕ(q) +i,1 ) = −d(2) +std(ϕ(q) +i ). By induction, we can construct elements ϕ(q) +i,0 = ϕ(q) +i +and ϕ(q) +i,ℓ ∈ D−q−ℓ,q+ℓ +2 +such that their sum +˜ϕ(q) +i += +p0−q +� +ℓ=0 +ϕ(q) +i,ℓ +satisfies d(2)( ˜ϕ(q) +i ) = 0. We now prove the main result of this section. +Theorem 4.9. The following hold. +a) Ψn(Vλ,µ +2 +) = 0 for n ̸= 0. +b) We have an isomorphism +Ψ0(Vλ,µ +2 +) +aΨ0(Vλ,µ +2 +) +≃ Ψ0(Wλ,µ +1 +) ≃ +� +ν +Ψ0(Vν +1) +where the sum ranges over all irreducible components V ν of V λ ⊗V µ, counted +with multiplicity. +c) The elements +� +˜ϕ(q) +i +� +are an A-basis of Ψ0(Vλ,µ). +Proof. From Lemma 4.8 we deduce that the classes of the elements ˜ϕ(q) +i +form an +A-basis of H0(D• +2), and that Hn(D• +2) = 0 for n ̸= 0. As the complex D• +2 is con- +centrated in non-negative degrees, by a standard homological argument we deduce +that Hn(Vλ,µ +2 +) is isomorphic to the n-th cohomology of the complex H0(D• +2)⊗AE• +2. +Using Lemma 4.5, we immediately obtain parts a) and c). +The second isomorphism appearing in part b) is clear, while the first follows from +a) and the long exact sequence associated with the isomorphism +C• +2 +aC• +2 +≃ C• +1(λ, µ). +□ +We will use the following Corollary in the next Section. +Corollary 4.10. The element [vλ ⊗ vµ] ∈ Ψ0(Vλ,µ) is indivisible. +Proof. By the previous theorem we can choose [vλ ⊗ vµ] as an element of a basis of +the free A module Ψ0(Vλ,µ). +□ +5. The action of the center +In this section we study the action of the center Z2 on the semi-infinite cohomo- +logy of the module Vλ,µ +2 +. +In this section we show that Vλ,µ +2 +is not a perfect analogue of the Weyl module +Vν +1. Indeed, we show that, as a Z2-module, the semi-infinite cohomology of Vλ,µ +2 +is +not isomorphic to Endˆg2(Vλ,µ +2 +) or to Funct(Opλ,µ +2 +). +We begin by observing that the module Ψ0(Vν +1) has no non-trivial Z1-equivariant +automorphisms. +First we notice, that by construction, the action of Z2 commutes with localization +and specialisation, as introduced before Equation (4.1). Concretely, we have: +Et(z · x) = Et(z) · Et(x), +Es(z · x) = Es(z) · Es(x), +Sp(z · x) = Sp(z) · Sp(x) +for all z ∈ Z2 and for all x ∈ Ψ0(Vλ,µ +2 +). +Lemma 5.1. If K : Endˆgt(Vλ +t )⊗QEndˆgs(Vµ +s ) −→ Ψ0(Vλ +t )⊗QΨ0(Vµ +s ) is a (Zt⊗Zs)- +equivariant isomorphism, then K(IdVλ +t ⊗ IdVµ +s ) = q[vλ] ⊗ [vµ] for some q ∈ Q ∖ {0}. + +SEMI-INFINITE COHOMOLOGY OF WEYL MODULES +17 +Proof. It follows from Theorem 4.1 that Endˆgt׈gs(Vλt ⊗Q Vµs) is isomorphic to +Funct(Opλ +t ×Spec Q Opµ +s ) = Funct(Opλ +t ) ⊗Q Funct(Opµ +s ) and this is a polynomial +ring in infinitely many variables over the field Q. In particular, its only invertible +elements are the non-zero scalars in Q. +Moreover, Theorem 4.1 also implies that Funct(Opλ +t ) is isomorphic as a Zt- +module to Ψ0(Vλ +t ), with an isomorphism given by z −→ Gt(z) · [vλ]. The claim +follows. +□ +Before proving that Vλ,µ +2 +does not have the “right” semi-infinite cohomology +we recall some properties of the modules Vν +1 that will be needed also in the next +section. +Remark 5.2. We denote by Zν +1 the coordinate ring of the scheme Opν +1. Recall +that the schemes Opν +1 for different values of ν are disjoint, so that the map Z1 −→ +Zν1 +1 × · · · × Zνk +1 +is surjective if the weights νi are distinct. Recall also that the ring +Zν +1 is a polynomial ring in infinitely many variables. This implies that +(1) There are no nontrivial ˆg1-morphisms between the ˆU1-modules Vν +1 and Vν′ +1 +if ν ̸= ν′. +(2) There are no nontrivial extensions between the ˆU1-modules Vν +1 and Vν′ +1 if +ν ̸= ν′. +(3) Assume that α : � Zνi −→ � Zνi is a map of Z-modules and that the +weights νi are distinct. If 1 is in the image of α then α is an isomorphism +and α(Zνi +1 ) = Zνi +1 . +By the Feigin-Frenkel Theorem (see [4] Theorem 5.2) the ring Funct(Op2) is +isomorphic to Z2. In the sequel we will identify these rings through this isomorph- +ism. +In particular the ring Funct(Opλ,µ +2 +) is a quotient of Z2. +We will denote +Funct(Opλ,µ +2 +) by Zλ,µ +2 +. +We now prove that Zλ,µ +2 +and Ψ0(Vλ,µ +2 +) are not isomorphic. +Proposition 5.3. Assume that V λ ⊗ V µ is not irreducible. +Then the two Z2- +modules Endˆg2(Vλ,µ +2 +) and Ψ0(Vλ,µ +2 +) are not isomorphic. +Similarly the two Z2- +modules Zλ,µ +2 +and Ψ0(Vλ,µ +2 +) are not isomorphic. +Proof. Suppose H : Endˆg2(Vλ,µ +2 +) −→ Ψ0(Vλ,µ +2 +) is a Z2-equivariant isomorphism. +Recall from Lemma 4.28 in [4] that Z2[1/a] is dense in Zt,s, and therefore the +localization of H is a (Zt ⊗Q Zs)-equivariant isomorphism +Endˆgt(Vλ +t ) ⊗Q Endˆgs(Vµ +s ) −→ Ψ0(Vλ +t ) ⊗Q Ψ0(Vµ +s ), +where we used the identification of the localization of Ψ0(Vλ,µ +2 +) with Ψ0(Vλ +t ) ⊗Q +Ψ0(Vµ +s ). +From Lemma 5.1 and 4.10 we deduce that H(IdVλ,µ +2 +) = [q vλ ⊗ vµ], where q ∈ A +and qvλ ⊗ vµ ∈ Vλ,µ +2 +. We set w = qvλ ⊗ vµ ∈ Vλ,µ +2 +. +By specialisation, H gives a Z1-equivariant isomorphism +H : Endˆg2(Vλ,µ +2 +) +a Endˆg2(Vλ,µ +2 +) +−→ Ψ0(Vλ,µ +2 +) +aΨ0(Vλ,µ +2 +) +. +(5.1) +This isomorphism sends IdVλ,µ +2 +to w. Now consider the decomposition V λ ⊗ V µ = +� V ν as g-modules. By Theorem 4.9, the target of the map H in (5.1) decomposes +as � Ψ0(Vν +1). The element w is a multiple of vλ ⊗ vµ hence its class belongs to +Ψ0(Vλ+µ +1 +). As H is Z1-equivariant and Vλ+µ +1 +is stable by the action of ˆg1, we get +that the image of H is contained in the direct summand Ψ0(Vλ+µ +1 +). In particular, +if V λ ⊗ V µ is not irreducible, the map H cannot be surjective. This proves the + +18 +FORTUNA, LOMBARDO, MAFFEI, MELANI +first claim. The second claim follows since the map from Zλ,µ +2 +to Ψ0(Vλ,µ +2 +) factors +through Endˆg2(Vλ,µ +2 +). +□ +6. A Weyl module for sl(2) +In this section, we propose an alternative Weyl module in the context of opers +with two singularities, in the case of g = sl(2). +We fix the following notation: +e, h, f is an sl(2)-triple such that h ∈ t and e ∈ n+, while ψ∗ ∈ n∗ ++ is the dual of +e. We identify dominant weights with natural numbers and we assume from now +on that λ ⩾ µ. In this case, the differential of the complex computing semi-infinite +cohomology takes the simpler form d(2) = ψ∗ + � ewn ⊗ ψ∗z−n−1/2. +Let �Vλ,µ +2 +be the ˆU2-submodule of Vλ,µ +2 +generated by the highest weight vector +1 ⊗ vλ ⊗ vµ ∈ A ⊗ V λ ⊗ V µ. We will prove that this module has the “correct” +semi-infinite cohomology and the “correct” endomorphism ring. +We start by giving a more explicit description of the module �Vλ,µ +2 +. If X is a +subspace of U(g) and Y is a subspace of a g-module Z we denote by X · Y the +subspace of Z generated by the products x · y with x ∈ X and y ∈ Y . We define +an increasing filtration F i of �Vλ,µ +2 +as follows +F i = U(g) · (C Id ⊗ Id + Id ⊗ g)i · (vλ ⊗ vµ). +This is an increasing filtration of V λ ⊗ V µ by g-modules and for i large enough we +have F i = V λ ⊗ V µ. Choose a g-stable complement Gi+1 of F i in F i+1 and set +G0 = F 0, so that F i = �i +j=0 Gj. If we set F i(V µ) = (CId + n−)ivµ, it is easy to +check by induction on i that +F i = U(g) · (Id ⊗ Id + Id ⊗ n−)i(vλ ⊗ vµ) = U(g) · +� +V λ ⊗ F i(V µ) +� +. +In the case of g = sl(2) we have Gi ≃ V λ+µ−2i and F µ(V µ) = V µ. +Let U − +2 +⊂ U(ˆg2) be the A-span of Poincar´e-Birkhoff-Witt monomials of the +form (x1wa1) · · · (xkwak) with xi ∈ g and ai < 0. This is a complement of U(ˆg+ +2 ) +in U(ˆg2), so that in particular we have +Vλ,µ +2 += U − +2 ⊗C (V λ ⊗ V µ). +Lemma 6.1. If λ ⩾ µ then +�Vλ,µ +2 += +µ +� +i=0 +aiU − +2 ⊗C F i = +µ +� +i=0 +aiU − +2 ⊗C Gi +Proof. To understand the module �Vλ,µ +2 +we need to compute the ˆg+ +2 -submodule of +A⊗C V λ ⊗C V µ generated by 1⊗vλ ⊗vµ. Notice that every element of the form xg, +with x ∈ g and g ∈ C[[t, s]] divisible by ts, acts trivially on A ⊗ V λ ⊗ V µ. Hence +we need to understand the action of elements of the form +z = x1 · · · xℓ · (y1t) · · · (ymt) · (vλ ⊗ vµ), +with xi, yi ∈ g. Moreover, elements of g act in the standard way on the tensor +product V λ ⊗ V µ, while elements of the form xt with x ∈ g act via −a(Id ⊗ x). +This implies the lemma. +□ +We now describe the specialisation of the module �Vλ,µ +2 +. We introduce the fol- +lowing decreasing filtration of �Vλ,µ +2 +: +Fi = �Vλ,µ +2 +∩ aiVλ,µ +2 +. +(6.1) + +SEMI-INFINITE COHOMOLOGY OF WEYL MODULES +19 +By Lemma 6.1 we have the following description of the terms of this filtration as +A-modules: +Fi = aiU − +2 ⊗C F i ⊕ +µ +� +j=i+1 +ajU − +2 ⊗C Gj +In particular we have F0 = �Vλ,µ +2 +, Fj = ajVλ,µ +2 +for j ⩾ µ. +Lemma 6.2. +a) Let ui ∈ Gi be the highest weight vector and set ˜wi = aiui ∈ +�Vλ,µ +2 +. Then ˜wi ∈ �Vλ,µ +2 +and ai−1ui /∈ �Vλ,µ +2 +. +b) There is an isomorphism of ˆU1-modules +Fi + a�Vλ,µ +2 +a�Vλ,µ +2 +≃ +µ +� +j=i +Vλ+µ−2j +1 +. +The quotient Fi+a�Vλ,µ +2 +a�Vλ,µ +2 +is generated as a ˆU1-module by the classes of ˜wi, . . . , ˜wµ. +In particular �Vλ,µ +2 +/a�Vλ,µ +2 +≃ Wλ,µ +1 +is generated by ˜w0, . . . , ˜wµ. +Proof. The first claim follows from Lemma 6.1. +We prove part b) by decreasing induction on i. By Lemma 6.1, for i > µ the +quotient is zero and the claim is true. For i ⩽ µ, consider the map +U − +2 ⊗ Gi −→ +Fi + ai+1Vλ,µ +2 +ai+1Vλ,µ +2 ++ Fi ∩ a�Vλ,µ +2 +≃ +(Fi + a�Vλ,µ +2 +)/a�Vλ,µ +2 +(Fi+1 + a�Vλ,µ +2 +)/a�Vλ,µ +2 +sending an element u ⊗ v to the class of aiu ⊗ v. This map induces an isomorphism +U − +2 +aU − +2 +⊗ Gi ≃ +(Fi + a�Vλ,µ +2 +)/a�Vλ,µ +2 +(Fi+1 + a�Vλ,µ +2 +)/a�Vλ,µ +2 +. +(6.2) +Moreover, notice that +U− +2 +aU− +2 ⊗ Gi ≃ U − +1 ⊗ Gi, where U − +1 = U(t−1g[t−1]) ⊂ U(ˆg1) = +U1, and that U − +1 ⊗ Gi has a natural structure of U1-module, as it can be identified +with Vλ+µ−2i +1 +. With this U1-action, the isomorphism 6.2 is U1-equivariant. Now +the claim follows by the inductive hypothesis, combined with the fact that there +are no nontrivial extensions between modules Vν +1 and Vν′ +1 if ν ̸= ν′ and that the +highest weight vector of V ν generates the module Vν +1 as an U1-module. +□ +Notice that, although the specialisations at a = 0 of Vλ,µ +2 +and �Vλ,µ +2 +are iso- +morphic, the specialisation of �Vλ,µ +2 +, is generated by vλ ⊗ vµ while in the first case +this vector generates the submodule Vλ+µ +1 +. +As a corollary, we get the following result. +Proposition 6.3. The following hold: +a) Ψn(�Vλ,µ +2 +) = 0 for n ̸= 0. +b) The inclusion of �Vλ,µ +2 +in Vλ,µ +2 +induces isomorphisms +Ψ0(�Vλ,µ +2 +)[a−1] ≃ Ψ0(Vλ,µ +2 +)[a−1] ≃ Ψ0(Vλ +t ) ⊗Q Ψ0(Vµ +s ). +c) Ψ0(�Vλ,µ +2 +) is torsion-free with respect to the action of A, and the natural pro- +jection induces isomorphisms +Ψ0(�Vλ,µ +2 +) +aΨ0(�Vλ,µ +2 +) +≃ Ψ0 +� �Vλ,µ +2 +a�Vλ,µ +2 +� +≃ Ψ0(Wλ,µ +1 +). +Proof. We use the filtration introduced in Equation (6.1). Notice that +Fi +Fi+1 += +aiU − +2 ⊗ F i +ai+1U − +2 ⊗ F i ≃ U − +1 ⊗C F i ≃ Indˆg1 +ˆg+ +1 F i, + +20 +FORTUNA, LOMBARDO, MAFFEI, MELANI +where we consider F i as a ˆg+ +1 -module on which tg[t] acts trivially. Notice that +Indˆg1 +ˆg+ +1 F i is a sum of modules of the form Vν +1, hence in particular has trivial non- +zero cohomology. +Hence, arguing by decreasing induction on i, starting from i = µ, it follows that +Fi has trivial semi-infinite cohomology in degree different from zero. Indeed for +i = µ we have Fµ = aµVλ,µ +2 +≃ Vλ,µ +2 +and this is the content of Theorem 4.9. For +i = 0 this implies claim a). +Part b) follows from the fact that semi-infinite cohomology commutes with local- +ization (see Lemma 3.3) combined with the isomorphism �Vλ,µ +2 +[a−1] = Vλ,µ +2 +[a−1] ≃ +Vλ +t ⊗Q Vµ +s . +To prove c), consider the exact sequence +0 +� �Vλ,µ +2 +·a +� �Vλ,µ +2 +� +�Vλ,µ +2 +a�Vλ,µ +2 +� 0 +By Lemma 6.2, the last module in this sequence is isomorphic to Wλ,µ +1 +. In par- +ticular, the semi-infinite cohomology groups Ψn of the modules appearing in this +sequence are zero for n ̸= 0, and c) follows. +□ +To prove that the semi-infinite cohomology of �Vλ,µ +2 +is isomorphic to Zλ,µ +2 +we will +use the action of a particular central element in Z2. Recall from [4] the definition +of the 2-Sugawara operator +S(2) +1/2 = +� +n∈ 1 +2 Z,b +: (Jbwn)(Jbz−n) : +(6.3) +where J1, J2, J3 are the basis elements e, h, f and J1, J2, J3 are the dual basis +elements f, h/2, e. As proved in [4], the element S(2) +1/2 is central. Its specialisation +is the Sugawara operator +S(1) +1 += +� +n∈Z,b +: (Jbtn) (Jbt−n) : +(6.4) +which is an element of Z1. It is straightforward to check that the action of S(1) +1 +on +the Weyl module Vν +1 is given by multiplication by ν(ν + 1). +Lemma 6.4. The element ˆwℓ = +� +et−1�ℓ ˜wℓ belongs to Z2 · (vλ ⊗ vµ) + a�Vλ,µ +2 +for +ℓ = 0, . . . , µ, +Proof. We notice first that the element vλ ⊗ f ℓvµ belongs to F ℓ \ F ℓ−1 and has +weight λ + µ − 2ℓ. Hence, up to a non-zero constant we have vλ ⊗ f ℓvµ = uℓ + u′ +ℓ, +where we recall that uℓ is the highest weight vector in Gℓ ≃ V λ+µ−2ℓ ⊂ V λ ⊗ V µ +and u′ +ℓ ∈ F ℓ−1. In particular, recall from Lemma 6.2 that aℓ−1F ℓ ⊂ �Vλ,µ +2 +, hence +aℓ � +et−1�ℓ vλ ⊗ f ℓvµ = +� +et−1�ℓ ˜wℓ + +� +et−1�ℓ (aℓu′ +ℓ) ≡ +� +et−1�ℓ ˜wℓ +mod a�Vλ,µ +2 +. +Hence, the lemma is equivalent to the fact that ˆwℓ = aℓ � +et−1�ℓ vλ ⊗ f ℓvµ is in +Z2 · vλ ⊗ vµ + a�Vλ,µ +2 +. We prove this statement by induction on ℓ. For ℓ = 0 it is +trivially true. Now assume ˆwℓ is in Z2 · vλ ⊗ vµ + a�Vλ,µ +2 +. We compute S(2) +1/2( ˆwℓ). In +order to do this, we notice that the action of xtisj on �Vλ,µ +2 +/a�Vλ,µ +2 +is equal to the +action of xti+j on the same module, and that vλ ⊗ e f ℓvµ is in F ℓ−1. We have +S(2) +1/2 ˆwℓ = 2 +� +n>0 +et−n · ftn · ˆwℓ + 2 +� +n>0 +ft−n · etn · ˆwℓ + +� +n>0 +ht−n · htn · ˆwℓ ++ e · f · ˆwℓ + e · f · ˆwℓ + 1 +2h · h · ˆwℓ. + +SEMI-INFINITE COHOMOLOGY OF WEYL MODULES +21 +In the second infinite sum above, the element etn commutes with et−1, hence etn · +ˆwℓ ∈ a�Vλ,µ +2 +for all n > 0. The summands of the third series are of the form +htn · (et−1)ℓ · ˆwℓ = (et−1)ℓhtn · ˆwℓ + 2ℓ(et−1)ℓ−1etn−1 · ˆwℓ, +hence they vanish for n ⩾ 3, while for n = 1, 2 they are easily checked to be elements +of a�Vλ,µ +2 +. The summands of the first series are given by +ftn·(et−1)ℓ· ˆwℓ = (et−1)ℓftn· ˆwℓ−ℓ(et−1)ℓ−1htn−1· ˆwℓ−ℓ(ℓ−1)(et−1)ℓ−2etn−2· ˆwℓ, +and all terms are zero or in a�Vλ,µ +2 +but for the case n = 1, for which we get +(et−1) · (ft) · (et−1)ℓ · ˆwℓ = aℓ+1(et−1)ℓ+1 · (vλ ⊗ f ℓ+1vµ) +− ℓ(et−1)ℓh · (vλ ⊗ f ℓvµ) − ℓ(ℓ − 1)(et−1)ℓ · ˆwℓ = ˆwℓ+1 + K1 ˆwℓ +for some constant K1. Finally, e·f · ˆwℓ+e·f · ˆwℓ+ 1 +2h·h· ˆwℓ belongs to K2 ˆwℓ+a�Vλ,µ +2 +for some constant K2. Hence we get +S(2) +1/2 ˆwℓ ≡ ˆwℓ+1 + K ˆwℓ mod a�Vλ,µ +2 +for some constant K, proving our claim. +□ +We now prove that the zero-th semi-infinite cohomology of the module �Vλ,µ +2 +is +isomorphic to Zλ,µ +2 +. +Theorem 6.5. For g = sl(2) the map Φ : Zλ,µ +2 +−→ Ψ0��Vλ,µ +2 +� +given by Φ(z) = +z · [vλ ⊗ vµ] is an isomorphism. +Proof. By [4], Theorem 6.4, the action of Z2 on Vλ,µ +2 +, hence on �Vλ,µ +2 +, factors through +Zλ,µ +2 +. Moreover vλ ⊗ vµ is a cycle, so the map Φ is well defined. Since we know +that both modules are torsion-free, to prove that Φ is an isomorphism it suffices to +prove that the localization Φa and the specialisation Φ are isomorphisms. +The fact that Φa is an isomorphism is the content of part b) of Proposition 6.3. +We need to prove that Φ is an isomorphism. By Lemma 6.2, Proposition 6.3 and +[4, Theorem 2.13] we have +Zλ,µ +2 +aZλ,µ +2 +≃ +µ +� +i=0 +Zλ+µ−2i +1 +and +Ψ0(�Vλ,µ +2 +) +aΨ0(�Vλ,µ +2 +) +≃ +µ +� +i=0 +Ψ0(Vλ+µ−2i +1 +). +In particular, by Theorem 4.1 these two Z1-modules are isomorphic, but we need to +prove that our specific map Φ provides an isomorphism between them. By Remark +5.2 it is enough to prove that Φ is surjective. We prove that the image of Φ contains +Ψ0(Fℓ+a�Vλ,µ +2 +/a�Vλ,µ +2 +) arguing by reverse induction on ℓ. For ℓ = 0 we get our claim. +For ℓ > µ there is nothing to prove. Now assume ℓ ⩽ µ. Consider again the exact +sequence +0 +� Fℓ+1+a�Vλ,µ +2 +a�Vλ,µ +2 +� Fℓ+a�Vλ,µ +2 +a�Vλ,µ +2 +�aℓU − +2 ⊗C Gℓ +�0. +We know that the last module is isomorphic to +aℓU − +2 ⊗C Gℓ ≃ Vλ+µ−2ℓ +1 += Indˆg1 +ˆg+ +1 (V λ+µ−2ℓ) +and that it is generated by the element ˜wℓ ∈ aℓGℓ. Notice this sequence of Z1- +modules splits by Remark 5.2. +Taking semi-infinite cohomology we get a short +exact sequence +0 +�Ψ0 � +Fℓ+1+a�Vλ,µ +2 +a�Vλ,µ +2 +� +�Ψ0 � +Fℓ+a�Vλ,µ +2 +a�Vλ,µ +2 +� +�Ψ0 � +aℓU − +2 ⊗C Gℓ� +�0. +and we know that the last Z2-module is generated by ˜wℓ. Hence it is enough to +prove that this element is in the image of Zλ,µ +2 +(vλ ⊗vµ) in Ψ0� +�Vλ,µ +2 +/Fℓ+1 +a�Vλ,µ +2 +� +. + +22 +FORTUNA, LOMBARDO, MAFFEI, MELANI +By Lemma 6.4 we know that ˆwℓ is in this image. Now we prove that ˜wℓ and ˆwℓ +define the same element in the semi-infinite cohomology of aℓU − +2 ⊗C Gℓ. This is a +claim about the cohomology of the module Vν +1 for ν = λ + µ − 2ℓ. For any ν we +prove that +� +et−1�hvν + +� +et−1�h−1vν is a coboundary. Indeed the boundary operator +in the case of sl(2) is equal to +d(1) = ψ∗ + +� +n∈Z +(etn) ⊗ ψ∗t−1−n, +so a simple computation shows +d(1) �� +et−1�h−1vν ⊗ (ψt−1)|0⟩Λ +� += +� +et−1�h−1vν ⊗ |0⟩Λ + +� +et−1�hvν ⊗ |0⟩Λ, +which implies our claim. +□ +Recall that in [4] we computed the endomorphism ring of Vλ,µ +2 +, showing that it +is isomorphic to Zλ,µ +2 +. We now prove the same result for the module �Vλ,µ +2 +. +Proposition 6.6. The action of the center Z2 on �Vλ,µ +2 +induces an isomorphism +Zλ,µ +2 +≃ Endˆg2(�Vλ,µ +2 +). +Proof. We already recalled at the beginning of the proof of Theorem 6.5 that the +action of Z2 on �Vλ,µ +2 +factors through Zλ,µ +2 +. We denote by α : Zλ,µ +2 +−→ End(�Vλ,µ +2 +) +this action. Since both modules have no A-torsion, in order to prove that α is +an isomorphism it suffices to show that its localization and its specialisation are +isomorphisms. +Moreover, since our modules are finitely generated and have no +torsion we have +Endˆg2 +� +�Vλ,µ +2 +� +[a−1] ≃ Endˆg2[a−1] +� +�Vλ,µ +2 +[a−1] +� +≃ Endˆgt,s +� +Vλ ⊗Q Vµ +s +� +≃ Zλ +t ⊗Q Zµ +t ≃ Zλ,µ +2 +[a−1], +hence the localization of α is an isomorphism. +Finally, we prove that the specialisation of α is also an isomorphism. We have +already recalled that by [4, Theorem 2.13] we have Zλ,µ +2 +/aZλ,µ +2 +≃ �µ +i=0 Zλ+µ−2i +1 +. +Hence by Theorem 4.1 we have the following abstract isomorphisms of Z1-modules: +Zλ,µ +2 +aZλ,µ +2 +≃ +µ +� +i=0 +Zλ+µ−2i +1 +≃ +µ +� +i=0 +Endˆg1(Vλ+µ−2i +1 +). +Moreover, since �Vλ,µ +2 +has no nontrivial A-torsion, by Lemma 6.2 and Remark 5.2 +part (1) we have the inclusion +Endˆg1 +� +�Vλ,µ +2 +� +a Endˆg1 +� +�Vλ,µ +2 +� ⊂ Endˆg1 +� �Vλ,µ +2 +a�Vλ,µ +2 +� +≃ +µ +� +i=0 +Endˆg1(Vλ+µ−2i +1 +). +Hence, composing the specialisation of the map α with this inclusion and the iso- +morphisms above we get a Z1-equivariant endomorphism of �µ +i=0 Zλ+µ−2i +1 +. Moreover, +α(1) = 1, hence we conclude by Remark 5.2 (3) that the specialisation of α is also +an isomorphism. +□ +References +[1] Casarin, L. A Feigin Frenkel theorem with n singularities. preprint, 2022. +[2] Feigin, B., and Frenkel, E. Affine Kac-Moody algebras at the critical level and Gelfand- +Diki˘ı algebras. In Infinite analysis, Part A, B (Kyoto, 1991), vol. 16 of Adv. Ser. Math. Phys. +World Sci. Publ., River Edge, NJ, 1992, pp. 197–215. +[3] Fortuna, G. The Beilinson-Bernstein Localization Theorem for the affine Grassmannian. +MIT, PhD thesis, 2013. + +SEMI-INFINITE COHOMOLOGY OF WEYL MODULES +23 +[4] Fortuna, G., Lombardo, D., Maffei, A., and Melani, V. Local opers with two singularities: +the case of sl(2). Comm. Math. Phys., 394 (2022), 1303–1360. +[5] Frenkel, E., and Ben-Zvi, D. Vertex algebras and algebraic curves, second ed., vol. 88 +of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, +2004. +[6] Frenkel, E., and Gaitsgory, D. Local geometric Langlands correspondence: the spherical +case. In Algebraic analysis and around, vol. 54 of Adv. Stud. Pure Math. Math. Soc. Japan, +Tokyo, 2009, pp. 167–186. +[7] Frenkel, E., and Gaitsgory, D. Weyl modules and opers without monodromy. In Arithmetic +and geometry around quantization, vol. 279 of Progr. Math. Birkh¨auser Boston, Boston, MA, +2010, pp. 101–121. +E-mail addresses: giorgiafortuna@gmail.com, davide.lombardo@unipi.it, +andrea.maffei@unipi.it, valerio.melani@unifi.it + diff --git a/FNFQT4oBgHgl3EQfRTbc/content/tmp_files/load_file.txt b/FNFQT4oBgHgl3EQfRTbc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ae2ffc93ad63b8e52e66d3131edf4078bcd14c6e --- /dev/null +++ b/FNFQT4oBgHgl3EQfRTbc/content/tmp_files/load_file.txt @@ -0,0 +1,882 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf,len=881 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='13286v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='RT] 30 Jan 2023 THE SEMI-INFINITE COHOMOLOGY OF WEYL MODULES WITH TWO SINGULAR POINTS GIORGIA FORTUNA, DAVIDE LOMBARDO, ANDREA MAFFEI, VALERIO MELANI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In their study of spherical representations of an affine Lie algebra at the critical level and of unramified opers, Frenkel and Gaitsgory introduced what they called the Weyl module Vλ corresponding to a dominant weight λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This object plays an important role in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In [4], we introduced a possible analogue Vλ,µ 2 of the Weyl module in the setting of opers with two singular points, and in the case of sl(2) we proved that it has the ‘correct’ endomorphism ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In this paper, we compute the semi-infinite cohomology of Vλ,µ 2 and we show that it does not share some of the properties of the semi- infinite cohomology of the Weyl module of Frenkel and Gaitsgory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For this reason, we introduce a new module ˜Vλ,µ 2 which, in the case of sl(2), enjoys all the expected properties of a Weyl module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Introduction Let g be a complex simple Lie algebra and let ˆg be its affinization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Choose a Borel subalgebra and a maximal toral subalgebra, and let G be a simply connected algebraic group with Lie algebra equal to g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As a particular case of a more general conjecture, Frenkel and Gaitsgory proved in [6] that the semi-infinite cohomology gives an isomorphism between the category ˆgcrit-modJG of spherical representations of ˆg at the critical level (that is, representations of ˆg at the critical level with a compatible action of JG = G(C[[t]])) and the category of quasi-coherent sheaves on the space of unramified opers Opunr 1 over gL, the Langlands dual of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As they explain, the space of unramified opers is the disjoint union of its connected components Opλ,unr 1 , and the category of spherical representations is the product of certain subcategories ˆgcrit-modJG,λ, where in both cases λ ranges over all dominant weights of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The equivalence given by semi-infinite cohomology specialises to an equivalence between ˆgcrit-modJG,λ and the category of quasi-coherent sheaves over Opλ,unr 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The space Opλ,unr 1 is a non-reduced indscheme, and its reduced version, denoted by Opλ 1, is an affine scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In this paper we will denote by Zλ 1 its coordinate ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In this theory, an important role is played by the Weyl module Vλ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This module enjoys the following fundamental properties: Endˆg(Vλ 1) ≃ Zλ 1 and Ψ0(Vλ 1) ≃ Zλ 1 , where Ψn is the n-th semi-infinite cohomology group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover the semi-infinite cohomology groups Ψn(Vλ 1) are trivial for n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Dennis Gaitsgory suggested to Giorgia Fortuna to study the space of unramified opers and spherical representations in a more general context, see [3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' in fact, the definition of unramified opers as well as the definition of spherical representations can be generalized in the presence of more than one singularity, raising the question on whether or not certain statements remain true and what happens when these singularities collide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 1 2 FORTUNA, LOMBARDO, MAFFEI, MELANI In [4] we took some steps in this direction, by studying the case of sl(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, we introduced a version of the Weyl module Vλ,µ 2 of critical level of the affine Lie algebra with two singularities ˆg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Thinking of t as a coordinate near the first singularity and s as a coordinate near the second singularity, this is the version of the affine Lie algebra over the ring A = C[[a]], where a = (t−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As an A module is equal to K2 ⊗C ⊕AC2 where K2 = C[[a, t]][1/t(t− a)] and C2 is a central element (see [4], Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 for the complete definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We also introduced reduced scheme over A of unramified opers Opλ,µ 2 which gener- alize the schemes Opλ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Both objects depend on two integral dominant weights λ, µ of G, and we proved that Endˆg2(Vλ,µ 2 ) ≃ Zλ,µ 2 , where Zλ,µ 2 is the coordinate ring of Opλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In this article we study the semi-infinite cohomology of Vλ,µ 2 in order to un- derstand what relation it has with the ring Zλ,µ 2 in order to understand how the equivalence Ψ0(Vλ 1) ≃ Zλ 1 generalizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This is done in Section 4, where we compute the cohomology of Vλ,µ 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' in Section 5 we study the action of Z2, the center of a completion ˆU2 of the enveloping algebra of ˆg2 at the critical level on this module (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, we prove that the specialisation at a = 0 and the localization at a ̸= 0 of the semi-infinite cohomology of Vλ,µ 2 are isomorphic to the specialisation and localization of Zλ,µ 2 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' However, in contrast to our intuition, we also show the following result which says that Ψ0(Vλ,µ 2 ) doesn’t exactly generalize the equivalence Ψ0(Vλ 1) ≃ Zλ 1 as expected: Theorem A (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='9 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We have Ψn(Vλ,µ 2 ) = 0 for n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, Ψ0(Vλ,µ 2 ) is not isomorphic to Zλ,µ 2 as a Z2-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For this computation, we rely on the formalism introduced by Casarin in [1], which makes it possible to use vertex algebras also in the context of opers with two singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Once this formalism is in place, for the computation of the semi- infinite cohomology we can follow closely the approach taken by Frenkel and Ben Zvi in [5, Chapter 15] for the case of one singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In the last section, we restrict our attention to the Lie algebra sl(2) and introduce a submodule �Vλ,µ 2 of Vλ,µ 2 , which is generated by the highest weight vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We prove that this module is the correct one to consider, in the sense that it has the expected cohomology groups and endomorphism ring, as the following result shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Theorem B (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5 and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' If g = sl(2) then we have Ψn(�Vλ,µ 2 ) = 0 for n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, we have Endˆg2(�Vλ,µ 2 ) ≃ Zλ,µ 2 and Ψ0(�Vλ,µ 2 ) ≃ Zλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now briefly explain the connection between these results and Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 in Fortuna’s Thesis [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As a particular case the conjecture predicts an equivalence between quasi-coherent sheaves over the space of unramified opers with two singu- larities and the category of spherical representations over ˆg2: that is the space of smooth representations of ˆg2 with a compatible action of J2G = G(C[[a, t]]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The conjecture stated in [3] predicts an equivalence of similar categories not only in the presence of two singularities but in the presence of n-possible singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular for any finite set with n elements I we can define the space of opers on the formal disc with n-singularities OpI and the subspace of unramified opers Opunr I (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5 in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These are spaces over the product of n-copies of the formal disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These are easily seen to be factorization spaces, which means that SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 3 this spaces specialise nicely when restricted along or outside the diagonals of this product (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5 in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' There are not substantial differences between the treatment we do here or in [4] of Op2 and the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The only minor difference is that we fix a singularity to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These spaces are indschemes, and so we can define the categories QCoh(OpI), and QCoh(Opunr I ) of quasi-coherent sheaves on OpI and Opur I (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 in [3] for the actual definition), and the nice factorization properties which make them factorization categories (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similarly, for a finite set I we can define a Lie algebra ˆgI and study its smooth representations at the critical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The objects constructed in this way live also on the product of n copies of the formal disc, and they also have nice factorization properties, in particular the collection of (completions of the) enveloping algebras specialized at the critical level ˆUI of the algebras ˆgI, is what is called a factorization algebra (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As a conseguence the collection of the categories of smooth representations at the critical of the Lie algebras ˆgI, denoted by ˆgI,crit-mod and their subcategories of spherical representations ˆgI,crit-modJG can be organized also in a factorization category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The semi-infinite cohomology can be defined also in this generality and defines a functor ΨI : ˆgI,crit-mod −→ D(QCoh(OpI)) compatible with the factorization properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' While in Fortuna’s thesis all these constructions are obtained somehow for free using the language of chiral algebras (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6 in [3]), in this paper we use the language of vertex algebras and the formalism introduced by Casarin [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let us notice that, from this point of view, there are no differences in treating the case with two singular points and the case with an arbitrary finite number of singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For example, the proof of Theorem A above can be repeated verbatim in the case of n singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' More generally we believe that all the technical difficulties in the study of this problem already appear in the case of two singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' It is easy to see from the factorization properties and the analogous statement for the case of one singularty by Frenkel and Gaitsgory (see [7]) that the semi- infinite cohomology of a ˆgI-spherical module is supported on Opunr I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence semi- infinite cohomology restricts to a functor ΨI : ˆgI,crit-modJG −→ D(QCoh(Opunr I )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 in [3] states that this functor is exact and Ψ0 I : ˆgI,crit-modJG −→ QCoh(Opunr I ) is an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 of [3], one possible strategy to prove this conjecture is sketched, using the factorization structure and the result proved in the case of one singularity to deduce the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, thanks to Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 and Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 of [3] and Theorems A and B above, a more careful study of the modules Vλ,µ 2 or �Vλ,µ 2 might help in finding a proof of [3, Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1], in the case of g = sl(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In the first section we recall some definitions from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In Section 3 we recall the formalism introduced by Casarin [1] and we use it to define semi-infinite cohomology and prove some of its basic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In Sections 3 and 4 we compute the semi-infinite cohomology of Vλ,µ 2 and in Section 5 we compute the semi-infinite cohomology of �Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We thank Luca Casarin for many useful discussions and in particular for explain- ing to us the formalism introduced in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' It seems to us that Casarin’s approach provides a natural framework to treat questions concerning opers with several singu- larities, making the theory much more transparent than it was in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, 4 FORTUNA, LOMBARDO, MAFFEI, MELANI the results of [1] allowed us to streamline several arguments and calculations which would have been quite hard to carry out using the direct approach of [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Basic constructions In this section we recall some basic constructions from [4], to which we refer for further details, and we introduce the notion of semi-infinite cohomology in the context of affine Lie algebras with more than one singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We follow [4, Section 1], to which the reader is referred for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We introduce the rings A = C[[a]], Q = C((a)), R2 = C[[t, s]], K2 = C[[t, s]][1/ts], where a = t − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall that we have expansion maps (given by suitable natural inclusions) and a specialisation map (which sends a to 0 and t, s to t, see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 in [4]) Et : K2[a−1] −→ Q((t)), Es : K2[a−1] −→ Q((s)), Sp : K2 −→ C((t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We also write E = Et × Es : K2[a−1] −→ Q((t)) × Q((s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall from [4, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1] that Sp induces an isomorphism K2/(a) ≃ C((t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These rings have natural topologies: with respect to these, the image of E is dense, and E(R2[a−1]) is dense in Q((t)) × Q((s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These rings are also equipped with residue maps Res2 : K2 → A Res1 : C((t)) → C, Rest : Q((t)) → Q, Ress : Q((s)) → Q, which behave nicely with respect to specialisation and expansion (see [4, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, we recall Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='10 in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 ([4], Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let M, N be two A-modules and ϕ : M −→ N be a morphism of A-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Then a) if M is flat and ϕa : M[a−1] −→ N[a−1] is injective, then ϕ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) if N is flat, ϕa : M[a−1] −→ N[a−1] is surjective, and ϕ : M/aM −→ N/aN is injective, then ϕ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, if M and N are flat, ϕa : M[a−1] −→ N[a−1] is an isomorphism, and ϕ : M/aM −→ N/aN is injective, then ϕ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Affine Lie algebras and completion of the enveloping algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We follow [4, Section 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let g be a finite-dimensional Lie algebra over the complex numbers and denote by κ the Killing form of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall from [4, Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3] that for each of the rings of the previous section we introduce an affine Lie algebra: ˆg1 is the usual affine Lie algebra (we take for convenience the version defined by Laurent polynomial and not Laurent series), ˆgt and ˆgs are also versions of the the usual affine Lie algebra, while ˆg2 is an A-Lie algebra having as underlying A-module the space ˆg2 = C[t, s][1/ts] ⊗C g ⊕ A C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We also introduce the Lie algebra ˆgt,s = ˆgt ⊕ ˆgs/(Ct − Cs) (see [4, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For each of these Lie algebras, we introduce the corresponding universal envel- oping algebra, which we suitably complete and then specialize at the critical level by imposing that the central element acts as −1/2 (see Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 in [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular ˆU2 = lim ←− n U(ˆg2) (C2 = −1/2, tnsnC[t, s] ⊗ g)left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall from [4, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4] that the expansion maps and the specialisation maps induce morphisms at the level of Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, the specialisation map Sp : ˆU2 −→ ˆU1 induces an isomorphism between ˆU2/a ˆU2 and ˆU1, while the SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 5 expansion map induces a morphism E : ˆU2[a−1] −→ ˆUt,s which is injective and has dense image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, the natural inclusions ˆgt ֒→ ˆgt,s and ˆgs ֒→ ˆgt,s induce a morphism ˆUt ⊗ ˆUs −→ ˆUt,s which is also injective and with dense image (see [4, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Weyl modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We follow [4, Section 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We choose a Borel subalgebra and a maximal toral subalgebra of g, which we denote by b and t respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This data induces a choice of weights, integral weights and dominant weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For every integral dominant weight λ, [7] introduced the Weyl module Vλ 1 over the affine Lie algebra ˆg1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The representation V = V0 1, which has a structure of vertex algebra, will play a particularly important role for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This vertex algebra enjoys the following universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let U be a vertex algebra such that there exists a linear map x �→ ux from g to U such that (ux)(0)(uy) = u[x,y] (ux)(1)(uy) = −1 2κ(x, y)|0⟩U (ux)(n)(uy) = 0 for all n ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' There exists a unique morphism of vertex algebras α : V → U such that α(xt−1|0⟩V) = ux for all x ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Weyl modules Vλ t and Vλ s can also be defined for the Lie algebras ˆgt and ˆgs, without any significant change from [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In [4], we introduced a generalization of these modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Given two dominant weights λ, µ, we consider the irreducible repres- entations V λ and V µ of the Lie algebra g having highest weights λ, µ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In [4, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2], given two dominant integral weights λ, µ we introduced the module Vλ,µ 2 = Indˆg2 ˆg+ 2 � A ⊗C V λ ⊗C V µ� , where ˆg+ 2 = C[t, s] ⊗ g ⊕ A C2 acts on A ⊗C V λ ⊗C V µ as f(t, s)x · (p(a) ⊗ u ⊗ v) = f(0, −a)p(a) ⊗ xu ⊗ v + f(a, 0)p(a) ⊗ u ⊗ xv, while C2 acts as −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In [4] we called this object the Weyl module of weights (λ, µ), although, as we will see, it does not have the the same properties as its 1-singularity analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We also define Wλ,µ 1 = Indˆg1 ˆg+ 1 � V λ ⊗C V µ� , where ˆg+ 1 = C[t] ⊗ g ⊕ C C1 acts on V λ ⊗C V µ as f(t)x · (u ⊗ v) = f(0)x · (u ⊗ v) and C1 acts as −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The specialisation and expansion maps are defined also for Weyl modules, and induce the following isomorphisms [4, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3]: Vλ,µ 2 aVλ,µ 2 ≃ Wλ,µ 1 , Vλ,µ 2 [a−1] ≃ Vλ t ⊗Q Vµ s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Clifford algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now define the Clifford algebra with two singularities, generalizing the construction of the classical case (see for example [5, Chapter 15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let n+ be the nilpotent radical of b and set X2 = K2 ⊗C n+ ⊕ K2 ⊗C n∗ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We equip X2 with the unique A-bilinear form such that K2 ⊗C n+ and K2 ⊗C n∗ + are isotropic subspaces and (f ⊗ x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' g ⊗ ϕ) = Res2(fg) ϕ(x) 6 FORTUNA, LOMBARDO, MAFFEI, MELANI for all f, g ∈ K2, x ∈ n+ and ϕ ∈ n∗ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by Cℓ2 the associated Clifford algebra over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' There are obvious variants of the same construction where we replace K2 with the ring C[t±1] or one of the rings Q[t±1], Q[s±1], Q[t±1] × Q[s±1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We obtain Clifford algebras that we denote by Cℓ1, Cℓt, Cℓs, Cℓt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The algebra CℓU in [5, Section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1] is a completion of Cℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These Clifford algebras have a natural grading called the charge and denoted by ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' It can be defined as follows: the elements of the base ring have charge 0, while for ψ ∈ n and ψ∗ ∈ n∗ we have ch ψ = −1, ch ψ∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2) The relations defining each Clifford algebra are homogeneous, hence the charge induces a well-defined grading on the Clifford algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now introduce completions of the tensor product ˆU2 ⊗A Cℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We define ˆU2 ˆ⊗ACℓ2 = lim ←− n ˆU2 ⊗A Cℓ2 � (ts)nR2g ⊗ 1, 1 ⊗ (ts)nR2n+, 1 ⊗ (ts)nR2n∗ + � left ideal and we notice that, as in the case of the algebra ˆU2, this A-module has a nat- ural structure of A-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We introduce the completed Clifford algebras ˆU1 ˆ⊗Cℓ1, ˆUt ˆ⊗QCℓt, ˆUs ˆ⊗QCℓs, and ˆUt,s ˆ⊗QCℓt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The specialisation and expansion map de- termine morphisms Sp : ˆU2 ˆ⊗ACℓ2 −→ ˆU1 ˆ⊗Cℓ1 and E : ( ˆU2 ˆ⊗ACℓ2)[a−1] −→ ˆUt,s ˆ⊗QCℓt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Arguing exacly as in [4, Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='7 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='9] we see that E is injective with dense image, while the specialisation map induces an isomorphism ˆU2 ˆ⊗ACℓ2/a( ˆU2 ˆ⊗ACℓ2) ≃ ˆU1 ˆ⊗Cℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, we have an injective map I : ˆUt ˆ⊗QCℓt → ˆUt,s ˆ⊗QCℓt,s induced by the natural inclusion Kt → Kt,s = Kt × Ks given by f �→ (f, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similarly, we have an injective map J : ˆUs ˆ⊗QCℓs → ˆUt,s ˆ⊗QCℓt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 of [4], the product of these maps I ⊗ J : ( ˆUt ˆ⊗QCℓt) ⊗Q ( ˆUs ˆ⊗QCℓs) → ˆUt,s ˆ⊗QCℓt,s is injective with dense image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Fock module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now describe the “fermionic” Fock spaces corresponding to the Clifford algebras defined in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As above, for the construction in the case of one singularity we refer to [5, Section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4]: here we mimic this definition in the case of two singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We define Cℓ+ 2 as the A-subalgebra of Cℓ2 generated by R2 ⊗ n+ and R2 ⊗ n∗ + and we define the Fock module Λ2 = Cℓ2 ⊗Cℓ+ 2 A |0⟩Λ2 where R2 ⊗ n+ and R2 ⊗ n∗ + acts trivially on |0⟩Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The charge (see equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2)) induces a grading on the Fock space by setting ch |0⟩Λ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by Λn 2 the subspace of elements of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similar constructions can be given for all the other Clifford algebras Cℓ1, Cℓt, Cℓs, and Cℓt,s, giving Fock modules Λ1, Λt, Λs, and Λt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Specialisation and expansion, induce maps also at the level of the Fock spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Arguing as in [4, Section 6] (where we considered the module Vλ,µ 2 ), it is easy to prove the following Lemma: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' a) The specialisation map Sp : Λ• 2 −→ Λ• 1 is homogeneous of degree zero and induces an isomorphism Λ• 2/aΛ• 2 ≃ Λ• 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) We have a homogeneous isomorphism of degree zero Λ• t,s ≃ Λ• t ⊗Q Λ• s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 7 c) The expansion map E : Λ• 2[a−1] −→ Λ• t ⊗Q Λ• s is a homogeneous isomorphism of degree zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall also that the Fock space Λ = Λ1 has a natural structure of vertex super- algebra with the following universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let U be a vertex superalgebra such that there exists a linear map x �→ ux from n∗ + ⊕ n∗ + to the space of odd elements of U such that (1) for all ϕ, ψ ∈ n and for all ϕ∗, ψ∗ ∈ n∗ + (uψ)(n)(uϕ) = (uψ∗)(n)(uϕ∗) = (uψ)(m)(uψ∗) = (uψ∗)(m)(uψ) = 0 for all n ⩾ 0 and for all m ⩾ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (2) (uψ)(0)(uψ∗) = (uψ∗)(0)(uψ) = ⟨ψ, ψ∗⟩|0⟩U for all ψ ∈ n and ψ∗ ∈ n∗ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Then there exists a unique morhism of vertex superalgebras α : Λ → U such that α(ψt−1|0⟩Λ) = uψ and α(ψ∗t−1|0⟩Λ) = uψ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For each of the objects introduced above – base rings, enveloping al- gebras, Clifford algebras, and Fock spaces – it is not hard to construct explicit bases (or topological bases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We give the details in the case of two singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The construction of a basis depends on the choice of a basis of C[t, s][1/ts] as an A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Following [4], Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 and Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1) we introduce the following bases, indexed by 1 2Z: for n ∈ Z we define � zn = tnsn zn+ 1 2 = tn+1sn � wn = tnsn wn+ 1 2 = tnsn+1 The elements zm for m ∈ 1 2Z form a basis of C[t, s][1/ts] as an A-module, and the elements wn are the dual basis with respect to the residue bilinear form: more precisely, one has Res2(znw−m− 1 2 ) = δn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This specific choice of basis is not particularly important, and several others would be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' However, some properties need to be satisfied for our approach to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particolar with our choice, the elements zm (or wm) with m ⩾ 0 form an A-basis of C[t, s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Since K2 is an A-free module, we deduce that the enveloping algebras of g2 and Cℓ2 are A-free modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, as R2 is a direct summand of K2, we also deduce that Vλ,µ 2 and Λ2 are also A-free modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Explicit bases of these modules, as well as an explicit topological basis of the algebra ˆU2 ˆ⊗ACℓ2, can be obtained using the Poincar´e-Birkhoff-Witt theorem and its analogue for Clifford algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Vertex algebras and semi-infinite cohomology In this section, we recall some results obtained by Casarin [1] which allow us to use the formalism of vertex algebras also in the context of several singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, using this formalism we develop a notion of semi-infinite cohomology for ˆU2-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Distributions and vertex algebra morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let R be a complete topo- logical associative A-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Following [1, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4], we denote by FA(K2, R) the space of continuous A-linear morphisms from K2 to R and call it the space of 2-fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We refer to [1] for the definitions of mutually local 2-fields (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1), of the n-products X(n)Y of two 2-fields (Definitions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='7) and of the derivative ∂(X) of a 2-field (before Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The definition in [1] applies also to the other ring we are considering: K1, Kt, Ks, Kt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 8 FORTUNA, LOMBARDO, MAFFEI, MELANI In particular to define n products it is necessary to choose what in [1] is called a global coordinate (see definition ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We choose always t as a global coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' More explicitly for the rings K2, K1, Kt and Ks we choose t = s + a as a global coordinate, and for the ring Kt,s = Kt × K2 we choose (t, t) = (t, s + a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We also use some foundational results proved in this context in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, the following result will be crucial for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 ([1], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let F be a C-linear subspace of FA(K2, R) of mutually local 2-fields closed under derivation and n-products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let 1 be a field such that 1(f) is central for every f ∈ K2, that ∂ 1 = 0 and such that 1(n)X = δn,−1X for all X ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Then the vector dpace F + C1, endowed with n-products and derivation T = ∂, is a C-vertex algebra with 1 as vacuum vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' It is straightforward to generalize the constructions and results in [1] to the case of superalgebras R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We are interested in the case where R is the superalgebra ˆU2 ˆ⊗ACℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For x ∈ g, ψ ∈ n+ and ψ∗ ∈ n∗ + we define the 2-fields x(2)(g) = (x⊗g)⊗1Cℓ2, ψ[2](g) = 1 ˆU2⊗(ψ⊗g), (ψ∗)[2](g) = 1 ˆU2⊗(ψ∗⊗g) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1) for all g ∈ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The first of these fields has even parity with respect to the superal- gebra structure, while the second and third ones are odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These fields are mutually local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We consider the minimal C-linear subspace F(2) of ˆU2 ˆ⊗ACℓ2 closed under n-products and derivation and containing the fields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, we define 12(f) = Res2(f) � 1 ˆU2 ⊗ 1Cℓ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' It is easy to check that this data satisfies the hypothesis of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Therefore, V(2) = F(2) + C12 has a structure of vertex superalgebra, and by the universal properties of the vertex algebra V (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2) and of the vertex superalgebra Λ• (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4) it follows that there exists a morphism of vertex superalgebras Φ(2) : V ⊗C Λ• −→ V(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2) This homomorphism will allow us to easily introduce many elements in V(2), hence also in ˆU2 ˆ⊗ACℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similar constructions apply if the algebra ˆU2 ˆ⊗ACℓ2 is replaced by the algebras ˆU1 ˆ⊗Cℓ1, ˆUt ˆ⊗QCℓt, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence, we construct the fields x(1), ψ[1], x(t), ψ[t], the vertex superalgebras V(1), V(t), and homomorphisms of vertex algebras Φ(1) : V⊗C Λ• −→ V(1), Φ(t) : V ⊗C Λ• −→ V(t), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Notice that we have a specialisation morphism SpF : FA(K2, ˆU2 ˆ⊗Cℓ2) −→ FC(K1, ˆU2 ˆ⊗Cℓ1) and an expansion map EF : FA(K2, ˆU2 ˆ⊗ACℓ2) −→ FQ(Kt,s, ˆU2 ˆ⊗QCℓt,s), determined by the conditions � SpF(X) � (Sp(f)) = Sp(ϕ(f)) and � EF(X) � (E(f)) = E(ϕ(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These maps commute with n-products and derivations and satisfy SpF(12) = 11 and EF(12) = 1t,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, by construction they satisfy SpF(x(2)) = x(1) and EF(x(2)) = x(t,s) for x ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similar relations hold for ψ[2] and (ψ∗)[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This implies in particular that the homomorphisms SpF and EF restrict to homomorphisms of vertex algebras Sp : V(2) −→ V(1) and E : V(2) −→ V(t,s) such that Sp ◦Φ(2) = Φ(1) E ◦ Φ(2) = Φ(t,s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We can also describe the morphism Φ(2) through the morphisms Φ(t) and Φ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall from the end of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4 the maps I, J from ˆUt ˆ⊗QCℓt and ˆUs ˆ⊗QCℓs to ˆUt,s ˆ⊗QCℓt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These maps induce maps at the level of fields IF : FQ(Kt, ˆUt ˆ⊗QCℓt) → SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 9 FQ(Kt,s, ˆUt,s ˆ⊗QCℓt,s) and JF : FQ(Ks, ˆUs ˆ⊗QCℓs) → FQ(Kt,s, ˆUt,s ˆ⊗QCℓt,s), given by IF(X)(f, g) = I(X(f)) and JF(X)(f, g) = J(X(g)) for all (f, g) ∈ Kt×Ks = Kt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The maps IF and JF preserve n-products, commute with derivations, and satisfy IF(1t) + JF(1s) = 1t,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover we notice that I(u) and J(v) commute for all u ∈ ˆUt ˆ⊗QCℓt and v ∈ ˆUs ˆ⊗QCℓs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By the discussion in [1, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2], this implies IF ◦ Φ(t) + JF ◦ Φ(s) = Φ(t,s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This is the only statement where it is relevant the choice of the global coordinate we have done in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Semi-infinite cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now define a notion of semi-infinite cohomo- logy for ˆU2-modules, in analogy with the analogous notion for ˆU1-modules described for example in [5, Chapter 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' To this end, we introduce some notation for elements in the vertex superalgebra V ⊗ Λ•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As in the the case of ˆU1, to describe these ele- ments we choose a basis Ja of g compatible with the decomposition g = n−⊕t⊕n+, where n+ is the nilpotent radical of b and n− is the radical of the opposite nilpotent borel subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by cb,d e the structure coefficients of the Lie bracket with respect to this basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by Φ ⊔ Γ the indexing set of the basis Ja, so that, if α ∈ Φ, then Jα = eα = f−α is a root vector of weight α and, if α ∈ Γ, then Jα ∈ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We also denote by ψ∗ α for α ∈ Φ+ the basis of n∗ + dual to the basis eα of n+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' With each element in n+ ⊗ · · · ⊗ n+ ⊗ n∗ + ⊗ · · · ⊗ n∗ + we associate an element in the vertex superalgebra Λ as follows: N(ψ1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ψℓ ⊗ ψ∗ 1 ⊗ · · · ⊗ ψ∗ m) = (ψ1t−1) · · · (ψℓt−1) · (ψ∗ 1t−1) · · · (ψ∗ mt−1) · |0⟩Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similarly, with an element in g ⊗ n∗ + we associate an element in the vertex superalgebra V ⊗ Λ∗ by setting M(x ⊗ ψ∗) = (xt−1) · |0⟩V ⊗ (ψ∗t−1) · |0⟩Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Following [5, Chapter 15] we define q =M(I) − 1 2|0⟩V ⊗ N(B) = � α∈Φ+ (eαt−1) · |0⟩V ⊗ (ψ∗ αt−1) · |0⟩Λ − 1 2 � α,β∈Φ+ cα,β α+β |0⟩V ⊗ (eα+βt−1) · (ψ∗ αt−1) · (ψ∗ βt−1) · |0⟩Λ, where I ∈ g ⊗ n∗ + represents the inclusion of n+ in g and B ∈ n+ ⊗ n∗ + ⊗ n∗ + is the Lie bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now define the boundary operator d(2) std ∈ ˆU2 ˆ⊗ACℓ2 as follows: d(2) std := � Φ(2)(q) � (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The boundary operator that we will use to define the semi-infinite cohomology is a deformation of d(2) std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let ψ∗ pr = � α simple ψ∗ α ∈ n∗ +, and define χ(2) = 1 ˆU2 ⊗ ψ∗ pr = Φ(2)(N(ψ∗ pr))(1) ∈ ˆU2 ⊗A Cℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similar constructions yield χ(s), χ(t), χ(s), and χ(s,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally set d(2) = d(2) std + χ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As we will check in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3, this is an element that squares to zero, and therefore, it can be used to define the semi-infinite cohomology of a ˆU2-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 10 FORTUNA, LOMBARDO, MAFFEI, MELANI Similarly we can define d(1) std, χ(1), d(1), d(t) std, χ(t), d(t), and so on, as elements of the corresponding superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By the discussion at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 we have Sp(d(2)) = d(1), E(d(2)) = d(t,s), and I(d(t)) + J(d(s)) = d(t,s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let M be an ˆU2 module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Consider the ˆU2 ˆ⊗ACℓ2-graded module M ⊗AΛ• 2, where the grading is given by charge on Λ• 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The element d(2) acts on this module as a boundary operator of degree one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Define Ψn(M) as the corresponding cohomology of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similar constructions apply to modules over the algebras ˆU1, ˆUt, ˆUs or ˆUt,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let Z2 be the center of the algebra ˆU2, and similarly introduce the center Z1 of ˆU1 and the centers Zt and Zs of ˆUt and ˆUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' If M is an ˆU2-module, the action of Z2 on M ⊗A Λ• 2 commutes with the differential d(2) and preserves the charge, hence induces an action of Z2 on the semi-infinite cohomology groups of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' A similar action is defined in the case of ˆU1-modules or ˆUt-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall that a module M over a topological algebra R is said to be smooth if the action of R on M is continuous with respect to the discrete topology on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Notice that, if M is a smooth ˆU2-module, then, since the map E has dense image, the action of ˆU2 on M extends to a smooth action of ˆUt,s on M[a−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similarly, if Mt is a smooth ˆUt-module and Ms is a smooth ˆUs-module, then there is an induced action of ˆUt,s on Mt ⊗Q Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In the next section we will use the following properties of the semi-infinite cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' a) Given a short exact sequence of ˆU2-modules, there is an induced long exact sequence in semi-infinite cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) Let M be an ˆU1-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The semi-infinite cohomology of M as an ˆU1-module is isomorphic to the semi-infinite cohomology of M considered as an ˆU2- module through the map Sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' c) Let M be an ˆUt,s-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The semi-infinite cohomology of M as an ˆUt,s- module is isomorphic to the semi-infinite cohomology of M considered as an ˆU2-module through the map E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, this applies to the case where M = N[a−1] is the localization of a smooth ˆU2-module N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' d) Let Mt be a smooth ˆUt-module, Ms be a smooth ˆUs-module, and let M := Mt ⊗Q Ms, regarded as a ˆUt,s-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The complex computing the semi- infinite cohomology of M is the total complex associated with the double com- plex given by the tensor product of the complex computing the semi-infinite cohomology of Mt and that of Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, being the base ring Q a field, if Mt and Ms have non zero semi-infinite cohomology only in degree zero, then M considered as an ˆUt,s-module has semi-infinite cohomology only in degree zero and the cohomology in degree zero is isomorphic to the product of the tensor product of Ψ0(Mt) and Ψ0(Ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Part a) follows from the fact that Λ2 is a free module over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Part b) follows from the fact that, since a ∈ A acts trivially on M, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 a) we have M ⊗A Λ• 2 ≃ M ⊗C Λ• 2 aΛ• 2 ≃ M ⊗C Λ• 1 and moreover, by construction, d(1) = Sp(d(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Part c) follows from the fact that, since the action of a on M is invertible, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 c) we have M ⊗A Λ• 2 = M ⊗A Λ• 2[a−1] = M ⊗A Λ• t,s SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 11 and, moreover, by construction, d(t,s) = E(d(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 c) we have (Mt ⊗Q Λ• t) ⊗Q (Ms ⊗Q Λ• s) ≃ M ⊗Q Λ• t,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Part d) then follows from the equality d(t,s) = I(d(t)) + J(d(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Commutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For their computation of the semi-infinite cohomo- logy of V, Frenkel and Ben Zvi (see [5] Chapter 15) relied on the choice of a clever basis of V ⊗ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For all x ∈ g, they define ˆx = xt−1 · |0⟩V ⊗ |0⟩Λ + N(αx), where αx ∈ n∗ +⊗n∗ + represents the linear map n+ → n+ obtained as the composition of adx : n+ −→ n+, the natural projection π : g −→ g/b−, and the inverse of the isomorphism n+ ∼= g/b− induced by π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Using the map Φ(2) from Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2) we define ˆx(2) = Φ(2)(ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' To compute the semi-infinite cohomology of Vλ,µ 2 we will need some information about the commutation relations among the elements ˆx(2), ψ[2], and (ψ∗)[2], and the boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These are easy to compute because all these objects are constructed through the map Φ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let us make this remark precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Given an element x in V⊗Λ, denote by x(z) the corresponding field in the vertex superalgebra and by x(2) : K2 −→ ˆU2 ˆ⊗Cℓ2 the 2-field Φ(2)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For any choice of elements x, y ∈ V ⊗ Λ, the commutator of the corresponding fields is given by [x(z), y(w)] = � n⩾0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (x(n)y)(w)∂n wδ(z − w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We have a similar Operator Product Expansion formula for 2-fields (see [1], Pro- position 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4) [x(2)(f), y(2)(g)] = � n⩾0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' � (x(2))(n)(y(2)) � (g ∂nf), where the product (x(2))(n)(y(2))) is the product of 2-fields defined in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' However, since Φ(2) is a map of vertex algebras we get (x(2))(n)(y(2)) = (x(n)y)(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence, if we know the commutator of x(z), y(w), we immediately deduce that of x(2) and y(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similar considerations apply when we want to compute [x(2)(1), y(2)(g)] assuming we know the commutator of x(0) and y(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In this case, the usual OPE formula gives [x(0), y(w)] = (x(0)y)(w), while the OPE formula for 2-fields gives [x(2)(1), y(2)(g)] = � (x(2))(0)(y(2)) � (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Using again the fact that Φ(2) is a map of vertex algebras, we get [x(2)(1), y(2)] = Φ(2) �� [x(0), y(w)](|0⟩V ⊗ |0⟩Λ) � |w=0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These formulas are enough to determine all commutation relations among the ele- ments ˆx(2), ψ[2], (ψ∗)[2] and the boundary operators from those obtained by Frenkel and Ben Zvi in [5, Chapter 15], without the need of any further computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We summarise these results in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4 below, which (in light of the above) follows from Sections 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='9 of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In the statement, we denote by epr, hpr, fpr the sl(2)-triple such that fpr = � α simple λαfα, κ(fpr, eα) = 1 for all simple root α and hpr ∈ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 12 FORTUNA, LOMBARDO, MAFFEI, MELANI Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' for all x ∈ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' y ∈ b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' z ∈ n+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' w ∈ b−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ψ ∈ n+ and ψ∗ ∈ n∗ + we have: a) (d(2) std)2 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [d(2) std,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' χ(2)]+ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) (χ(2))2 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (d(2))2 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' c) [χ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ψ[2]]+ = ⟨ψ∗ pr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ψ⟩ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [χ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (ψ∗)[2]]+ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' d) [χ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ˆz(2)] = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [χ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ˆw(2)] = � α ∈Φ+ κ([fpr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' z],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' eα)ψ∗ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' e) [d(2) std,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ψ[2]]+ = ˆψ(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [d(2) std,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (ψ∗)[2]]+ = −1 2Φ � 1 ˆU2 ⊗ N(ψ∗ ◦ B) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' f) [d(2) std,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ˆy(2)]+ = 0 where in the second formula of e) the element ψ∗ ◦ B ∈ n∗ + ⊗ n∗ + represents the composition of the bracket with the map ψ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, if we choose a basis Ja as at the beginnin of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2, for all γ ∈ Φ+ we have [d(2) std, ˆf (2) γ ]+ = � α∈Φ+,a∈Φ−⊔Γ cα,−γ a ( ˆJa)(2) (−1)(ψ∗ α)[2] − 1 2 κ(e−γ, fγ) ∂(ψ∗ −γ)[2] − � α,β∈Φ+, a∈Φ⊔Γ cα,a β cβ,−γ a ∂(ψ∗ α)[2] By specialisation and localization we obtain that similar formulas hold also in the case of our various other superalgebras ˆUt ˆ⊗Cℓt, ˆUt,s ⊗ Cℓt,s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The semi-infinite cohomology of Vλ,µ 2 In this section we compute the semi-infinite cohomology of Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by C• 2 = C• 2(λ, µ) the complex Vλ,µ 2 ⊗A Λ• 2 and similarly we introduce the complexes C• t = C• t (λ) = Vλ t ⊗Q Λ• t and C• s = C• s (µ) = Vµ s ⊗Q Λ• s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We further introduce the complexes C• 1(ν) = Vν 1 ⊗C Λ• 1 and C• 1(λ, µ) = Wλ,µ 1 ⊗C Λ• 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence, we have C• 1(λ, µ) ≃ ⊕C• 1(ν), where the sum ranges over the irreducible factors of V λ ⊗ V µ counted with multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by Op1 the indscheme of opers on the punctured disc and, for every integral dominant weight ν, we write Opν 1 for the associated connected component of the space of unramified opers without monodromy, equipped with its reduced structure (see, for example, [7] for a more complete definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We also denote by vν a highest weight vector in the g-module V ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Feigin and Frenkel [2] constructed an isomorphism F1 : Funct(Op1) −→ Z1 between the space of functions over Op1 and the center Z1 of ˆU1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall the following result, which combines Theorem 1, Theorem 2 and the proof of Proposition 1 in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 (Frenkel and Gaitsgory [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The action of Z1 on Vν 1 and the Feigin- Frenkel isomorphism induce an isomorphism G1 : Funct(Opν 1) −→ Endˆg1(Vν 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, the element vν ⊗|0⟩Λ is a cocycle in C• 1(ν) and the map z �→ [z ·vν ⊗|0⟩Λ from Z1 to Ψ0(Vν 1) induces isomorphisms of Z1-modules Funct(Opν 1) ≃ Endˆg1(Vν 1) ≃ Ψ0(Vν 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, Ψn(Vν 1) vanishes for all n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The result of Frenkel and Gaitsgory generalises easily to the case of the modules Vλ t and Vµ s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 13 By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3, as in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3, by the compatibility of boundary operators we get homomorphisms of complexes Sp : C• 2 → C• 1(λ, µ) and E : C• 2 → C• t (λ) ⊗Q C• s (µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These induce isomorphisms C• 2[a−1] ≃ C• t (λ) ⊗Q C• s (µ) and C• 2 aC• 2 ≃ C• 1(λ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1) From these isomorphisms and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 it follows easily that Ψn(Vλ,µ 2 ) is zero for n ̸= 0, 1, and we could also get information on the cohomology in degrees zero and one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' However, it is easier to compute these cohomology groups directly by adapting the strategy employed by Frenkel and Ben Zvi in [5, Chapter 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In order to do this, we now introduce certain subcomplexes of C• 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by 1V0,0 2 the element 1 ∈ A ⊗C C ⊗C C ⊂ V0,0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by E• 2 the subcomplex of C• 2(0, 0) spanned by elements of the form ˆx(2) 1 (g1) · · · ˆx(2) a (ga) · 1V0,0 2 ⊗ ψ(2) 1 (ℓ1) · · · ψ(2) b (ℓb) · |0⟩Λ2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2) where xi, ψi ∈ n+ and g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' , ga, ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' ℓb ∈ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By the commutation relations of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 we see that E• 2 is a subcomplex of C• 2(0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We define also analogous complexes E• t , E• s and E• 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These complexes were de- noted by C′ in [5] and by C0 in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By construction, these subcomplexes are compat- ible with specialisation and localization, and there are isomorphisms E• 2/aE• 2 ≃ E• 1 and E• 2[a−1] ≃ E• t ⊗Q E• s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by D• 2 = D• 2(λ, µ) the subcomplex of C• 2(λ, µ) spanned by elements of the form ˆy(2) 1 (h1) · · · ˆy(2) c (hc) · w ⊗ (ψ∗ 1)(2)(k1) · · · (ψ∗ d)(2)(kd) · |0⟩Λ2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3) where w ∈ V λ ⊗ V µ, yi ∈ b− = n− + t, ψ∗ i ∈ n∗ + and h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' , hc, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' , kd ∈ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By the commutation relations of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 we see that E• 2 is a subcomplex of C• 2(λ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We define also analogous complexes D• t (λ), D• s(µ) and D• 1(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These complexes were denoted by C0 in [5] and by C′ in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, we denote by D• 1(λ, µ) the analogous subcomplex of C• 1(λ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By construction, these subcomplexes are com- patible with specialisation and localization, and there are isomorphisms D• 2/aD• 2 ≃ D• 1(λ, µ) and D• 2[a−1] ≃ D• t (λ) ⊗Q D• s(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' There is an isomorphism of complexes E• 2 ⊗ D• 2 −→ C• 2 defined by � x · 1V0,0 2 ⊗ ψ · |0⟩Λ2 � ⊗ � y · w ⊗ ψ∗ · |0⟩Λ2 � �−→ x · y · w ⊗ ψ · ψ∗ · |0⟩Λ2, where x = ˆx(2) 1 (g1) · · · ˆx(2) a (ga) and ψ = ψ(2) 1 (ℓ1) · · · ψ(2) b (ℓb) are as in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2), y = ˆy(2) 1 (h1) · · · ˆy(2) c (hc) and ψ∗ = (ψ∗)(2)(k1) · · · (ψ∗)(2)(kd) are as in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3), and w is an element of V λ ⊗ V µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now compute the cohomology of the complex E• 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We will need the following result by Frenkel and Ben Zvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4 ([5, Section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hn(E• 1) = 0 for n ̸= 0 and Ψ0(E• 1) = C[|0⟩V ⊗ |0⟩Λ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This result generalizes easily to the case of E• t and E• s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Localizing and special- izing, we deduce the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hn(E• 2) = 0 for n ̸= 0 and H0(E• 2) = A[1V0,0 2 ⊗ |0⟩Λ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 14 FORTUNA, LOMBARDO, MAFFEI, MELANI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By definition, the complex E• 2 is concentrated in non-positive degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence, the long exact sequence induced by 0 � E• 2 a· � E• 2 � E• 1 � 0 implies that Hn(E• 2) is torsion free for every n, and that the specialisation of H0(E• 2) is isomorphic to H0(E• 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Since semi-infinite cohomology commutes with localization (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3), using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4 we get the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ We now compute the cohomology of D• 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The strategy is similar, but the ar- gument is less straightforward since we do not have an explicit representative for H0(D• 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Following the strategy in [5], we introduce the following bigraded struc- ture on D• 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall that the height ht(α) of a root α is equal to the sum of the coefficients of α when written as a sum of simple roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let also epr, hpr, fpr be an sl(2)-triple such that fpr = � α simple fα and hpr belongs to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We define a bidegree, with values in 1 2Z × 1 2Z and denoted by bideg, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' On elements of ˆg2, we set bideg(x ⊗ g) = (−n, n) if x ∈ g is such that [hpr, x] = 2 n x and g ∈ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We set also the bidegree of the central element C2 ∈ ˆg2 to be (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This induces a bidegree on U(ˆg2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' On the space X2 = K2 ⊗ n+ ⊕ K2 ⊗ n∗ + (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4) we define bideg eα ⊗ g = (− ht(α), −1 + ht(α)) bideg ψ∗ α ⊗ g = (ht(α), 1 − ht(α)) for α a positive root and g any element of K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This induces a bidegree on the Clifford algebra Cℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, if W is any finite-dimensional representation of g, then we set bideg w = (−n, n) if w ∈ W is such that hpr · w = 2 n w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' These choices induces a bidegree on the module C• 2(λ, µ), and the element ˆx(2)(g) is homogeneous of bidegree (−n, n) if [hpr, x] = 2 n x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, notice that if an element has bidegree (p, q), then it has charge p + q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, we introduce the submodule Dp,q 2 of elements of Dp+q 2 of bidegree (p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We notice also that bideg d(2) std = (0, 1) and that bideg χ(2) = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particu- lar, D•,• 2 is a double complex and D• 2 is the associated total complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Following Frenkel and Ben Zvi [5, Chapter 15], the cohomology of the rows of this double complex is easy to describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let a be the centralizer of fpr in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall from [5, Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 and Section 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='9] that the space spanned by monomials of the form (ˆp1)n1 · · · (ˆpk)nk ·|0⟩V ⊗|0⟩Λ with pi ∈ a generates a commutative vertex subalgebra F1 of V ⊗ Λ• isomorphic to S•(a ⊗ t−1C[t−1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3, it follows that for x, y ∈ a the fields ˆx(2) and ˆy(2) commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We define F2(λ, µ) as the span of elements of the form ˆx(2) 1 (g1) · · · ˆx(2) k (gk) · (v ⊗ |0⟩Λ2) ∈ Vλ,µ 2 ⊗A Λ• 2 with x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' , xk ∈ a and v ∈ V λ ⊗ V µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Notice that all these elements have charge equal to zero, and that the space F2(λ, µ) splits as a direct sum F2(λ, µ) = � q F −q,q 2 (λ, µ) according to the bidegree introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, by Propos- ition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4 d), these elements are annihilated by the action of χ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similarly we construct subspaces F −q,q 1 (ν) ⊂ Vν 1 ⊗C Λ• 1, F −q,q t (λ) ⊂ Vλ t ⊗Q Λ• t , F −q,q s (µ) ⊂ Vλ s ⊗Q Λ• s, and F −q,q 1 (λ, µ) ⊂ Wλ,µ 1 ⊗C Λ• 1, In particular, F −q,q 1 (λ, µ) = SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 15 � ν F −q,q 1 (ν) where the sum is over all irreducible factors of V λ ⊗C V µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By con- struction, the specialisation and localization maps induce isomorphisms F −q,q 2 (λ, µ) aF −q,q 2 (λ, µ) ≃ F −q,q 1 (λ, µ) and F −q,q 2 (λ, µ)[a−1] ≃ � b+c=q F −b,b t (λ) ⊗Q F −c,c s (µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall the following result on the cohomology of D•,q 1 with respect to the bound- ary χ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='7 ([5, Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='10] and [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let 2pν = ⟨ν, hpr⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' a) Dp,q 1 (ν) = 0 for q > pν and for p < −q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, Dp,q 1 = 0 for q > pλ+µ and for p < −q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) Hn(D•,q 1 (ν)) = 0 for n ̸= −q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, Hn(D•,q 1 (λ, µ)) = 0 for n ̸= −q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' c) The map v �→ [v] from F −q,q 1 (ν) to H−q(D•,q 1 (ν)) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, it follows from c) that the map v �→ [v] from F −q,q 1 (λ, µ) to H−q(D•,q 1 (λ, µ)) is also an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similar results hold for the complexes D•,q t (λ) and D•,q s (µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' From this result we deduce the cohomology of the complex D•,q q with respect to the boundary operator χ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let 2p0 = ⟨λ + µ, hpr⟩ as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' a) Dp,q 2 = 0 for q > p0 and for p < −q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) Hn(D•,q 2 ) = 0 for n ̸= −q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' c) The map v �→ [v] from F −q,q 2 (λ, µ) to H−q(D•,q 2 (λ, µ)) is an isomorphism of A-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Part a) is clear for the definition of Dp,q 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For parts b) and c), we start by studying the localization of the cohomology groups of D•,q 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Equivalently, we aim to compute the cohomology of the localization of the row D•,q 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This localization can be rewritten as � b+c=q D•,b t (λ) ⊗ D•,c s (µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, it follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='7 that its cohomology is concentrated in degree −q, and that its cohomology in this degree is given by � b+c=q F −b,b t (λ) ⊗ F −c,c s (µ), which is the localization of F −q,q 2 (λ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Since specialisation is compatible with bideg, we have an isomorphism D•,q 2 /aD•,q 2 ≃ D•,q 1 (λ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='7, the associated long exact sequence shows that Hn(D•,q 2 ) is torsion-free for n ̸= −q + 1, and that the map ι : H−q(D•,q 2 )/aH−q(D•,q 2 ) → H−q(D•,q 1 (λ, µ)) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now prove c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Notice that both F −q,q 2 (λ, µ) and H−q(D•,q 2 (λ, µ)) are torsion- free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We have already shown that the localization of the natural maps between them is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' To study its specialisation, we compose it with the injection ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This composition is the isomorphism of the last remark of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We conclude by applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In order to prove b), it is enough to notice that from the above discussion we know that, for n ̸= −q, the module Hn(D•,q 2 ) = 0 is torsion-free, and that its localization is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ 16 FORTUNA, LOMBARDO, MAFFEI, MELANI Let now be ϕ(q) i be an A-basis of F −q,q 2 (λ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Since the cohomology in degree −q of the complex D•,q+1 2 is zero, there exists an element ϕ(q) i,1 ∈ D−q−1,q+1 2 such that χ(2)(ϕ(q) i,1 ) = −d(2) std(ϕ(q) i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By induction, we can construct elements ϕ(q) i,0 = ϕ(q) i and ϕ(q) i,ℓ ∈ D−q−ℓ,q+ℓ 2 such that their sum ˜ϕ(q) i = p0−q � ℓ=0 ϕ(q) i,ℓ satisfies d(2)( ˜ϕ(q) i ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now prove the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' a) Ψn(Vλ,µ 2 ) = 0 for n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) We have an isomorphism Ψ0(Vλ,µ 2 ) aΨ0(Vλ,µ 2 ) ≃ Ψ0(Wλ,µ 1 ) ≃ � ν Ψ0(Vν 1) where the sum ranges over all irreducible components V ν of V λ ⊗V µ, counted with multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' c) The elements � ˜ϕ(q) i � are an A-basis of Ψ0(Vλ,µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='8 we deduce that the classes of the elements ˜ϕ(q) i form an A-basis of H0(D• 2), and that Hn(D• 2) = 0 for n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As the complex D• 2 is con- centrated in non-negative degrees, by a standard homological argument we deduce that Hn(Vλ,µ 2 ) is isomorphic to the n-th cohomology of the complex H0(D• 2)⊗AE• 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5, we immediately obtain parts a) and c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The second isomorphism appearing in part b) is clear, while the first follows from a) and the long exact sequence associated with the isomorphism C• 2 aC• 2 ≃ C• 1(λ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ We will use the following Corollary in the next Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The element [vλ ⊗ vµ] ∈ Ψ0(Vλ,µ) is indivisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By the previous theorem we can choose [vλ ⊗ vµ] as an element of a basis of the free A module Ψ0(Vλ,µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The action of the center In this section we study the action of the center Z2 on the semi-infinite cohomo- logy of the module Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In this section we show that Vλ,µ 2 is not a perfect analogue of the Weyl module Vν 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Indeed, we show that, as a Z2-module, the semi-infinite cohomology of Vλ,µ 2 is not isomorphic to Endˆg2(Vλ,µ 2 ) or to Funct(Opλ,µ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We begin by observing that the module Ψ0(Vν 1) has no non-trivial Z1-equivariant automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' First we notice, that by construction, the action of Z2 commutes with localization and specialisation, as introduced before Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Concretely, we have: Et(z · x) = Et(z) · Et(x), Es(z · x) = Es(z) · Es(x), Sp(z · x) = Sp(z) · Sp(x) for all z ∈ Z2 and for all x ∈ Ψ0(Vλ,µ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' If K : Endˆgt(Vλ t )⊗QEndˆgs(Vµ s ) −→ Ψ0(Vλ t )⊗QΨ0(Vµ s ) is a (Zt⊗Zs)- equivariant isomorphism, then K(IdVλ t ⊗ IdVµ s ) = q[vλ] ⊗ [vµ] for some q ∈ Q ∖ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' It follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 that Endˆgt׈gs(Vλt ⊗Q Vµs) is isomorphic to Funct(Opλ t ×Spec Q Opµ s ) = Funct(Opλ t ) ⊗Q Funct(Opµ s ) and this is a polynomial ring in infinitely many variables over the field Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, its only invertible elements are the non-zero scalars in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 also implies that Funct(Opλ t ) is isomorphic as a Zt- module to Ψ0(Vλ t ), with an isomorphism given by z −→ Gt(z) · [vλ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ Before proving that Vλ,µ 2 does not have the “right” semi-infinite cohomology we recall some properties of the modules Vν 1 that will be needed also in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by Zν 1 the coordinate ring of the scheme Opν 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall that the schemes Opν 1 for different values of ν are disjoint, so that the map Z1 −→ Zν1 1 × · · · × Zνk 1 is surjective if the weights νi are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall also that the ring Zν 1 is a polynomial ring in infinitely many variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This implies that (1) There are no nontrivial ˆg1-morphisms between the ˆU1-modules Vν 1 and Vν′ 1 if ν ̸= ν′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (2) There are no nontrivial extensions between the ˆU1-modules Vν 1 and Vν′ 1 if ν ̸= ν′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (3) Assume that α : � Zνi −→ � Zνi is a map of Z-modules and that the weights νi are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' If 1 is in the image of α then α is an isomorphism and α(Zνi 1 ) = Zνi 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By the Feigin-Frenkel Theorem (see [4] Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2) the ring Funct(Op2) is isomorphic to Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In the sequel we will identify these rings through this isomorph- ism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular the ring Funct(Opλ,µ 2 ) is a quotient of Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We will denote Funct(Opλ,µ 2 ) by Zλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now prove that Zλ,µ 2 and Ψ0(Vλ,µ 2 ) are not isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Assume that V λ ⊗ V µ is not irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Then the two Z2- modules Endˆg2(Vλ,µ 2 ) and Ψ0(Vλ,µ 2 ) are not isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Similarly the two Z2- modules Zλ,µ 2 and Ψ0(Vλ,µ 2 ) are not isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Suppose H : Endˆg2(Vλ,µ 2 ) −→ Ψ0(Vλ,µ 2 ) is a Z2-equivariant isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='28 in [4] that Z2[1/a] is dense in Zt,s, and therefore the localization of H is a (Zt ⊗Q Zs)-equivariant isomorphism Endˆgt(Vλ t ) ⊗Q Endˆgs(Vµ s ) −→ Ψ0(Vλ t ) ⊗Q Ψ0(Vµ s ), where we used the identification of the localization of Ψ0(Vλ,µ 2 ) with Ψ0(Vλ t ) ⊗Q Ψ0(Vµ s ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='10 we deduce that H(IdVλ,µ 2 ) = [q vλ ⊗ vµ], where q ∈ A and qvλ ⊗ vµ ∈ Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We set w = qvλ ⊗ vµ ∈ Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By specialisation, H gives a Z1-equivariant isomorphism H : Endˆg2(Vλ,µ 2 ) a Endˆg2(Vλ,µ 2 ) −→ Ψ0(Vλ,µ 2 ) aΨ0(Vλ,µ 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1) This isomorphism sends IdVλ,µ 2 to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Now consider the decomposition V λ ⊗ V µ = � V ν as g-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='9, the target of the map H in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1) decomposes as � Ψ0(Vν 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The element w is a multiple of vλ ⊗ vµ hence its class belongs to Ψ0(Vλ+µ 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As H is Z1-equivariant and Vλ+µ 1 is stable by the action of ˆg1, we get that the image of H is contained in the direct summand Ψ0(Vλ+µ 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, if V λ ⊗ V µ is not irreducible, the map H cannot be surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This proves the 18 FORTUNA, LOMBARDO, MAFFEI, MELANI first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The second claim follows since the map from Zλ,µ 2 to Ψ0(Vλ,µ 2 ) factors through Endˆg2(Vλ,µ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' A Weyl module for sl(2) In this section, we propose an alternative Weyl module in the context of opers with two singularities, in the case of g = sl(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We fix the following notation: e, h, f is an sl(2)-triple such that h ∈ t and e ∈ n+, while ψ∗ ∈ n∗ + is the dual of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We identify dominant weights with natural numbers and we assume from now on that λ ⩾ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In this case, the differential of the complex computing semi-infinite cohomology takes the simpler form d(2) = ψ∗ + � ewn ⊗ ψ∗z−n−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let �Vλ,µ 2 be the ˆU2-submodule of Vλ,µ 2 generated by the highest weight vector 1 ⊗ vλ ⊗ vµ ∈ A ⊗ V λ ⊗ V µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We will prove that this module has the “correct” semi-infinite cohomology and the “correct” endomorphism ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We start by giving a more explicit description of the module �Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' If X is a subspace of U(g) and Y is a subspace of a g-module Z we denote by X · Y the subspace of Z generated by the products x · y with x ∈ X and y ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We define an increasing filtration F i of �Vλ,µ 2 as follows F i = U(g) · (C Id ⊗ Id + Id ⊗ g)i · (vλ ⊗ vµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This is an increasing filtration of V λ ⊗ V µ by g-modules and for i large enough we have F i = V λ ⊗ V µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Choose a g-stable complement Gi+1 of F i in F i+1 and set G0 = F 0, so that F i = �i j=0 Gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' If we set F i(V µ) = (CId + n−)ivµ, it is easy to check by induction on i that F i = U(g) · (Id ⊗ Id + Id ⊗ n−)i(vλ ⊗ vµ) = U(g) · � V λ ⊗ F i(V µ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In the case of g = sl(2) we have Gi ≃ V λ+µ−2i and F µ(V µ) = V µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Let U − 2 ⊂ U(ˆg2) be the A-span of Poincar´e-Birkhoff-Witt monomials of the form (x1wa1) · · · (xkwak) with xi ∈ g and ai < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This is a complement of U(ˆg+ 2 ) in U(ˆg2), so that in particular we have Vλ,µ 2 = U − 2 ⊗C (V λ ⊗ V µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' If λ ⩾ µ then �Vλ,µ 2 = µ � i=0 aiU − 2 ⊗C F i = µ � i=0 aiU − 2 ⊗C Gi Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' To understand the module �Vλ,µ 2 we need to compute the ˆg+ 2 -submodule of A⊗C V λ ⊗C V µ generated by 1⊗vλ ⊗vµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Notice that every element of the form xg, with x ∈ g and g ∈ C[[t, s]] divisible by ts, acts trivially on A ⊗ V λ ⊗ V µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence we need to understand the action of elements of the form z = x1 · · · xℓ · (y1t) · · · (ymt) · (vλ ⊗ vµ), with xi, yi ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, elements of g act in the standard way on the tensor product V λ ⊗ V µ, while elements of the form xt with x ∈ g act via −a(Id ⊗ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This implies the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ We now describe the specialisation of the module �Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We introduce the fol- lowing decreasing filtration of �Vλ,µ 2 : Fi = �Vλ,µ 2 ∩ aiVλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1) SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 19 By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 we have the following description of the terms of this filtration as A-modules: Fi = aiU − 2 ⊗C F i ⊕ µ � j=i+1 ajU − 2 ⊗C Gj In particular we have F0 = �Vλ,µ 2 , Fj = ajVλ,µ 2 for j ⩾ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' a) Let ui ∈ Gi be the highest weight vector and set ˜wi = aiui ∈ �Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Then ˜wi ∈ �Vλ,µ 2 and ai−1ui /∈ �Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) There is an isomorphism of ˆU1-modules Fi + a�Vλ,µ 2 a�Vλ,µ 2 ≃ µ � j=i Vλ+µ−2j 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The quotient Fi+a�Vλ,µ 2 a�Vλ,µ 2 is generated as a ˆU1-module by the classes of ˜wi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' , ˜wµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular �Vλ,µ 2 /a�Vλ,µ 2 ≃ Wλ,µ 1 is generated by ˜w0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' , ˜wµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The first claim follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We prove part b) by decreasing induction on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1, for i > µ the quotient is zero and the claim is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For i ⩽ µ, consider the map U − 2 ⊗ Gi −→ Fi + ai+1Vλ,µ 2 ai+1Vλ,µ 2 + Fi ∩ a�Vλ,µ 2 ≃ (Fi + a�Vλ,µ 2 )/a�Vλ,µ 2 (Fi+1 + a�Vλ,µ 2 )/a�Vλ,µ 2 sending an element u ⊗ v to the class of aiu ⊗ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This map induces an isomorphism U − 2 aU − 2 ⊗ Gi ≃ (Fi + a�Vλ,µ 2 )/a�Vλ,µ 2 (Fi+1 + a�Vλ,µ 2 )/a�Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2) Moreover, notice that U− 2 aU− 2 ⊗ Gi ≃ U − 1 ⊗ Gi, where U − 1 = U(t−1g[t−1]) ⊂ U(ˆg1) = U1, and that U − 1 ⊗ Gi has a natural structure of U1-module, as it can be identified with Vλ+µ−2i 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' With this U1-action, the isomorphism 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 is U1-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Now the claim follows by the inductive hypothesis, combined with the fact that there are no nontrivial extensions between modules Vν 1 and Vν′ 1 if ν ̸= ν′ and that the highest weight vector of V ν generates the module Vν 1 as an U1-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ Notice that, although the specialisations at a = 0 of Vλ,µ 2 and �Vλ,µ 2 are iso- morphic, the specialisation of �Vλ,µ 2 , is generated by vλ ⊗ vµ while in the first case this vector generates the submodule Vλ+µ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As a corollary, we get the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The following hold: a) Ψn(�Vλ,µ 2 ) = 0 for n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' b) The inclusion of �Vλ,µ 2 in Vλ,µ 2 induces isomorphisms Ψ0(�Vλ,µ 2 )[a−1] ≃ Ψ0(Vλ,µ 2 )[a−1] ≃ Ψ0(Vλ t ) ⊗Q Ψ0(Vµ s ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' c) Ψ0(�Vλ,µ 2 ) is torsion-free with respect to the action of A, and the natural pro- jection induces isomorphisms Ψ0(�Vλ,µ 2 ) aΨ0(�Vλ,µ 2 ) ≃ Ψ0 � �Vλ,µ 2 a�Vλ,µ 2 � ≃ Ψ0(Wλ,µ 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We use the filtration introduced in Equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Notice that Fi Fi+1 = aiU − 2 ⊗ F i ai+1U − 2 ⊗ F i ≃ U − 1 ⊗C F i ≃ Indˆg1 ˆg+ 1 F i, 20 FORTUNA, LOMBARDO, MAFFEI, MELANI where we consider F i as a ˆg+ 1 -module on which tg[t] acts trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Notice that Indˆg1 ˆg+ 1 F i is a sum of modules of the form Vν 1, hence in particular has trivial non- zero cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence, arguing by decreasing induction on i, starting from i = µ, it follows that Fi has trivial semi-infinite cohomology in degree different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Indeed for i = µ we have Fµ = aµVλ,µ 2 ≃ Vλ,µ 2 and this is the content of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For i = 0 this implies claim a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Part b) follows from the fact that semi-infinite cohomology commutes with local- ization (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3) combined with the isomorphism �Vλ,µ 2 [a−1] = Vλ,µ 2 [a−1] ≃ Vλ t ⊗Q Vµ s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' To prove c), consider the exact sequence 0 � �Vλ,µ 2 a � �Vλ,µ 2 � �Vλ,µ 2 a�Vλ,µ 2 � 0 By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2, the last module in this sequence is isomorphic to Wλ,µ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In par- ticular, the semi-infinite cohomology groups Ψn of the modules appearing in this sequence are zero for n ̸= 0, and c) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ To prove that the semi-infinite cohomology of �Vλ,µ 2 is isomorphic to Zλ,µ 2 we will use the action of a particular central element in Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Recall from [4] the definition of the 2-Sugawara operator S(2) 1/2 = � n∈ 1 2 Z,b : (Jbwn)(Jbz−n) : (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3) where J1, J2, J3 are the basis elements e, h, f and J1, J2, J3 are the dual basis elements f, h/2, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' As proved in [4], the element S(2) 1/2 is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Its specialisation is the Sugawara operator S(1) 1 = � n∈Z,b : (Jbtn) (Jbt−n) : (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4) which is an element of Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' It is straightforward to check that the action of S(1) 1 on the Weyl module Vν 1 is given by multiplication by ν(ν + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The element ˆwℓ = � et−1�ℓ ˜wℓ belongs to Z2 · (vλ ⊗ vµ) + a�Vλ,µ 2 for ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' , µ, Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We notice first that the element vλ ⊗ f ℓvµ belongs to F ℓ \\ F ℓ−1 and has weight λ + µ − 2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence, up to a non-zero constant we have vλ ⊗ f ℓvµ = uℓ + u′ ℓ, where we recall that uℓ is the highest weight vector in Gℓ ≃ V λ+µ−2ℓ ⊂ V λ ⊗ V µ and u′ ℓ ∈ F ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, recall from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 that aℓ−1F ℓ ⊂ �Vλ,µ 2 , hence aℓ � et−1�ℓ vλ ⊗ f ℓvµ = � et−1�ℓ ˜wℓ + � et−1�ℓ (aℓu′ ℓ) ≡ � et−1�ℓ ˜wℓ mod a�Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence, the lemma is equivalent to the fact that ˆwℓ = aℓ � et−1�ℓ vλ ⊗ f ℓvµ is in Z2 · vλ ⊗ vµ + a�Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We prove this statement by induction on ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For ℓ = 0 it is trivially true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Now assume ˆwℓ is in Z2 · vλ ⊗ vµ + a�Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We compute S(2) 1/2( ˆwℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In order to do this, we notice that the action of xtisj on �Vλ,µ 2 /a�Vλ,µ 2 is equal to the action of xti+j on the same module, and that vλ ⊗ e f ℓvµ is in F ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We have S(2) 1/2 ˆwℓ = 2 � n>0 et−n · ftn · ˆwℓ + 2 � n>0 ft−n · etn · ˆwℓ + � n>0 ht−n · htn · ˆwℓ + e · f · ˆwℓ + e · f · ˆwℓ + 1 2h · h · ˆwℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 21 In the second infinite sum above, the element etn commutes with et−1, hence etn · ˆwℓ ∈ a�Vλ,µ 2 for all n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The summands of the third series are of the form htn · (et−1)ℓ · ˆwℓ = (et−1)ℓhtn · ˆwℓ + 2ℓ(et−1)ℓ−1etn−1 · ˆwℓ, hence they vanish for n ⩾ 3, while for n = 1, 2 they are easily checked to be elements of a�Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The summands of the first series are given by ftn·(et−1)ℓ· ˆwℓ = (et−1)ℓftn· ˆwℓ−ℓ(et−1)ℓ−1htn−1· ˆwℓ−ℓ(ℓ−1)(et−1)ℓ−2etn−2· ˆwℓ, and all terms are zero or in a�Vλ,µ 2 but for the case n = 1, for which we get (et−1) · (ft) · (et−1)ℓ · ˆwℓ = aℓ+1(et−1)ℓ+1 · (vλ ⊗ f ℓ+1vµ) − ℓ(et−1)ℓh · (vλ ⊗ f ℓvµ) − ℓ(ℓ − 1)(et−1)ℓ · ˆwℓ = ˆwℓ+1 + K1 ˆwℓ for some constant K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, e·f · ˆwℓ+e·f · ˆwℓ+ 1 2h·h· ˆwℓ belongs to K2 ˆwℓ+a�Vλ,µ 2 for some constant K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence we get S(2) 1/2 ˆwℓ ≡ ˆwℓ+1 + K ˆwℓ mod a�Vλ,µ 2 for some constant K, proving our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ We now prove that the zero-th semi-infinite cohomology of the module �Vλ,µ 2 is isomorphic to Zλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For g = sl(2) the map Φ : Zλ,µ 2 −→ Ψ0��Vλ,µ 2 � given by Φ(z) = z · [vλ ⊗ vµ] is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By [4], Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4, the action of Z2 on Vλ,µ 2 , hence on �Vλ,µ 2 , factors through Zλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover vλ ⊗ vµ is a cycle, so the map Φ is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Since we know that both modules are torsion-free, to prove that Φ is an isomorphism it suffices to prove that the localization Φa and the specialisation Φ are isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The fact that Φa is an isomorphism is the content of part b) of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We need to prove that Φ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='3 and [4, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='13] we have Zλ,µ 2 aZλ,µ 2 ≃ µ � i=0 Zλ+µ−2i 1 and Ψ0(�Vλ,µ 2 ) aΨ0(�Vλ,µ 2 ) ≃ µ � i=0 Ψ0(Vλ+µ−2i 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In particular, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 these two Z1-modules are isomorphic, but we need to prove that our specific map Φ provides an isomorphism between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' By Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 it is enough to prove that Φ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We prove that the image of Φ contains Ψ0(Fℓ+a�Vλ,µ 2 /a�Vλ,µ 2 ) arguing by reverse induction on ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For ℓ = 0 we get our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For ℓ > µ there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Now assume ℓ ⩽ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Consider again the exact sequence 0 � Fℓ+1+a�Vλ,µ 2 a�Vλ,µ 2 � Fℓ+a�Vλ,µ 2 a�Vλ,µ 2 �aℓU − 2 ⊗C Gℓ �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We know that the last module is isomorphic to aℓU − 2 ⊗C Gℓ ≃ Vλ+µ−2ℓ 1 = Indˆg1 ˆg+ 1 (V λ+µ−2ℓ) and that it is generated by the element ˜wℓ ∈ aℓGℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Notice this sequence of Z1- modules splits by Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Taking semi-infinite cohomology we get a short exact sequence 0 �Ψ0 � Fℓ+1+a�Vλ,µ 2 a�Vλ,µ 2 � �Ψ0 � Fℓ+a�Vλ,µ 2 a�Vλ,µ 2 � �Ψ0 � aℓU − 2 ⊗C Gℓ� �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' and we know that the last Z2-module is generated by ˜wℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence it is enough to prove that this element is in the image of Zλ,µ 2 (vλ ⊗vµ) in Ψ0� �Vλ,µ 2 /Fℓ+1 +a�Vλ,µ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 22 FORTUNA, LOMBARDO, MAFFEI, MELANI By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='4 we know that ˆwℓ is in this image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Now we prove that ˜wℓ and ˆwℓ define the same element in the semi-infinite cohomology of aℓU − 2 ⊗C Gℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' This is a claim about the cohomology of the module Vν 1 for ν = λ + µ − 2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' For any ν we prove that � et−1�hvν + � et−1�h−1vν is a coboundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Indeed the boundary operator in the case of sl(2) is equal to d(1) = ψ∗ + � n∈Z (etn) ⊗ ψ∗t−1−n, so a simple computation shows d(1) �� et−1�h−1vν ⊗ (ψt−1)|0⟩Λ � = � et−1�h−1vν ⊗ |0⟩Λ + � et−1�hvν ⊗ |0⟩Λ, which implies our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ Recall that in [4] we computed the endomorphism ring of Vλ,µ 2 , showing that it is isomorphic to Zλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We now prove the same result for the module �Vλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The action of the center Z2 on �Vλ,µ 2 induces an isomorphism Zλ,µ 2 ≃ Endˆg2(�Vλ,µ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We already recalled at the beginning of the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='5 that the action of Z2 on �Vλ,µ 2 factors through Zλ,µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We denote by α : Zλ,µ 2 −→ End(�Vλ,µ 2 ) this action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Since both modules have no A-torsion, in order to prove that α is an isomorphism it suffices to show that its localization and its specialisation are isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, since our modules are finitely generated and have no torsion we have Endˆg2 � �Vλ,µ 2 � [a−1] ≃ Endˆg2[a−1] � �Vλ,µ 2 [a−1] � ≃ Endˆgt,s � Vλ ⊗Q Vµ s � ≃ Zλ t ⊗Q Zµ t ≃ Zλ,µ 2 [a−1], hence the localization of α is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Finally, we prove that the specialisation of α is also an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' We have already recalled that by [4, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='13] we have Zλ,µ 2 /aZλ,µ 2 ≃ �µ i=0 Zλ+µ−2i 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='1 we have the following abstract isomorphisms of Z1-modules: Zλ,µ 2 aZλ,µ 2 ≃ µ � i=0 Zλ+µ−2i 1 ≃ µ � i=0 Endˆg1(Vλ+µ−2i 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, since �Vλ,µ 2 has no nontrivial A-torsion, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 and Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 part (1) we have the inclusion Endˆg1 � �Vλ,µ 2 � a Endˆg1 � �Vλ,µ 2 � ⊂ Endˆg1 � �Vλ,µ 2 a�Vλ,µ 2 � ≃ µ � i=0 Endˆg1(Vλ+µ−2i 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Hence, composing the specialisation of the map α with this inclusion and the iso- morphisms above we get a Z1-equivariant endomorphism of �µ i=0 Zλ+µ−2i 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Moreover, α(1) = 1, hence we conclude by Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='2 (3) that the specialisation of α is also an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' □ References [1] Casarin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' A Feigin Frenkel theorem with n singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' preprint, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [2] Feigin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', and Frenkel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Affine Kac-Moody algebras at the critical level and Gelfand- Diki˘ı algebras.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', River Edge, NJ, 1992, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 197–215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [3] Fortuna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' The Beilinson-Bernstein Localization Theorem for the affine Grassmannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' MIT, PhD thesis, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' SEMI-INFINITE COHOMOLOGY OF WEYL MODULES 23 [4] Fortuna, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', Lombardo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', Maffei, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', and Melani, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Local opers with two singularities: the case of sl(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', 394 (2022), 1303–1360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [5] Frenkel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', and Ben-Zvi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Vertex algebras and algebraic curves, second ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 88 of Mathematical Surveys and Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' American Mathematical Society, Providence, RI, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [6] Frenkel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', and Gaitsgory, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Local geometric Langlands correspondence: the spherical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In Algebraic analysis and around, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 54 of Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Pure Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Japan, Tokyo, 2009, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 167–186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' [7] Frenkel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=', and Gaitsgory, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Weyl modules and opers without monodromy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' In Arithmetic and geometry around quantization, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 279 of Progr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' Birkh¨auser Boston, Boston, MA, 2010, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' 101–121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content=' E-mail addresses: giorgiafortuna@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='com, davide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='lombardo@unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='it, andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='maffei@unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='it, valerio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='melani@unifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} +page_content='it' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFQT4oBgHgl3EQfRTbc/content/2301.13286v1.pdf'} diff --git a/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf b/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf new file mode 100644 index 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b/GNE2T4oBgHgl3EQfTQeG/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:275349d6c2c75874893cc9023b50bc2aa01b7dea1c0b4c8734f7ad94e9aabf0a +size 77261 diff --git a/ItAyT4oBgHgl3EQfTPfP/content/tmp_files/2301.00103v1.pdf.txt b/ItAyT4oBgHgl3EQfTPfP/content/tmp_files/2301.00103v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a58247d10ede6f892a0e1464566bee5a9d259c9d --- /dev/null +++ b/ItAyT4oBgHgl3EQfTPfP/content/tmp_files/2301.00103v1.pdf.txt @@ -0,0 +1,4571 @@ +Renormalization of Ising cage-net model and generalized foliation +Zongyuan Wang,1 Xiuqi Ma,1 David T. Stephen,2, 1 Michael Hermele,2 and Xie Chen1 +1Department of Physics and Institute for Quantum Information and Matter, +California Institute of Technology, Pasadena, California 91125, USA +2Department of Physics and Center for Theory of Quantum Matter, +University of Colorado, Boulder, CO 80309, USA +(Dated: January 3, 2023) +A large class of type-I fracton models, including the X-cube model, have been found to be fixed +points of the foliated renormalization group (RG). The system size of such foliated models can be +changed by adding or removing decoupled layers of 2D topological states and continuous deformation +of the Hamiltonian. In this paper, we study a closely related model – the Ising cage-net model – +and find that this model is not foliated in the same sense. In fact, we point out certain unnatural +restrictions in the foliated RG, and find that removing these restrictions leads to a generalized +foliated RG under which the Ising cage-net model is a fixed point, and which includes the original +foliated RG as a special case. The Ising cage-net model thus gives a prototypical example of the +generalized foliated RG, and its system size can be changed either by condensing / uncondensing +bosonic planon excitations near a 2D plane or through a linear depth quantum circuit in the same +plane. We show that these two apparently different RG procedures are closely related, as they lead +to the same gapped boundary when implemented in part of a plane. Finally, we briefly discuss the +implications for foliated fracton phases, whose universal properties will need to be reexamined in +light of the generalized foliated RG. +I. +INTRODUCTION +The renormalization group (RG) plays a fundamental +role in the characterization and classification of quantum +phases of matter.1–3 It is a piece of conventional wisdom +that each phase – defined as a deformation class of +quantum systems – is characterized by a unique RG fixed +point, which encodes the universal long-distance and low- +energy properties of the phase. Moreover, the existence +of such a fixed point underlies the key role played by +continuum quantum field theory as a tool to describe +universal properties of phases (and phase transitions) +while discarding extraneous non-universal information. +Fracton models in three spatial dimensions (3D)4,5 pro- +vide exceptions to this conventional wisdom, and accord- +ingly challenge our understanding of the relationships +among quantum phases of matter, the renormalization +group, and quantum field theory. +This is nicely illus- +trated in the X-cube model,6 perhaps the simplest frac- +ton model. The defining characteristic of a fracton model +is the presence of excitations of restricted mobility, and +the X-cube model supports point-like excitations mobile +in planes (planons), along lines (lineons), and for which +an isolated excitation is fully immobile (fractons). The +model is exactly solvable and has zero correlation length, +so we might expect it to be a fixed point of the RG, as is +the case for toric code and string-net models.7,8 +However, the X-cube model on a lattice of linear size +L is equivalent (under the application of a finite-depth +circuit) to an X-cube model on a smaller lattice stacked +with 2D toric code layers.9 Therefore, when trying to +coarse-grain the X-cube model, non-trivial 2D layers are +left behind. +These layers cannot be integrated out or +otherwise removed, thus preventing the model from being +a fixed point of any conventional RG procedure. This +behavior is closely related to the striking system-size +dependence of certain properties, such as the ground +state degeneracy (GSD) and the number of types of +fractional excitations, both of which grow exponentially +in the linear system size.9,10 Similar phenomena occur in +other fracton models, including Haah’s cubic code11. +It is interesting to ask whether some fracton models are +fixed points of a suitably generalized RG. While there +are many schemes and procedures for carrying out RG +in different settings, it is important to emphasize that +simply finding a new RG scheme is not enough. Instead, +a more radical generalization of what we usually mean by +RG is needed, because, for instance, any RG procedure +that can have the fracton models as fixed points must +allow for the increase / decrease in GSD and the addition +/ removal of fractional excitations in the process. +Along these lines, it was found the X-cube model +is a fixed point of a foliated RG procedure.9,12–14 It +is helpful to recall the conventional RG procedure for +gapped phases2,3, which allows, in each RG step, for +continuous deformations of the Hamiltonian that keep +the gap open, and for the addition/removal of trivial +gapped systems (those whose ground state is a product +state). In the foliated RG, one also allows addition or +removal of decoupled, gapped 2D systems. +Such 2D +systems can be topologically ordered and thus carry +non-trivial GSD and fractional excitation types, hence +allowing for these properties to change under RG. In +the case of the X-cube model, we can remove 2D toric +code layers under the foliated RG, thus making the model +into a fixed point. More generally, a large class of type- +I fracton models6 – those where some of the fractional +excitations are mobile – are fixed points of the foliated +RG. +The foliated RG leads to the closely related notion +arXiv:2301.00103v1 [cond-mat.str-el] 31 Dec 2022 + +2 +of foliated fracton phases.10,15 Foliated fracton phases, +which we define in Appendix A, are a coarser equivalence +relation on ground states than ordinary phases, and +each foliated fracton phase contains a fixed point of the +foliated RG. This fixed point captures certain universal +properties that are the same everywhere in the foliated +phase, and these properties are referred to as foliated +fracton order. When a model belongs to a foliated fracton +phase, it is a convenient shorthand terminology to refer +to the model as being foliated. +An interesting type-I fracton model that has not been +investigated from this perspective is the Ising cage-net +model.16 The Ising cage-net model is very similar to +the X-cube model in many ways. +Both are exactly +solvable models that can be obtained from a coupled layer +construction, based on toric code layers in the X-cube +case,17,18 and doubled-Ising string-net layers in the cage- +net case.16 Both have fracton excitations that are created +at the corners of a rectangular membrane operator. Both +have lineon excitations (abelian in the X-cube model and +non-abelian in the cage-net model) that move in the x, +y and z directions. Both have other planon excitations +that move in xy, yz or zx planes. +Despite these similarities, it has not been clear whether +the Ising cage-net model is foliated in the sense defined +above. +It is important to emphasize that, while both +involve a layer structure, the coupled-layer constructions +of X-cube and cage-net models are very different from +foliated RG and from the notion of foliated fracton +phases. +In particular, there is no obvious relationship +between whether a model can be obtained by a coupled- +layer construction and whether it is foliated. By analogy +with the X-cube model, it is natural to guess that +the Ising cage-net model is a foliated RG fixed point +upon adding/removing doubled-Ising string-net layers. +However, this cannot be the case, because the doubled- +Ising string-net model contains non-abelian excitations +with quantum dimension +√ +2, while the cage-net model +has excitations with integer quantum dimension only.16 +While this argument does not rule out the possibility +of a foliated RG fixed point with other 2D topological +states as resources, in fact the Ising cage-net model is +not foliated. This can be seen by studying the model’s +GSD, which has been computed by some of the authors +in a separate paper.19 It is found that the GSD does not +grow by integer multiples when the system size grows by +unity in the x, y or z directions. +The question is then open again: +can we think of +the Ising cage-net model as a fixed point of a suitably +generalized RG? More specifically, can the foliated RG +be generalized somehow to include the Ising cage-net +model? In fact, we argue in this paper that the foliated +RG should be extended, independent of the Ising cage-net +example. We do this by re-examining foliated RG from +two complementary perspectives, one based on planon +condensation, and the other based on quantum circuits, +and point out that in both these pictures, the foliated +RG has unnatural restrictions. These observations lead +us to a generalized foliated RG under which, remarkably, +the Ising cage-net model is a fixed point. +finite depth +circuit +condensation/ +linear depth +circuit +foliated +fracton +model +2D topological order +Lx × Ly × Lz +Lx × Ly × (Lz − 1) +FIG. 1: Top: the foliated RG scheme, where a layer +of topologically ordered state (shown in orange) can be +added into or removed from a foliated fracton model via +a finite depth circuit. Bottom: generalized foliated RG +scheme realized by condensation of bosonic planons or a +sequential linear depth circuit around the plane. +The generalized foliated RG can be carried out +either by condensing or uncondensing bosonic planon +excitations supported near a 2D plane, or by acting +with a quantum circuit, supported near a 2D plane, +whose depth scales with the linear size of the plane. +We show that either of these operations can be used to +decrease or increase the system size of the Ising cage- +net model, which is thus a generalized foliated RG fixed +point. The two apparently different ways of carrying out +the generalized foliated RG are closely related, through a +connection that we explain between anyon condensation +and a class of linear depth circuits that we refer to as +sequential circuits. +We note that the original foliated RG arises as a special +case of the generalized procedure introduced here. +In +particular, for the X-cube model, instead of decoupling +a toric code layer and removing it to decrease system +size, we can condense the bosonic planon that effectively +comes from the toric code layer (either e or m), which +has the same effect as removing the layer. Alternatively, +we can act with a certain linear-depth circuit (more +specifically, +a sequential circuit) whose effect is to +condense the same bosonic planon. Therefore, we can use +generalized foliation to study the X-cube model, the Ising +cage-net model and many other type-I fracton models +within a single framework. Just as foliated RG comes +with the notion of foliated fracton phases and foliated +fracton order, we expect that the generalized foliated RG +comes with corresponding notions of generalized foliated + +3 +fracton phases and generalized foliated fracton order. It +will be interesting to study these notions in future work. +The paper is structured as follows: In Sec. II, we review +the original foliated RG by focusing on the X-cube model. +In Sec. III, we review the Ising cage-net model, which is +not foliated according to the original scheme. Section IV +then briefly points out some unnatural restrictions within +the original foliated RG, and proposes a generalized +foliated RG where these restrictions are removed. +In +Sec. V, we show that the Ising cage-net model is foliated +in terms of a generalized foliated RG defined by planon +condensation. Then, in Sec. VI, we demonstrate that the +generalized foliated RG can also be implemented by a +planar linear depth circuit. The linear depth circuit has +a special structure, and we dub it a sequential circuit; +in Sec. VII we show how the sequential circuit we use is +closely related to the condensation of planons via gapped +boundaries. +Finally, in Sec. VIII, we conclude with a +brief discussion on the implications of and outlook for +the generalized foliated RG. +II. +FOLIATION IN X-CUBE +Before our discussion of the ‘generalized foliation’, it +is instructive to review the original notion of foliation +and see how the corresponding RG procedure is carried +out for the X-cube. +The X-cube model has a foliated +structure, where layers of the toric code can be added to +or removed from the X-cube via a finite depth circuit S.9 +Given an X-cube ground state |ΨX.C.⟩ of the system size +Lx × Ly × Lz and a toric code ground state |ΨT.C.⟩, S +yields a |ΨX.C.⟩ of the size Lx × Ly × (Lz + 1). In rest of +this section, we review the finite depth circuit S on the +three-torus. +(a) +(b) +FIG. 2: (a) The three types of vertex terms in the X-cube +Hamiltonian Ax +v, Ay +v, and Az +v, which are tensor products +of Pauli-Z operators. (b) The cube term Bc. +Let us consider the X-cube Hamiltonian defined on a +cubic lattice on the three-torus; and one copy of the toric +code Hamiltonian defined on a square lattice on the two- +torus. For both models, the local qubit DOFs are placed +on the edges. The X-cube Hamiltonian6 +HX.C. = − +� +v +(Ax +v + Ay +v + Az +v) − +� +c +Bc +(1) +contains three types of vertex terms Ax +v, Ay +v, and Az +v; and +one type of cube term Bc, as shown in Fig. 2. The toric +(a) +(b) +FIG. 3: +(a) The vertex term Qv in the toric code +Hamiltonian. (b) The plaquette term Bp. +code Hamiltonian20 +HT.C. = − +� +v +Qv − +� +p +Bp +(2) +is a sum of local terms as shown in Fig. 3. +FIG. 4: The insertion of a layer of toric code living on an +xy-plane (blue colored square lattice) into a cubic lattice, +which hosts the X-cube. The inserted layer bisects an +edge i near the inserted plane into edges labeled by i′ +and k′. For every bisected edge, the X-cube Hamiltonian +is modified by replacing Zi → Zi′ and Xi → Xi′. The +new edges k′ are product states with the Hamiltonian of +H0 = − � +{k′} Zk′. +To construct the circuit, we first insert a decoupled +toric code into the X-cube. As depicted in Fig. 4, when +the inserted toric code lies in the xy-plane, it bisects the +z-direction edges in the X-cube model, thus creating new +qubit edges k′ colored in orange. These new k′ edges are +added to the system as product states whose Hamiltonian +is chosen to be H0 = − � +{k′} Zk′. For each bisected edge +i in the X-cube Hamiltonian, we substitute Zi → Zi′ and +Xi → Xi′. +The circuit S is a product of two finite depth circuits +S2 and S1, S = S2S1. Each is a product of the controlled- +NOT (CNOT) gates. The circuit S1 acts on the edges of +the modified X-cube Hamiltonian, as shown in Fig. 5a. +Every CNOT gate in S1 has an i′ edge serving as the +controlled qubit and the corresponding k′ edge as the +target. On the other hand, S2 acts on both edges of the +X-cube and those of the toric code. Every edge of the + +X +X +X +X +X +X +X +X +X +X +X2 +ZX +X +X +XZ +x +k'4 +(a) +(b) +FIG. 5: An illustration of the finite depth circuit S = +S2S1. (a) The action of the circuit S1 when focus on an +elementary cube of the original cubic lattice. The arrows, +representing the CNOT gates, point from the controlled +qubits to the targets. (b) S2’s action viewed at a cube. +toric code serves as the controlled qubit for the CNOT +gates whose targets are edges in the modified X-cube. An +illustration of S2 is given in Fig. 5b. The CNOT gate, +acting by conjugation, has the actions of +ZI �→ ZI, +IZ ↔ ZZ, +XI ↔ XX, +IX �→ IX, +(3) +where the first qubit is the control and the second is the +target. All the CNOT gates in S1 or S2 commute with +each other. Therefore, S is a finite depth circuit. By +direct computation, we see that +S +� +˜H(Lx,Ly,Lz) +X.C. ++ HT.C. + H0 +� +S† ∼= H(Lx,Ly,Lz+1) +X.C. +, (4) +where ˜HX.C. is the modified X-cube Hamiltonian, and the +symbol ∼= denotes that the L.H.S. and the R.H.S. share +the same ground space. +III. +ISING CAGE-NET +In this section, we review the basic definition and +properties of the Ising cage-net model. +The Ising cage-net is an exactly solvable model +obtained from the coupled layer construction16, in which +decoupled layers of the doubled-Ising string-net21–25 +are coupled together through the particle-loop (p-loop) +condensation. +Specifically, we take three stacks of the doubled-Ising +string-net defined on a square-octagon lattice (see Fig. 6), +and stack them together to form a truncated cubic lattice, +as shown in Fig. 7. Each of the six faces of a cube is an +octagonal plaquette. We call an edge l, parallel to the µ- +direction for µ ∈ {x, y, z}, a µ-principal edge, and denote +it by lµ. +As a 2D lattice model, the doubled-Ising string-net is +built from the Ising unitary modular tensor category26,27, +which consists of an index set {0, 1, 2} and a set of +symbols (δijk, ds, F ijm +kln , Rij +k ). +The model has a three- +dimensional local Hilbert space of spanC{|0⟩ , |1⟩ , |2⟩} for +each edge of the square-octagon lattice. The states |0⟩, +Qv +Bs +p +W fluxon +l +FIG. 6: A square-octagon lattice, where the doubled- +Ising string-net is defined. Each edge has a local Hilbert +space with a basis {|0⟩ , |1⟩ , |2⟩}. Qv is defined for every +trivalent vertex, and Bp = �2 +s=0(ds/D)Bs +p is defined +for each square and octagonal plaquette. +The string- +operator for a fluxon excitation W fluxon +l +violates the two +Bp terms containing the edge l and no Qv term anywhere. +FIG. 7: +A truncated cubic lattice. +It is formed by +intersecting layers of the square-octagon lattice. Every +cube has six octagonal faces. +At the corners of each +cube are octahedrons (see Fig. 9). The edges l, parallel +to µ direction for µ +∈ +{x, y, z}, are called the µ- +principal edges, which are denoted by lµ. +For the +system of decoupled layers, a µ-principal edge has a nine- +dimensional local space given by the tensor product of +( spanC{|0⟩ , |1⟩ , |2⟩} )⊗2. +|1⟩, |2⟩ are dubbed as 0-string, 1-string, and 2-string +respectively. The commuting projector Hamiltonian +HD.I. = − +� +v +Qv − +� +p +Bp +(5) +consists of the vertex projector Qv and the plaquette +projector Bp = �2 +s=0(ds/D)Bs +p (see Fig. 6). The symbol +ds takes values in d0 = d2 = 1, and d1 = +√ +2. +D = +� +s(ds)2 is the total quantum dimension of the model. + +Z5 +Qv’s action is defined by +Qv +����� +j +i +k +� += δijk +����� +j +i +k +� +, +(6) +where the symbol δijk is symmetric under permutation +of its indices. The non-zero elements are δ000 = δ011 = +δ211 = δ022 = 1, up to permutations. +The subspace +where all the vertex terms Qv are satisfied is called the +stable vertex subspace HD.I. +Qv .25 The plaquette operator +Bs +p’s action are evaluated by the graphical rules, which +are defined via the d- and F-symbols (Appendix B). Bs +p +acts on a plaquette by fusing a loop of s into the edges +as, for example, +Bs +p +����� +ℓ6 +ℓ5 +ℓ8 +ℓ7 +ℓ2 +ℓ1 +ℓ4 +ℓ3 +e2 +e4 +e6 +e8 +e3 +e5 +e7 +e1 +� += +����� +ℓ6 +ℓ5 +ℓ8 +ℓ7 +ℓ2 +ℓ1 +ℓ4 +ℓ3 +e2 +e4 +e6 +e8 +e3 +e5 +e7 +e1 +s +� += +� +e′ +1,e′ +2,...,e′ +8 +� 8 +� +i=1 +F ℓi ei+1 ei +s e′ +i e′ +i+1 +� ����� +ℓ6 +ℓ5 +ℓ8 +ℓ7 +ℓ2 +ℓ1 +ℓ4 +ℓ3 +e′ +2 +e′ +4 +e′ +6 +e′ +8 +e′ +3 +e′ +5 +e′ +7 +e′ +1 +� +. +(7) +For every ground state |ΨD.I.⟩, which is a superposition +of different configurations of closed loops satisfying Qv +at each vertex, Bs +p acts as +Bs +p |ΨD.I.⟩ = ds |ΨD.I.⟩ . +(8) +Moreover, the Bs +p operators form a commutative fusion +algebra of +Bi +pBj +p = +2 +� +k=0 +δijkBk +p. +(9) +The doubled-Ising string-net has nine topological +excitations {1, ψ, ¯ψ, σ, ¯σ, σ ¯ψ, ψ¯σ, σ¯σ, ψ ¯ψ}. +In terms of +the theory of anyons, these excitations come from a copy +of the chiral Ising anyon {1, σ, ψ}, and an anti-chiral copy +{1, ¯σ, ¯ψ}. The fusion rules for the chiral Ising anyon are +× +1 +σ +ψ +1 +1 +σ +ψ +σ +σ +1 + ψ +σ +ψ +ψ +σ +1 +(10) +The anti-chiral Ising anyon obeys the same fusion rules; +we simply replace the anyon labels above with the barred +version. Among the nine excitations, the non-abelian σ¯σ +and the abelian ψ ¯ψ are bosons. They are also the only +non-trivial pure fluxon excitations. A fluxon excitation +violates exactly one Bp term and none of the Qv terms. +A fluxon string-operator W fluxon +l +creates the fluxon and +its anti-particle on the two adjacent plaquettes sharing +the edge l (see Fig. 6). In particular, the ψ ¯ψ has a string- +operator +W ψ ¯ +ψ +l += (−1)n1(l), +(11) +where n1(l) = 1 if the edge l is in the state |1⟩, and +n1(l) = 0 otherwise. +FIG. 8: An elementary ψ ¯ψ particle-loop (p-loop), the red +loop, created by the coupling operator Vlµ shown by the +green tube. We represent a flux by a line segment normal +to the hosting plaquette. Joining the segments together, +we have the red loop. +To couple the stacks of the doubled-Ising string-net +layers together, we condense the ψ ¯ψ p-loop. Illustrated +in Fig. 8 is the smallest ψ ¯ψ p-loop created by the coupling +operator +Vlµ = W (ψ ¯ +ψ)µν +lµ +W (ψ ¯ +ψ)µρ +lµ +, +(12) +which is a product of ψ ¯ψ string-operators, from the +µν- and µρ-planes, acting on the edge lµ. +We add +−Vlµ for every principal edge to the Hamiltonian of +the decoupled layers. +−Vlµ penalizes the presence of +the states |01⟩, |10⟩, |21⟩, and |12⟩ on lµ. +Using the +Brillouin-Wigner degenerate perturbation theory and +treating doubled-Ising string-nets as perturbations, we +arrive at the Ising cage-net. Hence, on a principal edge, +the Ising cage-net has a five-dimensional local Hilbert +space of spanC{|00⟩ , |11⟩ , |02⟩ , |20⟩ , |22⟩}. Other edges +are unchanged. +The Ising cage-net has a commuting Hamiltonian of +HI.C. = − +� +µν,v +Aµν +v − +� +ps +Bps− +� +po +1 +2 +� +B0 +po + B2 +po +� +− +� +c +Bc, +(13) +where Aµν +v +is the vertex projector in a µν-plane; Bps is the +doubled-Ising string-net plaquette projector for a square +plaquette; 1 +2 +� +B0 +po + B2 +po +� +is a plaquette term associated +with each octagonal plaquette po; and +Bc = +� +po∈c +√ +2 +2 B1 +po +(14) + +少小 +少小6 +is the cube term. The vertex term acts as +Aµν +v +����� +j ℓ +i +k +� += δijkδ(j,ℓ) +����� +j ℓ +i +k +� +, +(15) +where we have used the doubled line to represent a +principal edge in the state |jℓ⟩, and +δ(j,ℓ) = +� +1 +for (j, ℓ) ∈ I +0 +otherwise +(16) +with the index set I = {(0, 0), (1, 1), (0, 2), (2, 0), (2, 2)}. +Mobility +Type +Excitations +Planons +Abelian +(ψ ¯ψ)µν +ψµν +¯ψµν +Non-abelian +(σ¯σ)µν +Lineons +Abelian +— +Non-abelian +σµνσµρ +¯σµνσµρ +σµν ¯σµρ +¯σµν ¯σµρ +TABLE I: Excitations in the Ising cage-net for each µν- +plane, written in terms of the doubled-Ising excitations. +Amongst, the only composite excitation, (ψ ¯ψ)µν, is a +fracton dipole, a planon in the µν-plane. +A lineon +can only move along the line specified by the repeated +position index. +For example, σµνσµρ is mobile along +a line in the µ direction. +Moreover, pairs of lineons +from different planes can form a lineon dipole, which is +a planon. +FIG. 9: The vacuum fusion channel of three different +lineons at an octahedron (Ref. 16). +The colored solid +lines represent the labeled string-operators in the same +color. The dashed lines represent edges without string- +operators. +The quasi-particle excitations of the Ising cage-net +follow directly from the constituent doubled-Ising layers. +Excitations that survive the condensation must have +string-operators that commute with Vlµ. Thus, some of +the doubled-Ising planons must now exist together with +some other doubled-Ising planons from a perpendicular +plane, hence the emergence of lineons. A lineon can turn +at a corner and become another lineon at the cost of +emitting a third one (Fig. 9). +The ψ ¯ψ, on the other +hand, splits into two fractons, where each fracton is +immobile as there is no operator that can annihilate an +individual fracton and create it at a different location. +We summarize the excitations in Table I. +FIG. 10: A cage configuration, as dictated by Av. The +orange colored cage is formed by a loop of 1 on each +octagonal face. The purple lines represent strings of 2. +A ground state of the Ising cage-net is a superposition +of different configurations of cages, as illustrated in +Fig. 10. Bps, B0 +po, B2 +po, and Bc all have the eigenvalue of +1 on the ground state. In a separate paper19, we find the +GSD of a Lx × Ly × Lz Ising cage-net to be +GSD(Lx, Ly, Lz) = 1 +8 +� +A + B + 5C + 45 +� +, +(17) +where A = 9Lx+Ly+Lz, B = 9Lx+Ly + 9Ly+Lz + 9Lz+Lx, +and C = 9Lx + 9Ly + 9Lz. +We immediately see that +GSD(Lx, Ly, Lz + 1)/GSD(Lx, Ly, Lz) is not an integer. +Thus, the Ising cage-net is ‘not’ foliated according to the +foliation10,15 introduced previously for the X-cube and +other models. Nevertheless, as we will see, the Ising cage- +net is foliated in a generalized sense. +IV. +GENERALIZING THE NOTION OF +FOLIATION +The calculation of the GSD for Ising cage-net model +shows that it is not foliated in the usual sense. However, +from its construction in terms of stacks of 2D topological +orders, it is reasonable to expect that it may be foliated +in some generalized sense. +Indeed, once we examine +the original defnition of foliation in more detail, we can +uncover two parallel ways in which it is unnaturally +restrictive. +First, let us formulate the original foliated RG process +purely in terms of quantum circuits. Recall that foliated +RG in the X-cube model involves adding a topologically +ordered layer and then coupling it to the X-cube bulk + +Z +OxzO yz +X +Oyz: +xy2-string +L-string7 +with a finite-depth quantum circuit. +The topological +layer cannot itself be created with a finite-depth circuit +from a product state. However, it is now well-understood +that it can be created with a linear-depth circuit28,29. +Therefore, if we view foliated RG as a generalization of +usual entanglement RG2,3, in which one is allowed to +add ancillary degrees of freedom in a product state and +then apply finite-depth circuits, moving to foliated RG +corresponds to additionally allowing linear-depth circuits +within a 2D subsystem of the 3D model. However, from +this perspective, the current definition of foliated RG is +restricted, in that we only allow the linear-depth circuit +to act on the ancillae qubits and not on the 3D bulk. +A more natural definition would be to allow the linear- +depth circuit to act arbitrarily within a 2D layer on +both the ancillae and the bulk. +We remark that the +kinds of linear-depth circuits involved here have a special +structure that preserves the area law of entanglement, as +discussed in more detail in Sec. VII. +Second, we can also view foliated RG in terms of +condensation. Namely, suppose we want to implement +the inverse process of removing a single layer from the X- +cube model, reducing its size in one direction. This can +be achieved by condensing a planon within a single layer, +corresponding to disentangling the toric code layer and +then trivializing that layer by condensing a boson. In this +case, the planon which we condense is very special: it can +be viewed as being part of a 2D theory that is decoupled +from the rest of the excitation spectrum of the 3D bulk. +To be more general, if we allow condensation of planons in +RG, we should allow condensation of arbitrary planons, +not only those that are part of decoupled 2D theories. +In light of the above, there are two natural ways to +extend the notion of foliated RG: linear-depth circuits +and planon condensation. In what follows, we will show +that both approaches lead to a generalized foliated RG +that is applicable to the Ising cage-net model. +Then, +in Sec. VII, we argue that these two approaches, while +seemingly distinct, are in fact very closely related to each +other. +V. +RG VIA CONDENSATION +How can the system size of the Ising cage-net model +be increased / decreased? +In this section, we show +that it can be changed through condensation and un- +condensation of bosonic planons. This is closely tied to +the topic of anyon condensation in 2D systems, and we +refer the reader to Ref. 30 and references therein for a +review. +Let us begin by considering the process of condensing +planons in an xy-plane to decrease the system size in the +z direction by one (Fig. 11). Recall from the last section +that for each xy-plane there is a bosonic planon ψ ¯ψ which +can be condensed. When ψ ¯ψ in plane z = 0 is condensed, +the quasi-particle content of the model changes as follows: +1. Since ψ ¯ψ is the fracton dipole, fractons between +z = −1 +z = 0 +z = 1 +FIG. 11: An illustration of the relevant xy-planes of a +Lx×Ly×Lz Ising cage-net. Via the condensation process +described in the text, we remove the z = 0 plane and +obtain a Lx × Ly × (Lz − 1) Ising cage-net. +planes z = 0 and z = 1 are identified with the +corresponding fracton between planes z = −1 and +z = 0. +2. The planons ψ and ¯ψ on the z = 0 plane are +identified. +3. The σ¯σ planon on the z = 0 plane splits into two +abelian bosonic planons e and m with a mutual −1 +braiding statistics. +4. The lineons in the z = 0 plane composed of σxyσxz, +¯σxyσxz, σxy¯σxz, and ¯σxy¯σxz are all confined. +5. Planons and lineons on other planes are unchanged. +After this step, we can further condense either e or m. +This gets rid of the remaining planons on the z = 0 plane +without affecting other quasi-particle excitations. Now, +we see that the quasi-particle content of the model is the +same as that of an Ising cage-net model with the z = 0 +plane removed. The planons and lineons on planes other +than z = 0 are left intact. Moreover, the fracton between +z = 0 and z = 1, which is now identified with the fracton +between z = −1 and z = 0, becomes the new fracton +between z = −1 and z = 1. Therefore, the size of the +Ising cage-net model can be decreased by one in the z +direction by first condensing the ψ ¯ψ planon in a plane, +and then by condensing one of the split channels of the +σ¯σ planon on the same plane. +We see that if we allow condensation of bosonic planons +as a RG operation, we obtain a generalized foliated RG +under which the Ising cage-net model is a fixed point. As +noted in Sec. IV, the original foliated RG for the X-cube +model can also be viewed in terms of such condensation. +The condensation of planons is, of course, a singular +process where the bulk gap needs to close and then +reopen, corresponding to a phase transition between + +z +X8 +different standard phases (see Appendix A for the +definition of standard phases). +This means that, +similar to the original foliated RG, the generalized +foliated RG operations can move across certain phase +boundaries. +However, only certain phase boundaries +can be crossed; +the singularity involved in planon +condensation is localized to a selected plane and is hence +a “subsystem” singularity, not one in the full 3D bulk. +A useful way to think about the condensation process +is to use the fact that the Ising cage-net model can +be obtained by gauging the planar Z2 symmetries of +a subsystem symmetry protected topological (SSPT) +model protected by such symmetries31. +The planons +being condensed correspond to the symmetry charges of +the planar symmetries in the SSPT model. Hence the +condensation of the planons in a given plane corresponds +to breaking / removing that planar symmetry and +reducing the size of the model. On the other hand, if +we want to increase the size of the system by adding a +plane at z = 0, we need to add the planar symmetry and +the corresponding planar state back to the SSPT model +and ‘re-gauge’ the planar symmetry. +VI. +RG VIA PLANAR LINEAR DEPTH +CIRCUIT +The planar linear depth circuit we construct for the +Ising cage-net model is a direct generalization of a RG +scheme that maps product states to ground states of +a string-net model, introduced by Liu Y. et al.29. +In +Sec. VI A, we review this RG procedure for the string-net +models. We describe carefully an initialization step that +is nontrivial for non-abelian string-net models, which was +not discussed in detail in Ref. 29. +In Sec. VI B, we +describe the RG scheme as a linear depth circuit for the +Ising cage-net model. We will see that the initialization +step is also important and nontrivial. +A. +String-net RG +In this section, we will first describe an important step +in the RG procedure – the ‘controlled gate’ which adds +a plaquette to the string-net wave-function. After that, +we will describe the full RG procedure starting from the +string-net wave-function on the minimal lattice on a torus +and then adding plaquettes row by row. A brief review +of the string-net models is given in Appendix B 1. +1. +Adding plaquettes via the controlled gate +The controlled gate can be used to add a plaquette +to +the +string-net +wave-function. +We +present +the +definition and properties of the gate in this sub-section. +Computational details of the results discussed here can +be found in Appendix D. +Suppose that on a trivalent lattice, a plaquette is added +by adding an edge (the red edge in the diagrams below), +and we want to extend the string-net wave-function from +the original lattice to that including this new plaquette. +When the edge is added, it is not entangled with the rest +of the lattice and is in the state |0⟩. To merge the added +edge into the lattice, first, map it to � +s +ds +√ +D|s⟩ where D +is the total quantum dimension of the string-net. +|0⟩ �→ +� +s +ds +√ +D +|s⟩ +(18) +Then, we use this edge as the control to draw loops +around the added plaquette. More specifically, we can +represent the controlled gate Gp = � +s Gs +p graphically as +in Eq. (19). The action of Gs +p is similar to the action +of Bs +p which adds a loop s to a plaquette, but for the +graphical evaluation of Gs +p, we treat the control edge as +if it is in the state |0⟩, i.e. +Gs +p +����� +¯¯ℓ3 +¯ℓ3 +ℓ4 +¯ℓ4 +¯ℓ1 +ℓ1 +¯¯ℓ2 +¯ℓ2 +ℓ∗ +1 +ℓ2 +ℓ3 +ℓ4 +a +b +c +s′ +� += δss′ +����� +¯¯ℓ3 +¯ℓ3 +ℓ4 +¯ℓ4 +¯ℓ1 +ℓ1 +¯¯ℓ2 +¯ℓ2 +s +ℓ∗ +1 +ℓ2 +ℓ3 +ℓ4 +a +b +c +0 +� += δss′ +� +α,β,γ, +δ,ε,η,τ +F ℓ1ℓ∗ +10 +ss∗α F +¯ℓ1aℓ1 +sα∗β F +¯ℓ2ℓ2a∗ +sβ∗γ F +¯¯ℓ2bℓ∗ +2 +sγ∗δ F +¯ℓ3ℓ3b∗ +sδ∗ε +F +¯¯ℓ3cℓ∗ +3 +sε∗η F +¯ℓ4ℓ4c∗ +sη∗τ +����� +¯¯ℓ3 +¯ℓ3 +ℓ4 +¯ℓ4 +¯ℓ1 +ℓ1 +¯¯ℓ2 +¯ℓ2 +α +γ +ε +τ +β +δ +η +s +� +, +(19) + +9 +where the red line with an arrow marks the control +edge. +We carry out the explicit graphical evaluation +in Appendix D 1. Note that Gs +p can be defined on any +polygonal plaquette. +Gs +p is not a unitary on the full Hilbert space, but only +between subspaces. More specifically, it is an isometry +from VSN +p,s to HSN +p,s, both of which involve the DOF around +a plaquette p. In VSN +p,s, the control edge is set to |s⟩ while +the other edges come from the string-net wave-function +on the lattice with the control edge missing (pretending +that it is set to |0⟩). The vertices containing the control +edge, then, involve configurations like +ℓ∗ +ℓ +s +(20) +In HSN +p,s, all edges, including the control edge, come from +the string-net wave-function with the control edge set to +|s⟩. +In Appendix D 2, we prove that Gs +p is an isometry from +VSN +p,s to HSN +p,s by demonstrating +Gs +p +†Gs +p +����� +¯¯ℓ3 +¯ℓ3 +ℓ4 +¯ℓ4 +¯ℓ1 +ℓ1 +¯¯ℓ2 +¯ℓ2 +ℓ∗ +1 +ℓ2 +ℓ3 +ℓ4 +a +b +c +s′ +� += δss′ +����� +¯¯ℓ3 +¯ℓ3 +ℓ4 +¯ℓ4 +¯ℓ1 +ℓ1 +¯¯ℓ2 +¯ℓ2 +ℓ∗ +1 +ℓ2 +ℓ3 +ℓ4 +a +b +c +s +� +. +(21) +The controlled gates commute with each other +� +Gs +p, Gs′ +p′ +� +=0 = +� +Gs +p +†, Gs′ +p′ +� +, +(22) +as long as they do not act on each other’s controlled edge. +Moreover, we can show +� +Gs +p, Bs′ +p′ +� += 0 = +� +Gs +p +†, Bs′ +p′ +� +, +(23) +provided that Bs′ +p′ does not act on the control edge of Gs +p. +We prove these commutation relations in Appendix D 3. +In Appendix D 4, we prove a useful equation, which we +call the central equation +Gs +p (|s⟩ ⟨s′|)ct Gs′ +p +† = P s +ct +�� +k +dk +dsds′ Bk +p +� +P s′ +ct , +(24) +where (|s⟩ ⟨s′|)ct acts on the control edge and P s +ct = |s⟩ ⟨s| +is a projector on the control edge. +With the central +equation, we can show that the controlled gate does +what we claimed – it adds a plaquette to the string-net +wave-function. In particular, we show below that under +conjugation by Gp = � +s Gs +p, the projector on the control +edge Pct = � +s,s′ +dsds′ +D |s⟩⟨s′| is mapped to the plaquette +projector Bp = � +s +ds +D Bs +p. +GpPctG† +p = +� +s,s′ +dsds′ +D +Gs +p (|s⟩⟨s′|)ct Gs′ +p +† += +� +s,s′,k +dk +D P s +ctBk +pP s′ +ct += +� +k +dk +D Bk +p = Bp +(25) +2. +The RG circuit +Using the controlled gate as a building block, we +can construct the full linear depth circuit that maps +a product state to the string-net wave-function. +We +present the linear depth circuit in two steps: 1. +from +a product state to a string-net wave-function on the +minimal lattice on torus; 2. from the string-net wave- +function on the minimal lattice to the full lattice by +adding plaquettes. We are going to focus on the trivalent +square-octagon lattice, although the general procedure +applies to other trivalent graphs as well. +The minimal lattice on the torus consists of three +edges, two vertices, and one plaquette, as shown in +Fig. 23. +On the square-octagon lattice, we start from +the product state ⊗l|0⟩l. +Pick three edges around a +vertex as shown in Fig. 12. +Apply a local unitary +transformation on the three edges so that they become +one of the ground states on the minimal lattice. Note +that for abelian string-net states, the ground states can +be chosen to be a product state of the three edges. In +fact, the ⊗l|0⟩l state is a legitimate state already, because +it satisfies the vertex term while the plaquette term is +trivial for abelian strings on the minimal lattice (for proof +see Appendix B 2). +However, for non-abelian string- +nets, the Bs +p term for a non-abelian s-string acts non- +trivially in the stable vertex subspace, and the ground +states generally become entangled. +In the case of the +doubled-Ising on the minimal lattice, ten configurations +satisfy the vertex constraints. +Of this ten-dimensional +space, only nine dimensions belong to the ground space, +where B0 +p = 1, B1 +p = +√ +2, and B2 +p = 1. The remaining +one dimension carries a ψ ¯ψ fluxon excitation such that +B0 +p = 1, B1 +p = − +√ +2, and B2 +p = 1. One possible choice +of the nine doubled-Ising ground states on the minimal +lattice is given in Appendix C 1. +Now, we need to grow this minimal structure so that +it reaches the full extent of the lattice. To do this, we +‘copy’ the states on the i and j edges along the non- +contractible loops in the y and x directions. To achieve +this, we use controlled gates of the form � +i |i⟩|i⟩⟨i|⟨0|, +and apply them sequentially along the non-contractible +loops, as shown in Fig. 12. As this step has to be done +sequentially along the loop, its depth increases linearly +with the size of the lattice. This completes step 1 of the +linear depth circuit, which we call initialization. + +10 +i +j +k +i +j +k +y +x +FIG. 12: The initialization step in the RG circuit for +generating the string-net wave-function. Left: pick three +edges around a vertex and map them into one of the +ground states of the string-net on the minimal lattice. +Right: grow the minimal structure by copying the string +states |i⟩ and |j⟩ along non-contractible loops so that +they reach the full extent of the lattice. +(2) +(1) +(2) +(Ly − 1) +(1) +{1} +(Lx − 1) +˜p +y +x +FIG. 13: Adding loops to plaquettes in step 2 of the RG +circuit for generating the string-net wavefunction. The +state has been initialized into one of the ground states on +the minimal lattice (black lines). First, loops are added +to the square plaquettes (shown in red) in a single step. +Then, loops are added to octagon plaquettes in row (1), +(2), ... +(ly − 1) sequentially. +For the last row, loops +are added to octagon plaquette in column (1), (2), ...., +(Lx − 1) sequentially. +No action is needed in the last +plaquette ˜p. +Step 2 is also of linear depth. The minimal lattice has +only one plaquette. In step 2, we add more plaquettes +to the lattice using the controlled gate introduced in +Sec. VI A 1. The plaquettes cannot be added all at once, +because the controlled gates commute only when they +do not act on each other’s control edge. A linear depth +circuit is hence needed to add all the plaquettes to the +square-octagon lattice. A particular sequence for adding +these plaquettes is shown in Fig. 13. +Firstly, all the +square plaquettes (red circles) can be added at the same +time because they do not overlap with each other. The +small circle indicates the control edge while the big circle +indicates the action of Gs +p. Secondly, we add the square- +octagon lattice in row one (labeled (1) in Fig. 13). All +controlled gates in row one commute with each other so +they can be added in one step. Then we add row two, row +three, etc., until the next to last row (labeled (Ly − 1) +in Fig. 13). +For the last row, we need to choose the +control edges side ways because we need un-entangled +edges to be used as control edge. Due to this change, the +plaquettes in the last row need to be added sequentially +as the controlled gates do not commute any more. As +shown in the figure, we can add them in the order of +(green labels) (1), (2), ..., (Lx − 1). We do not need to +act in the last plaquette (labeled ˜p) as the constraint due +to the last plaquette is already implied by that of the +largest plaquette that we started from combined with +all the small plaquettes added so far. Therefore, at this +point, we have finished the linear depth RG procedure +that starts from a product state and maps it to the the +string-net wave-function on the square-octagon lattice. +B. +Ising cage-net +In this section, we use the controlled gate of Eq. (19) +to build up the RG circuit to enlarge an Ising cage-net +ground state on the three-torus by one layer. We will +start, in Sec. VI B 1, by introducing finite depth circuits +that grow cages on the cage-net ground state. They serve +as the building blocks of the full planar linear depth RG +circuit, which we discuss in Sec. VI B 2. +1. +Adding cages via the controlled gate +In 2D, we have seen that a plaquette can be added +to the string-net wave function, via the controlled gates, +after an edge is added to the lattice. We can extend this +procedure to 3D cage-net states. +Suppose that we start with the Ising cage-net ground +state on the truncated cubic lattice (Fig. 7) and add +a plane in the xy direction. +At each point where the +added plane bisects the z direction edges, an octahedron +is added, as shown in Fig. 14, to ensure the trivalent +structure in each of the coupled planes. +In the added +plane, octagonal plaquettes fill in the space between +the octahedrons. Every edge of the added octahedrons +carries a three dimensional Hilbert space spanned by +{|0⟩, |1⟩, |2⟩}. +We start with these edges all set to +the state |0⟩. +The principal edges on the octagons +each carry a five dimensional Hilbert space spanned by +{|00⟩, |02⟩, |20⟩, |22⟩, |11⟩}, which is a subspace of the +tensor product Hilbert space of two three dimensional +DOFs {|0⟩, |1⟩, |2⟩} ⊗ {|0⟩, |1⟩, |2⟩} that come from the +two intersecting planes. +We start with these principal +edges in the state |00⟩. + +11 +FIG. 14: Insertion of an xy-plane bisects a cube in the +original cage-net lattice into two cubes. Each intersection +point between the xy-plane and the z-principal edges is +expanded into an octahedron to preserve the trivalent +structure in the xy, yz and zx planes. +FIG. 15: ‘Copying’ the states on the bisected z-principal +edges onto edges of the added octahedron to satisfy +vertex rules in the xz and yz planes. +The copying +process can be performed by controlled gates of the form +� +xy |xy⟩⟨xy|⊗|x⟩⟨0| and � +xy |xy⟩⟨xy|⊗|y⟩⟨0|, indicated +by the arrows pointing from the control to the target. +We describe first the process to add one cube into +the new layer, which consists of two steps: 1. add the +octahedrons to the cage-net wave-function; 2. +grow a +cage structure in the upper truncated cube of Fig. 14. In +step one, we first need to copy the state of the bisected +z-principal edge onto some of the octahedron edges so +that the vertex rules are satisfied at the octahedrons’ +vertices. Suppose the bisected edge is in the state |xy⟩. +The copying process can be achieved with the controlled +gates � +xy |xy⟩⟨xy| ⊗ |x⟩⟨0| and � +xy |xy⟩⟨xy| ⊗ |y⟩⟨0| +as indicated by the blue and green arrows in Fig. 15. +Then, we add the square plaquettes to the cage-net wave- +function. This can be done as described in the previous +section on how to add a square plaquette to the doubled- +Ising string-net wave function, as the square plaquettes +remain unaffected when the doubled-Ising layers are +coupled into Ising cage-net. More specifically, for each +square plaquette, we pick an edge in the state |0⟩ as +the control edge, map it to � +s +ds +√ +D|s⟩, and use it as the +control in the controlled gate Gp that adds loops into the +plaquette. +(a) +(b) +FIG. 16: Growing a cage structure in an added cube. +(a) First, using an edge from the bottom face (colored +green) as control, add loops to the bottom and top faces, +(b) then use the edges on the side faces (colored green) +as control to add loops to the side face. +Step 2, which adds a cage structure to the cube, is +more complicated. +As shown in Fig. 16, first we add +loops to the bottom and top faces and then to the side +faces. More specifically, first we pick a principal edge on +the bottom face in the state |00⟩ as the control. We will +use the convention where the first |0⟩ comes from the xy +plane while the second |0⟩ comes from the vertical xz and +yz planes. Map the control edge as +|00⟩ �→ +� +s +ds +√ +D +|s0⟩ , +(26) +Note that this takes the controlled edge out of the five +dimensional subspace of {|00⟩, |02⟩, |20⟩, |22⟩, |11⟩} but +keeps it in the nine dimensional space of {|0⟩, |1⟩, |2⟩}⊗2. +This will also happen to other principal edges as we +implement the procedure, but at the end of the process +of growing a cube, all principal edges will be back to the +five dimensional subspace. +Now, using the |s⟩ state as the control, apply the +controlled gate to the bottom face pb and top face pt +as +G0 +pb + G2 +pb + 1 +√ +2G1 +pbB1 +pt +(27) +as shown in Fig. 16 (a). Note that Gs +pb and Bs +pt act on +the first part of the principal edges (the part that comes +from horizontal planes). After these controlled gates, the +projector on the control edge |0⟩⟨0| (the first part) gets +mapped to +(|0⟩⟨0|)ct �→ +� +ss′ +dsds′ +D +(|s⟩⟨s′|)ct +�→ B0 +pb + B2 +pb + B1 +pbB1 +pt, +(28) +where in deriving the last line, we used the fact that the +top face is part of the original cage-net wave-function + ++ +ZZ +. +■ +■ +■.12 +and B0 +pt = B2 +pt = 1. Note that it might seem that the +operator in Eq. (27) is not unitary as B1 +p is not. But since +B1 +ptB1† +pt = B0 +pt + B2 +pt = 2, the action of the operator +restricted to the ground space of the original cage-net +model is indeed unitary. +Next, we need to add loops to the side faces. To do this, +we take the principal edges on the bottom face, which are +now in the states |s0⟩ and send them to |sαs⟩, where αs +comes from the xz or yz planes and αs = 0 if s is even, +αs = 1 if s is odd. This brings the principal edges on the +bottom face back to the five dimensional Hilbert space. +Then map the |αs⟩ states to +|0⟩ �→ +1 +√ +2 (|0⟩ + |2⟩) , |1⟩ �→ |1⟩ +(29) +Use the |αs⟩ states as the control to draw loop on the +side faces by applying � +αs Gαs +ps as shown in Fig. 16 +(b) to each side face. Let us see how the Hamiltonian +terms in Eq. (28) transforms. +We show the step by +step calculation for the third term B1 +pbB1 +pt. +The B1 +pt +part is not affected by the transformation and will be +omitted from the following equation. Let us focus on the +transformation induced by on principal edge. We label +the two three-dimensional DOFs on the principal edge +as 1 and 2 respectively, where 1 comes from the bottom +face whose state is labeled by s and 2 comes from the +side face whose state is labeled by αs. +� +(P 0 +1 + P 2 +1 )B1 +pbP 1 +1 + P 1 +1 B1 +pb(P 0 +1 + P 2 +1 ) +� +⊗ (|0⟩⟨0|)2 +�→ 1 +√ +2(P 0 +1 + P 2 +1 )B1 +pbP 1 +1 ⊗ (|0⟩2 + |2⟩2) 2⟨1| ++ 1 +√ +2P 1 +1 B1 +pb(P 0 +1 + P 2 +1 ) ⊗ |1⟩2 (2⟨0| + 2⟨2|) +�→ 1 +√ +2(P 0 +1 + P 2 +1 )B1 +pbP 1 +1 ⊗ +� +P 0 +2 + P 2 +2 +� +B1 +psP 1 +2 ++ 1 +√ +2P 1 +1 B1 +pb(P 0 +1 + P 2 +1 ) ⊗ P 1 +2 B1 +ps +� +P 0 +2 + P 2 +2 +� +(30) +The result is the product of B1 +pb and B1 +ps projected +onto the five dimensional subspace of the principal edge, +as promised. +This works for all side faces. +Similar +calculations can be carried out for the first two terms +in Eq. (28). +If we put everything together and omit +the projection onto the five dimensional subspace of the +principal edges, we see the Hamiltonian terms in Eq. (28) +becomes +� +B0 +pb + B2 +pb +� � +ps +� +B0 +ps + B2 +ps +� ++ B1 +pbB1 +pt +� +ps +B1 +ps, +(31) +which is a sum over the desired plaquette terms on the +bottom and side faces as well as the cube term on the +cube. +In the RG circuit to be discussed in the next section, +we need to grow cubes in the same row at the same time. +This works in a similar way as growing a single cube and +we describe the procedure here. First, as shown in Fig. 17 +FIG. 17: Adding a row of cubes to the cage-net state, +step 1: the inserted xy-plane bisects the cubes into two; +octahedrons are added at the intersection point. +(a) +(b) +FIG. 18: Adding a row of cubes to the cage-net state, step +2: (a) first, we simultaneously add loops to the bottom +and the top faces of all cubes in the row; (b), use the +edges on the side face (colored green) as control to add +loops to all the side faces at the same time. +which illustrates the situation with two cubes in the row, +a new plane is added which bisects the row of cubes into +two. Octahedrons are added to the intersection points +to preserve the trivalent structure in the coupled xy, +yz and zx planes. The ‘copying’ process illustrated in +Fig. 15 is then used to restore vertex rules at the vertices +of the octahedrons and then the square plaquettes in +the octahedrons are added to the cage-net wave-function. + +区 +区区 +Z13 +The next step is illustrated in Fig. 18, which adds cage +structures to a whole row of cubes at the same time. This +is done by first picking the principal edge in, for example, +the x direction and use them as controls to add loops +in the bottom and top faces as described above for each +cube in the row (Fig. 18 (a)). The operations in each cube +commute with that in another cube, and hence they can +be done all at the same time. Next, loops are added to +the side faces using the principal edges on the bottom face +as control, as shown in Fig. 18 (b). Again, the operations +on each side face commute with each other, so they can +be done at the same time. As a result of this process, all +the cubes in the row are now added to the cage-net wave- +function. +Note that the process illustrated in Fig. 18 +applies to the first row in the added plane. When we +try to add subsequent rows, some of the side faces would +have been added to the cage-net state already. +Those +side faces can be treated in the same way as the top +face. That is, apply B1 +ps in step Fig. 18 (a) when the +x-principal edge is in the state |10⟩, instead of applying +� +αs Gαs +ps controlled by the bottom principal edge of the +side face in the state |sαs⟩. A similar procedure applies +to the cubes in the last row of the added plane as well, +which have to be added one by one. +2. +RG circuit – Ising cage-net +The processes for adding single cubes and a row of +cubes are building blocks for the full RG circuit that +adds a full plane to the cage-net state. Similar to the +case of the doubled-Ising, we first need to initialize the +added plane into proper eigenstates of the non-local +logical operators before adding the local structures of +cubic cages (plaquettes in the case of doubled-Ising). +A commuting set of logical operators of the Ising cage- +net ground space can be chosen to be generated by the +string-operators of ψ, ¯ψ planons in each µν plane along +the µ and ν directions respectively. We can choose the +original cage-net state (before adding the plane) to be +an eigenstate of all such logical operators. The added +xy plane can be initialized into an eigenstate of ψx, ψy, +¯ψx and ¯ψy on that plane. The circuit described in the +last section on how to add cubic cages and plaquette +terms to the wave-function does not affect these nonlocal +logical operators. Therefore, the resulting cage-net state +after the RG circuit remains an eigenstate of all the ψ, ¯ψ +logical operators. +But the choice of the eigenvalue for the ψ, ¯ψ logical +operators is not arbitrary as the operators are related to +each other and hence their eigenvalues are constrained. +In Ref. 19, we study carefully the relations among these +operators, which allowed us to derive the ground state +degeneracy of the Ising cage-net model. +The relations +are listed below. +For derivation, see the discussion in +section VII of Ref. 19. For {µ, ν, λ} = {x, y, z} +� +i +� +ψ ¯ψ +�µ +µλ (ν = i) +� +j +� +ψ ¯ψ +�ν +νλ (µ = i) = 1 +rµν(λ = i)¯rµν(λ = i) = 1, ∀i, ∀{µ, ν} +rµν(λ = i)rµν(λ = i + 1) = 1, ∀i, ∀{µ, ν} +(32) +where +rµν += +1 +2 +� +1 + ψµ +µν + ψν +µν − ψµ +µνψν +µν +� +, +¯rµν += +1 +2 +� +1 + ¯ψµ +µν + ¯ψν +µν − ¯ψµ +µν ¯ψν +µν +� +. +As we started from a +ground state of the cage-net model, the original set of +ψ, ¯ψ operators satisfy the relations in Eq. (32). When +we add a new xy-plane, we need to make sure that after +the new ψx +xy, ψy +xy, ¯ψx +xy, ¯ψy +xy operators are added to the +original set, the total set still satisfy the relations in +Eq. (32). This can be guaranteed when the added string- +operators satisfy +ψx +xy ¯ψx +xy = 1, ψy +xy ¯ψy +xy = 1 +(33) +rxy = ¯rxy = ±1 +(34) +The choice of ±1 in the last relation depends on whether +rxy(z = i) = 1 or −1 in the original set. Compared to +the eigenstates listed in Appendix C 1, |ΨD.I. +min⟩1, |ΨD.I. +min⟩5, +|ΨD.I. +min⟩9 satisfy the relations in Eq. (33) and rxy = 1 while +|ψ ¯ψ +D.I. +min⟩ satisfies the relations in Eq. (33) and rxy = −1. +Therefore, we can initialize the added layer into one of +these states. +FIG. 19: Inserting an xy-plane into the original cage-net +lattice. Each red ball represents an octahedron. The new +principal edges are shown in blue. +In particular, consider the added xy-plane in Fig. 19. +Each red ball represents an octahedron. The added DOF +are initially set to be either in state |0⟩ (on edges of the +octahedron) or |00⟩ (on principal edges). Now initialize +the trivalent lattice in the xy-plane into one of |ΨD.I. +min⟩1, +|ΨD.I. +min⟩5, |ΨD.I. +min⟩9 and |ψ ¯ψ +D.I. +min⟩ following the procedure +described in Fig. 12. This linear depth process set up +the stage for the next step of the RG circuit: adding +cage structures to the cubes. + +Z +x +1 +114 +FIG. 20: Adding cage structures to the cubes in step +2 of the RG circuit for the cage-net state. +The red +lines indicate the minimal lattice state determined by +the initialization step. Cage structures are added to the +cubes in the 1st row, the 2nd row, ... the (Ly − 1)th row +in each step. In the last row, cage structures are added to +the cube in the 1st column, 2nd column, ..., (Lx − 1)th +column in each step. No action is required in the last +cube. +Now we can use the procedure described in the last +section to add cage structures to the cubes. As shown +in Fig. 20, on top of the minimal structure set up in the +initialization step (red lines), cage structures are added to +the cubes in the 1st row, the 2nd row, ... the (Ly − 1)th +row in each step. +In the last row, cage structures are +added to the cube in the 1st column, 2nd column, ..., +(Lx − 1)th column in each step. No action is required in +the last cube. This process has depth ∼ (Lx + Ly) and +completes the addition of a new layer into the cage-net +wave-function. +VII. +RELATING CONDENSATION AND +LINEAR-DEPTH CIRCUITS VIA GAPPED +BOUNDARIES +A. +General discussion +In Sec. V, we discussed the RG process in terms of +condensation of planons. In Sec. VI, we discussed the RG +process in terms of a linear depth circuit. In this section, +we show that these two are closely related to each other +by understanding each in terms of gapped boundaries. +We first consider a gapped boundary between a 2D +topological order and vacuum. +If an excitation moves +from the bulk to the boundary, it may become trivial in +the sense that it can be destroyed by a local operator +on the boundary. +This phenomenon is referred to as +condensation at the boundary. +On the other hand, +some excitations remain non-trivial as they approach +the boundary. +These phenomena can be characterized +precisely in a category-theoretic language32–35; in the +abelian case, this amounts to specifying a maximal +subset of bosons that can simultaneously condense at +the boundary36–39. It is believed the universality class of +a gapped boundary is fully determined by its category- +theoretic characterization. +The above discussion allows us to define distinct types +of anyon condensation (to vacuum) in a precise way, +as distinct types of gapped boundaries (to vacuum). +Such a definition is natural if we view the vacuum as a +condensate of certain anyons in the 2D topological order. +For instance, creating a puddle of anyon condensate +within the bulk 2D topological order amounts to creating +a puddle of trivial state (vacuum) separated from the +bulk by a gapped boundary. +This discussion, and the +definition of anyon condensation in terms of gapped +boundaries, can be generalized to gapped boundaries +between arbitrary 2D topological orders. +In the context of generalized foliated RG, we consider +condensation of planons. Condensation of a single planon +can similarly be associated with – and defined in terms +of – certain gapped boundaries between two fracton +orders, with the property that the boundary should be +transparent to mobile excitations away from the selected +plane where the condensation occurs. +It will be an +interesting problem for future work to fully characterize +those boundaries between fracton phases that correspond +to planon condensation. +We note that there has been +some related prior work discussing gapped boundaries of +fracton models in terms of condensation40,41. +It turns out that the kind of linear-depth circuits +considered here can also be associated with a type of +gapped boundary. A linear depth circuit has the general +form U += �K +ℓ=1 Uℓ where each layer Uℓ consists of +a number of local unitary gates with non-overlapping +support, and the number of layers K is proportional to +the linear system size L. In general, Uℓ can contain gates +acting across the entire system. However, for the circuits +we employed for RG, each layer Uℓ only contains gates +acting in a lower dimensional subsystem of the entire +system, such as the rows in Figs. 13 and 20. Such circuits +are much more restrictive than generic dense linear-depth +circuits, particularly because they preserve the area law +when acting on a state. +We call this class of circuits +sequential circuits. +Again we first focus on the 2D case, +where as +we have discussed, sequential circuits can be used to +generate topologically ordered ground states from an +initial product state (the topological “vacuum”). In order +to avoid complications associated with periodic boundary +conditions, we make a simplification as compared to the +circuits discussed in Sec. VI; namely, we work with an +infinite system and consider circuits that generate a disc + +Z +MOL +1 +- +1st col.15 +of 2D topological order from vacuum. If desired, the size +of the disc can later be taken to infinity. This allows us +to drop the initialization step, whose role is to take care +of the non-trivial ground state degeneracy on a 2-torus. +We can also drop the final linear-depth sequence of gates +needed to stitch two gapped boundaries together in a +manner consistent with periodic boundary conditions. +With these simplifications, the circuits operate in the +following way. We slice the 2D space into 1D concentric +circles surrounding the center of the disc, and order these +subspaces according to their radial coordinate. The ℓth +layer of the circuit is assumed to be supported near (but +not entirely within) the ℓth circle. After applying some +number of layers of the circuit, one is left with a disc +of topological order which has a gapped boundary to +the vacuum region which has not yet been acted on by +the circuit. +Then, the next layer in the circuit acts +only within the vicinity of the one-dimensional gapped +boundary between the topological order and the vacuum. +The action of the unitary in this layer is to “grow” the +topological order by a small amount, pushing the gapped +boundary further into the vacuum region. Continuing in +this way allows one to grow the topologically ordered +region arbitrarily. +Based on the above, given a sequential circuit, we can +associate the universality class of the gapped boundary +to vacuum which emerges when the circuit is truncated +at some radius. This association is well-defined in the +following sense. We can define a truncation of the circuit +¯U = �K0 +ℓ=1 Uℓ where K0 < K. This will create a disc +of topological order with a particular gapped boundary +to vacuum. Now, consider a different truncation ¯U′ = +�K0 +ℓ=1 Vℓ where each Vℓ again consists of non-overlapping +gates such that Vℓ = Uℓ for ℓ sufficiently less than +K0, but the layers near the boundary may differ. +By +definition, the two truncated circuits differ only by a +finite-depth circuit near the boundary. But a 1D finite +depth circuit cannot change the universality class of the +gapped boundary, i.e. it cannot change the set of anyons +which can condense on the boundary. +So the gapped +boundary type is independent of how the sequential +circuit is truncated. We note this conclusion only holds +for truncations that are compatible with the 1D layer +structure of concentric circles; the key property is that +the truncation only cuts through a finite number of 1D +layers, which is bounded above as the size of the disc +increases. +We emphasize that this discussion can be generalized +to gapped boundaries between two different 2D topologi- +cal orders. That is, given two topological orders referred +to as A and B that admit a gapped boundary, an A- +ground-state can be converted into a B-ground-state by +applying a sequential circuit. +Or, if we apply a trun- +cated version of the same sequential circuit, we can cre- +ate a puddle of B within the bulk topological order A, +separated by a gapped boundary whose universality class +does not depend on how the circuit is truncated. +In formulating the generalized foliated RG in terms +of quantum circuits, we apply sequential circuits within +2D layers of a 3D fracton model. +Truncating such a +sequential circuit (along its 1D layer structure) results in +a gapped boundary between two different fracton orders, +where some of the mobile excitations may condense +along the layer where the circuit is applied. +This is +how we described planon condensation above, and thus +we propose that planon condensation and applying 2D +sequential circuits are different ways to realize the same +operation in generalized foliated RG. +B. +Condensation in the Ising cage-net circuit +In accordance with the above discussion, we now +identify the type of gapped boundary that is associated +with the sequential circuits used to create Ising cage- +net model. To accomplish this, we are going to apply +the circuit only to a finite disc-shaped region within +a plane; we will not take the limit that the size of +the disc goes to infinity. Inside the region, we get the +fracton order as expected. +Outside of the region, the +added degrees of freedom remain unentangled. +There +is a gapped boundary between the two sides. We show +that the gapped boundary and the region outside can be +obtained by condensing bosonic planons starting from a +complete fractonic state. +First, let’s see how a similar relation works in the +doubled-Ising string-net state. We imagine a very large +disc of string-net state, and we ignore the curvature of +the disc’s boundary to simplify the following discussion. +Recall that in the RG circuit, the plaquettes are added +row by row. Suppose that we stop the process at row +i. +The boundary between row i and row i + 1 is a +smooth boundary on the lattice. +As the Hamiltonian +terms remain commuting throughout the process, the +boundary is gapped. +The gapped boundary can be induced by the condensa- +tion of ‘fluxon excitations’22 ψ ¯ψ and σ¯σ on the boundary +and beyond. To see that, consider a string-operator of the +form shown in Fig. 21, which consists of a string segment +above the lattice, a parallel segment under the lattice +and the two are connected by segments that vertically +go through the lattice plane. Note that, while embed- +ded in the 3D space, the string-operator is a closed loop, +from the 2D perspective, it ends at the locations where +the string goes through the lattice plane and can create +excitations at those points. In particular, such string- +operators in general violate the plaquette term at their +ends, as the plaquette terms correspond to a loop oper- +ator that links with the string-operator and the linking +generates nontrivial action. Therefore, in the bulk of the +string-net state, the string-operator generates ‘fluxon ex- +citations’ at its ends. In the doubled-Ising model, there +are two string-operators of this type, corresponding re- +spectively to a loop of string type 1 and a loop of string +type 2. The two string-operators generate the ψ ¯ψ and +σ¯σ excitations, respectively. If the string-operator ends + +16 +s-loop +i +i − 1 +i + 1 +p +p′ +FIG. 21: Condensation of the ψ ¯ψ and the σ¯σ fluxons on +the smooth boundary of the doubled-Ising model. The +vertex details are omitted. The dashed lines represent +the unentangled edges. +An open ended fluxon string- +operator is constructed from a loop of s-string that passes +through the lattice plane vertically at a plaquette. If the +plaquette (for example, the one labeled p) lies within the +doubled-Ising region, it creates a fluxon excitation. If the +plaquette (for example, the one labeled p′) falls outside +the string-net region, then no excitation is generated. +Thus, all fluxons condense on the smooth boundary. +For computational details on the condensation, +see +Appendix C 2. +(goes vertically through the lattice plane) outside of the +smooth boundary (Fig. 21), there are no more plaquette +terms to violate and the string-operator does not gener- +ate any excitations. Detailed calculations can be found +in Appendix C 2. Therefore, the ψ ¯ψ and σ¯σ excitations +condense on the boundary and beyond, thus demonstrat- +ing the connection between anyon condensation and the +linear depth circuit for the doubled-Ising string-net state. +The situation is very similar in the Ising cage-net +model. The RG circuit is again implemented row by row +in a sequential manner. Suppose that we stop the process +at row i, there will be a gapped boundary between row +i and row i + 1. +As shown in Fig. 22, like for the +string-nets, a vertical loop operator that goes through the +lattice plane at two points generates planon excitations +ψ ¯ψ and σ¯σ in the bulk of the cage-net state (in rows +j ≤ i). Beyond row i, however, it does not generate any +excitations and hence the ψ ¯ψ and σ¯σ are condensed. This +agrees with the RG procedure driven by condensation +described in Sec. V. Therefore, the process of sequential +application in the linear depth circuit can be interpreted +as moving the boundary between the cage-net state and +the condensed state, hence enlarging or shrinking the +fracton order in the plane. +(s-loop)xy +i +i + 1 +p +p′ +FIG. 22: Condensation of the ψ ¯ψ and the σ¯σ fluxon +excitations in the half xy-plane (shown in blue) in the +Ising cage-net. If the end of the the fluxon string operator +falls within the Ising cage-net region (for example at the +plaquette p), a fluxon excitation is created. If the end +falls outside of the Ising cage-net region (for example +at the plaquette p′), then no excitation is generated. +Therefore, both ψ ¯ψ and σ¯σ planons condense on the +boundary. +VIII. +SUMMARY AND DISCUSSION +In this paper, we studied the renormalization group +transformation for the Ising cage-net model and found +that the system size of the Ising cage-net model can +be decreased / increased by condensing / uncondensing +planon excitations near a 2D plane, or correspondingly +through a so-called sequential circuit which preserves the +area law and whose depth scales with the linear size of +the plane. We argued that these two ways of carrying out +the RG are closely related through gapped boundaries. +We call this procedure the generalized foliated RG, +because the previously defined foliated RG, under which +the X-cube and related models are fixed points,9 fits +into this new definition as a special case. On the one +hand, the system size of the X-cube can be decreased / +increased by condensing / uncondensing a lineon dipole +or fracton dipole on a given plane (both these excitations +are planons). Or, the RG procedure can be carried out +with a linear depth circuit in the same plane. One way +to construct the linear depth circuit is to use the finite +depth circuit discussed for the original foliation scheme9 +to decouple a layer of toric code out of the X-cube model, +and then disentangled the toric code into product state + +Z1 +1 +1 +1 +1 +1 +1 +1 +1 +1- +1 +1 +1 +1 +1 +1 +1 +1 +- +117 +with a linear depth circuit. Altogether this is a linear +depth circuit. Alternatively, we can use a circuit similar +to that discussed in Sec. VI to remove cage structures in +a plane row by row and hence removing a plane from the +X-cube model. +On the other hand, the generalized foliated RG allows +a broader class of RG operations. Indeed, the Ising cage- +net model is not a fixed point of the original foliated +RG as can be seen from its ground state degeneracy +calculation19. +We recall that the original foliated RG +led to an associated notion of foliated fracton phases +(see Appendix A for a definition), with the key property +that two systems related by a foliated RG operation lie +within the same foliated fracton phase. +Similarly, we +expect that there exists a notion of generalized foliated +fracton phase (GFF phase), again with the key property +that two systems related by a generalized foliated RG +operation lie in the same GFF phase. GFF phases should +be a coarser equivalence relation on quantum systems +than foliated fracton phases, because a broader class of +RG operations are allowed. We do not currently know +how to give a definition of GFF phases along the lines of +those in Appendix A; however, one possibility is to give a +definition based on circuit equivalence of ground states, +where one allows certain linear depth circuits supported +on planes. +In Sec. IV, we pointed out that the original foliated RG +contains certain unnatural restrictions, while the gener- +alized foliated RG seems to be more natural. +There- +fore, we expect that GFF phases are correspondingly a +more natural concept than foliated fracton phases as orig- +inally defined, so it will be important to revisit what we +have learned about foliated fracton phases. In particu- +lar, several invariants have been devised for foliated frac- +ton phases as originally defined, including those based +on fractional excitations and entanglement entropy10,15. +Now, with a new notion of GFF phases, we need to re- +consider the question of what quantities remain invariant +under the new equivalence relation, and which models +belong to the same GFF phase and which do not. For +example, we can ask whether the twisted foliated fracton +model proposed in Ref. 13 is still in a different phase than +the X-cube model or not under the new definition. +Finally, we want to comment that the generalized +foliation defined in this paper makes the discussion of +type I fracton models more in-line with that of Subsystem +Symmetry Protected Topological (SSPT) phases with +planar symmetry in e.g. Ref. 42–44. In the definition +of ‘strong SSPT’ in these papers, when a decoupled +layer with planar symmetry is added to the bulk of +the system, the planar symmetry can be combined +with an existing planar symmetry in the system, which +corresponds to the condensation of the composite of +the symmetry charges from the decoupled plane and a +planar symmetry charge in the bulk of the system. The +‘strong SSPT’ orders discussed in these papers hence may +become nontrivial (twisted) foliated fracton orders when +the planar symmetries are gauged. +ACKNOWLEDGMENTS +We are indebted to inspiring discussions with Dave +Aasen, Kevin Slagle, Nathan Seiberg, and Dominic +Williamson, +and helpful correspondence with Fiona +Burnell and Michael Levin. +Z.W., X.M. and X.C. are +supported by the National Science Foundation under +award number DMR-1654340, the Simons Investigator +Award (award ID 828078) and the Institute for Quantum +Information and Matter at Caltech. +X.C. is also +supported by the Walter Burke Institute for Theoretical +Physics at Caltech. The research of MH is supported by +the U.S. Department of Energy (DOE), Office of Science, +Basic Energy Sciences (BES) under Award number DE- +SC0014415. This work is also partly supported by the +Simons Collaboration on Ultra-Quantum Matter, which +is a grant from the Simons Foundation (651438, XC and +ZW; 651440, MH and DTS). 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Wang, Topological quantum computation, 112 (American +Mathematical Soc., 2010). +Appendix A: Definition of foliated fracton phases +Here we give a definition of foliated fracton phases, and +in the process provide a framework for thinking about the +relationship among fracton phases, universality and RG +fixed points. Here we focus on foliated fracton phases +as introduced in earlier works; in Sec. VIII, we briefly +comment on a possible definition of generalized foliated +fracton phases associated with the generalized foliated +RG. An important point is that notions of standard and +foliated phases both play important, but different, roles +in fracton physics. For ease of presentation, we do not +consider symmetry in this discussion. +First we recall the definition of standard gapped phases +and make some comments on the physical basis of this +definition. Phases are equivalence classes of systems; by a +system we mean a specification of the degrees of freedom +on some d-dimensional spatial lattice, together with a +local Hamiltonian H. In a slight abuse of notation we +use H to denote the system and not just its Hamiltonian. +Two systems H and H′ are in the same phase (considered +equivalent) if there exist resource systems R and R′ so +that there is a continuous path between the Hamiltonians +for the systems H ⊗ R and H′ ⊗ R′. Here “⊗” denotes +the operation of stacking two systems. +Each “trivial” +resource system is a collection of gapped, decoupled zero- +dimensional systems arranged in d-dimensional space, +i.e. R and R′ have product ground states. The energy +gap is required to remain open along the continuous path, +which must also avoid first-order phase transitions.45 A +special case of the above definition is that H and H′ +are equivalent if there is a continuous path between their +Hamiltonians (without stacking with resource systems). +Typically we are interested in the universal properties +of phases, which we define simply as those properties that +are the same everywhere within a phase. In many cases, a +standard phase contains within it a representative system +that is a RG fixed point under some (conventional) +scheme for carrying out the RG. When this occurs, +the universal properties of a phase are encapsulated +in the properties of the RG fixed point. +This holds +because two (infinite) systems related by a RG step are +in the same phase; for instance, this property is clear in +“entanglement RG” schemes. +Why do we consider stacking with trivial resource + +19 +systems? +This certainly leads to nice mathematical +properties, +and has the advantage of allowing for +comparison between systems with different local degrees +of freedom. However, there is a more fundamental reason, +namely to obtain a definition of phases that can be +distinguished in experiments, at least in principle. The +key point is that lattice models are always idealizations +of continuum systems, where some degrees of freedom +are deemed unimportant and left out of the model (e.g. +atomic core levels). Any physically measurable notion of +phases cannot depend on which degrees of freedom we +choose to include or ignore in a theoretical model, and +this issue is addressed by including stacking with trivial +resource systems in the equivalence relation. +More generally, we emphasize that the equivalence +relation used to define standard phases is not arbitrary. +Given the idea that gapped phases should be connected +components of parameter space where the gap remains +open, standard phases are the finest equivalence classes +that can be distinguished (in principle) by experiments. +Therefore, it is always physically relevant to consider +standard phases, even when the universal properties +of a standard phase are not captured in a RG fixed +point, as occurs in fracton models. +Put another way, +we cannot achieve a complete understanding of fracton +physics if we ignore standard phases. However, this does +not preclude the relevance of other notions of phases to +fracton physics. +Before +defining +foliated +fracton +phases, +we +first +introduce the closely related notion of F-phases. In order +to talk about F-phases (and foliated fracton phases), we +need to introduce a foliation of 3D space, which is a +certain geometrical structure. In particular, a foliation +consists of one or more decompositions of space into +parallel 2D layers. If we have k separate decompositions, +we sometimes speak more specifically about a k-foliation. +An important example of a 3-foliation is given by the sets +of all xy, yz and xz planes. +Similar to standard phases, two systems H and H′ +are considered to be in the same F-phase (or to be F- +equivalent) if there is a continuous path between the +Hamiltonians for H ⊗ R and H′ ⊗ R′. +The difference +from standard phases is that the resource systems R and +R′ are allowed to consist of decoupled gapped 2D systems +on any layers of the foliation structure. For instance, R +can consist of decoupled 2D toric codes lying on a set +of xy planes (as long as the foliation structure includes +xy planes). +Here, as in the foliated RG, we view 2D +layers as a free resource, analogous to product states in +the definition of standard phases. +The relation of F-equivalence is obviously coarser than +standard phase equivalence, because the set of allowed +resource systems contains those allowed for standard +phases. Moreover, it is strictly coarser; for instance, a +stack of 2D topologically ordered layers is non-trivial as +a standard phase, but is in the trivial F-phase (with +appropriate foliation structure). Therefore each F-phase +can contain multiple standard phases. +The reason we define F-phases is to be able to define +foliated fracton phases, which are those F-phases that +contain a representative system that is a fixed point of +the foliated RG. Because two (infinite) systems related by +a foliated RG step are in the same F-phase, it is expected +that the fixed point of a foliated fracton phase captures +certain universal properties that are the same throughout +the foliated fracton phase, and are referred to as its +foliated fracton order. It is important to emphasize that +the foliated fracton order consists of properties that are +the same even within different standard phases, so long +as these standard phases are F-equivalent and belong to +a foliated fracton phase. +Foliated fracton phases are a useful concept in the +study of fracton physics because, +in some fracton +systems, they restore a connection between universal +properties of a phase and a RG fixed point. +This +connection fails when we study fracton models using +standard phases. +It should be noted that not every F-phase is a foliated +fracton phase. +For instance, the F-phase containing +Haah’s cubic code model is not a foliated fracton phase +for any choice of foliation structure. It is not clear that +such F-phases are interesting objects of study. +Appendix B: Review of string-net models +In Appendix B 1, we review the basics of the string- +net models that are relevant for our purposes. We follow +the original construction as introduced in Ref. 21. For +more comprehensive introductions, we refer the readers +to Ref. 21, 22, 24, and 25. In Appendix B 2, we discuss +the string-net models on the minimal lattice on the torus. +1. +String-net models +The input data of a string-net model is a unitary fusion +category46, which includes an index set {0, 1, ..., N} and +the associated data set (δijk, ds, F ijm +kℓn ). +A string-net +model is defined on a trivalent lattice, where the local +DOF live on the edges. Each edge has a Hilbert space of +spanC{|0⟩ , |1⟩ ..., |N⟩}. Usually, an edge of the string-net +is represented by a directed line. For a directed edge, i∗ +represents the edge in the state i pointing in the opposite +direction. That is +i∗ +i += +. +(B1) +In particular, 0∗ = 0. +The δ-symbol specifies the vertex rules. +δijk takes +values in {0, 1} and it is symmetric under permutation of +the indices. δijk determines the allowed states on edges + +20 +at a trivalent vertex. A vertex is stable25 if +i +j +k +(B2) +satisfies δijk = 1. A vertex is not stable if δijk = 0. +The d- and F-symbols define the graphical rules. The +d-symbols evaluate loops to real numbers as +s += ds = ds∗ = +s∗ , +(B3) +where d0 = 1. They satisfy the equation +didj = +� +k +δijk∗dk. +(B4) +The F-symbols define the transformations +i +j +m ℓ +k += +� +n +F ijm +kℓn +i +ℓ +n +j +k +, +(B5) +where the F-symbols are nonzero if all the vertices satisfy +the vertex rules. They are normalized as +F ijk +j∗i∗0 = +� +dk +didj +δijk . +(B6) +For the cases of interest in this paper, the F-symbols +satisfy the tetrahedral symmetry +F ijm +kℓn = F ℓkm∗ +jin += F jim +ℓkn∗ = F imj +k∗nℓ +� +dmdn +djdℓ +, +(B7) +as well as the pentagon equation +� +n +F ijm +kℓn F psℓ∗ +inq F pqn +jkr∗ = F psℓ∗ +m∗kr∗F rsm∗ +ijq +. +(B8) +From the pentagon equation, we can derive the orthogo- +nality relation of the F-symbols that +� +n +F ijm′ +kℓn +� +F ijm +kℓn +�∗ += δmm′, +(B9) +where the complex conjugation on the F-symbol is given +by +� +F ijm +kℓn +�∗ += F i∗j∗m∗ +k∗ℓ∗n∗ . +(B10) +The ground state wave-function of the string-net model +|Ψ⟩ is given by +|Ψ⟩ = +� +|X⟩∈HSN +Qv +Ψ(X) |X⟩ , +(B11) +where Ψ(X) = ⟨X|Ψ⟩, and |X⟩ denotes a string-net +configuration in the stable vertex subspace HSN +Qv. +A +vector |X⟩ is a product state. +Note that the set of +all different |X⟩’s form an orthonormal basis for this +subspace. The graphical rules define a set of relations +between the amplitudes +Ψ +� +i +i′ +k +j +� += +�� +dkdj +di +�∗ +δii′Ψ +� +i +� +, +Ψ +� +i +j +ℓ +k +m +� += +� +n +� +F ijm +kℓn +�∗ +Ψ +� +i +j +ℓ +k +n +� +. +(B12) +Moreover, the graphical rules can be used to define +transformations for a generic string-net configuration ket- +vector. Eq. (7) and Eq. (19) are examples. +The commuting projector Hamiltonian, which has the +above wave-function as the ground state, is given by +HSN = − +� +v +Qv − +� +p +Bp, +(B13) +where Qv is the vertex projector enforcing the vertex +rules δijk, and Bp = � +s(ds/D)Bs +p with D = � +s(ds)2 +being the total quantum dimension is the plaquette +projector. +Each Bs +p adds a counter-clockwise directed +loop of s in the interior of a plaquette. Its action can +be evaluated by the F-symbols as illustrated in Eq. (7). +Eq. (B4) implies that the Bs +p’s satisfy +Bi +pBj +p = +� +k +δijk∗Bk +p. +(B14) +The ground state satisfies +Qv |Ψ⟩ = |Ψ⟩ , +Bp |Ψ⟩ = |Ψ⟩ +(B15) +for all v and p. +2. +The minimal lattice +A string-net model can be defined on the minimal +trivalent lattice on the torus. +The minimal lattice +consists of three edges, two vertices, and one plaquette as +shown in Fig. 23. The ground states first have to satisfy +the vertex constraints Qv. Recall that a vertex is called +stable if the vertex constraint is satisfied. We denote a +basis vector of the stable vertex subspace on the minimal +lattice by +|ijk⟩ = +����� +i +i +j +j +k +� +∈ HSN +Qv. +(B16) + +21 +i +j +k +y +x +FIG. 23: The minimal trivalent lattice on the torus. The +lattice has three edges colored by red, blue, and green; +two vertices; and one plaquette. +i, j, and k are state +labels on the colored edges. +i +i +j +j +k +Cx +Cy +y +x +FIG. 24: An illustration of the two non-contractible loops +Cx and Cy on the minimal lattice, taken by the logical +operators W Ci +α +where α is the excitation label. +The logical operators act within HSN +Qv. +Illustrated in +Fig. 24 are the two different paths taken by the logical +operators {W Ci +α } where i ∈ {x, y} and α ∈ {excitations}. +The action of the logical operators on a basis vector |ijk⟩ +can be computed by the method introduced in Ref. 21, +which we will not discuss in this paper. We will review +a string-operator construction for the double-Ising in +Appendix C 2. +Next, the ground states need to satisfy the one +plaquette term on the minimal lattice. +Consider the +action of Bs +p on a basis vector |abc⟩. Instead of directly +fusing the s-loop into the edges, we can first fuse different +parts of the s-loop together and map it into a trivalent +diagram +Bs +p +����� +a +a +b +b +c +� += +����� +a +a +b +b +c +s +� += +����� +a +a +b +b +c +s +� += +� +h,g +F s∗s0 +s∗sh∗F s∗s0 +s∗sg +����� +a +a +b +b +c +s +g +h +� += +� +h,g +F s∗s0 +s∗sh∗F s∗s0 +s∗sg +����� +a +a +b +b +c +s +g +g +h +h +� += +� +h,g, +t1,t2 +F s∗s0 +s∗sh∗F s∗s0 +s∗sg F s∗hs +gst1 F sg∗s∗ +h∗s∗t∗ +2 +����� +a +a +b +b +c +s +g +g +h +h +t2 +t1 +� += +� +h,g,t1 +ds +(ds)2 +� +dhdg +dt1 +F s∗hs +gst1 F sg∗s∗ +h∗s∗t∗ +1 +����� +a +a +b +b +c +g +g +h +h +t1 +� +, +(B17) +where in the second equality, we have brought the s-loop +over the lattice. We can also bring the loop below the +lattice. The choice does not matter. The action of the +Bs +p term can then be determined by fusing the trivalent +diagram (orange) into the underlying lattice (black). +This will help us to show that for abelian string-net +models, the plaquette term on the minimal lattice is +trivial. +For abelian string-net models, the fusion of s + +22 +and s∗ is 0. So, the above equation reduces to +Bs +p +����� +a +a +b +b +c +� +abelian += +ds +(ds)2 +����� +a +a +b +b +c +0 +0 +0 +0 +0 +� +abelian += ds +����� +a +a +b +b +c +� +abelian +, +(B18) +where to get to the last line, we have used (ds)2 = 1 for +abelian models. Hence, we see that Bp acts as identity on +HSN +Qv in abelian models. In other words, on the minimal +lattice, HSN +Qv is the ground space of the abelian models. +Appendix C: Other details of the doubled-Ising +In Appendix C 1, we discuss the ground states of +the doubled-Ising string-net on the minimal lattice +(introduced in Appendix B 2). In Appendix C 2, we show +that when we put the doubled-Ising state on a lattice +with smooth boundary, ψ ¯ψ and σ¯σ are condensed on the +boundary. +1. +Doubled-Ising on minimal lattice +The doubled-Ising string-net on the minimal lattice has +a 10-dimensional stable vertex subspace HD.I. +Qv , spanned +by |000⟩, |220⟩, |202⟩, |022⟩, |110⟩, |112⟩, |101⟩, |011⟩, +|211⟩, and |121⟩. +One of the dimensions is not part of the ground space, +because the projector Bp has the eigenvalue of 0 on this +state. To find the ground space, we calculate the action +of B1 +p. Following from Eq. (B17) (the actions of B0 +p and +B2 +p are trivial), we find +B1 +p +����� +a +a +b +b +c +� += +1 +√ +2 +����� +a +a +b +b +c +0 +0 +0 +0 +0 +� ++ 1 +√ +2 +����� +a +a +b +b +c +2 +0 +0 +2 +2 +� ++ 1 +√ +2 +����� +a +a +b +b +c +2 +2 +2 +0 +0 +� ++ 1 +√ +2 +����� +a +a +b +b +c +0 +2 +2 +2 +2 +� +. +(C1) +That is, +B1 +p |abc⟩ = +1 +√ +2 +� +I + W Cy +ψ ++ W Cx +ψ +− W Cx +ψ W Cy +ψ +� +|abc⟩ += +1 +√ +2 +� +I + W Cy +¯ +ψ ++ W Cx +¯ +ψ +− W Cx +¯ +ψ W Cy +¯ +ψ +� +|abc⟩ , +(C2) +where we have identified the logical operators by the +construction discussed in Appendix C 2, and the second +equality follows from that we can bring the s-loop below +the lattice in Eq. (B17). We can further compute the +action of B1 +p by fusing the orange trivalent diagram into +the underlying lattice. At this point, it is clear that a +product state |abc⟩ is generally not an eigenstate of B1 +p. +An +explicit +calculation +shows +that +|ψ ¯ψ +D.I. +min⟩ += +− 1 +2 |000⟩ + 1 +2 |202⟩ + 1 +2 |022⟩ + 1 +2 |220⟩ has the eigenvalue +− +√ +2 under B1 +p. Hence, it is an excited state, which car- +ries a ψ ¯ψ fluxon. The other 9 dimensions have the eigen- +value +√ +2 under B1 +p and are, thus, ground states on the +minimal lattice. +An orthonormal basis for the nine-dimensional ground +space can be chosen to be the common eigenstates of +logical operators W Cx +ψ , W Cy +ψ , W Cx +¯ +ψ , and W Cy +¯ +ψ , which all +commute with each other. The nine common eigenstates +are +|ΨD.I. +min⟩1 = 1 +2 |000⟩ + 1 +2 |202⟩ + 1 +2 |022⟩ − 1 +2 |220⟩ , +|ΨD.I. +min⟩2 = +1 +√ +2 |011⟩ + +i +√ +2 |211⟩ , +|ΨD.I. +min⟩3 = +1 +√ +2 |101⟩ − +i +√ +2 |121⟩ , +|ΨD.I. +min⟩4 = +1 +√ +2 |011⟩ − +i +√ +2 |211⟩ , +|ΨD.I. +min⟩5 = 1 +2 |000⟩ − 1 +2 |202⟩ + 1 +2 |022⟩ + 1 +2 |220⟩ , +|ΨD.I. +min⟩6 = e− iπ +8 +√ +2 |110⟩ + ie− iπ +8 +√ +2 +|112⟩ , +|ΨD.I. +min⟩7 = +1 +√ +2 |101⟩ + +i +√ +2 |121⟩ , +|ΨD.I. +min⟩8 = e +iπ +8 +√ +2 |110⟩ − ie +iπ +8 +√ +2 |112⟩ , +|ΨD.I. +min⟩9 = 1 +2 |000⟩ + 1 +2 |202⟩ − 1 +2 |022⟩ + 1 +2 |220⟩ . +The 10th dimension |ψ ¯ψ +D.I. +min⟩ is also a common eigenstate +of W Cx +ψ , W Cy +ψ , W Cx +¯ +ψ , and W Cy +¯ +ψ . In this basis, the logical + +23 +operators takes the diagonal form +W Cx +ψ += +� +�� +� +� +� +1 0 +0 +0 1 +0 +0 0 −1 +� +� +� ⊗ I3×3 +� +�� ⊕ (−1), +(C3) +W Cy +ψ += +� +�� +� +� +� +1 +0 +0 +0 −1 0 +0 +0 +1 +� +� +� ⊗ I3×3 +� +�� ⊕ (−1), +(C4) +W Cx +¯ +ψ += +� +��I3×3 ⊗ +� +� +� +1 0 +0 +0 1 +0 +0 0 −1 +� +� +� +� +�� ⊕ (−1), +(C5) +W Cy +¯ +ψ += +� +��I3×3 ⊗ +� +� +� +1 +0 +0 +0 −1 0 +0 +0 +1 +� +� +� +� +�� ⊕ (−1). +(C6) +2. +Condensation on smooth boundary +FIG. 25: +An illustration of an open ended fluxon +string-operator W path +fluxon acting on a wave-function of +the +doubled-Ising +string-net. +W path +fluxon +creates +the +fluxon and its antiparticle on the plaquettes p1 and +p2. +It does not create any excitations along the +path. W path +fluxon is constructed by a loop of s-string that +vertically penetrates the square-octagon lattice at the +two plaquettes. +The string-operators of ψ ¯ψ and σ¯σ +are given by a loop of 2-string and a loop of 1-string +respectively. +Using the string-operator construction discussed in +Ref. 22 and 23, we can readily show that on the smooth +boundary, the bosonic fluxons ψ ¯ψ and σ¯σ condense. +Let us start with a review of the fluxon string- +operators. Consider a large square-octagon lattice placed +on the xy-plane as shown in Fig. 25. +An open ended +fluxon string-operator is given by a loop of s-string which +vertically penetrates the lattice through the center of the +plaquettes p1 and p2. The fluxon excitations are created +at p1 and p2 respectively. The ψ ¯ψ string-operator is given +by a loop of 2-string, and that of σ¯σ is given by a loop +of 1-string. +An open ended string-operator of a chargeon is +constructed by a line segement, which can be either above +or below the lattice, as shown in Fig. 26. +FIG. 26: The open ended chargeon string-operators of +the doubled-Ising string-net. +They are constructed by +line segements of the strings, which can be either above or +below the lattice depending on the type of the chargeon +excitations. On a ground state of the doubled-Ising, An +open chargeon string-operator creates a vertex violation +at each end. The ψ and σ string-operators correspond to +a 2-string and a 1-string above the lattice respectively. +The string-operators for ¯ψ and ¯σ correspond to those +below the lattice. +To compute the action of the string-operators, we need +the R-symbols and the S-matrix26,27. +The R-symbols +define the braiding transformations +b +a +c += Rba +c +b +a +c +, +(C7) +where Rba +c ∈ C. The inverse transformations are defined +by +b +a +c += +� +Rba +c +�∗ +b +a +c +. +(C8) +Same as the F-symbols, Rba +c ̸= 0 if δbac ̸= 0. Elements of +the S-matrix are given by +Sab = +1 +√ +D +� +c +dcRba +c Rab +c , +(C9) +where +√ +D = +�� +s(ds)2 = 2 is the total quantum +dimension of the Ising unitary modular tensor category. +The full S-matrix is +S = 1 +2 +� +� +� +1 +√ +2 +1 +√ +2 +0 +− +√ +2 +1 +− +√ +2 +1 +� +� +� . +(C10) +Using the R- and F-symbols, we can always fuse the +chargeon string-operators into the lattice at the cost of + +Z +do0l-s +p1 +p2Z +S-string +X24 +violating the vertex constraints at the ends. +On the +other hand, the fluxon string-operators can be fused into +the lattice without introducing any vertex violations. +For example, consider the action of a non-trivial fluxon +string-operator on the string-net wave-function +����� +ℓ1 +ℓ4 +e +ℓ2 +ℓ3 +s +� += +� +j +F ss0 +eej +����� +ℓ1 +ℓ4 +e +e +ℓ2 +ℓ3 +s +j +� += +� +j +F ss0 +eej +� +Res +j +�∗ � +Rse +j +�∗ +����� +ℓ1 +ℓ4 +e +e +ℓ2 +ℓ3 +s +j +� += +� +j +dj +de +� +Res +j +�∗ � +Rse +j +�∗ +����� +ℓ1 +ℓ4 +e +ℓ2 +ℓ3 � += +√ +D +de +Sse +����� +ℓ1 +ℓ4 +e +ℓ2 +ℓ3 � +. +(C11) +Note that the expression above differs from that of Ref. 22 +by a normalization factor, which is not important for our +purposes. For an example of a chargeon string-operator, +let us consider a line segment below the lattice. +We +compute +����� +ℓ1 +ℓ4 +e +ℓ2 +ℓ3 +s +� += +� +j +F ss0 +eej +����� +ℓ1 +ℓ4 +e +e +ℓ2 +ℓ3 +s +s +j +� += +� +j +F ss0 +eej +� +Res +j +�∗ +����� +ℓ1 +ℓ4 +e +e +ℓ2 +ℓ3 +s +s +j +� += +� +j +F ss0 +eej +� +Res +j +�∗ +����� +ℓ1 +ℓ4 +e +e +ℓ2 +ℓ3 +j +� +, +(C12) +where in the last step we have removed the s-strings. +So, we see that an open chargeon string-operator always +creates violations to the vertex constraints. +Moreover, different fluxons correspond to different sets +of eigenvalues of Ba +p. To determine these eigenvalues, we +need the graphical rule +a +b += Sab +S0b +b . +(C13) +As an example, let us compute +Ba +pW path +ψ ¯ +ψ +|ΨD.I.⟩ += Ba +p +����� +2 � +ΨD.I. += +����� +2 +a +� +ΨD.I. += Sa2 +S02 +����� +2 � +ΨD.I. +, +(C14) +from which we see that the ψ ¯ψ fluxon has the eigenvalues +of 1, − +√ +2, and 1 for B0 +p, B1 +p, and B2 +p respectively. This +result is exactly what we found for the ψ ¯ψ fluxon state +in Appendix C 1. It is easy to see that the σ¯σ fluxon has +B0 +p = 1, B1 +p = 0, and B2 +p = −1. +We now show that, on the smooth boundary, the +fluxons ψ ¯ψ and σ¯σ are condensed. +Without loss of +generality, let us consider the doubled-Ising string-net +with a single plaquette and everywhere else is set to +|0⟩. Consider the action of an open ended fluxon string- +operator passing through the lattice just outside the +plaquette +s +Since the s-loop does not pass through the region +enclosed by the plaquette, the string-operator does not +create any fluxon excitation on the plaquette. Via the F- +and R-symbols, we can fuse the s-loop into the edges of +the plaquette without changing any of the edges outside +the plaquette or introducing any vertex violations. +Therefore, we see that the fluxons all condense on the +smooth boundary. +On the other hand, because the +chargeon string-operators necessarily introduce vertex +violations, the chargeons remain as excitations on the +boundary. Thus, we reach the conclusion that, on the +smooth boundary, the condensed excitations are the +fluxons ψ ¯ψ and σ¯σ. +Appendix D: Controlled gate details +In this appendix, we present the details of the graphical +definition of Gs +p, its inverse, the commutation relations, + +25 +and the proof for the central equation. +1. +Graphical definition +We perform the graphical calculation that leads to the +last line in Eq. (19). We compute +Gs +p +����� +¯¯ℓ3 +¯ℓ3 +ℓ4 +¯ℓ4 +¯ℓ1 +ℓ1 +¯¯ℓ2 +¯ℓ2 +ℓ∗ +1 +ℓ2 +ℓ3 +ℓ4 +a +b +c +s′ +� += δss′ +����� +¯¯ℓ3 +¯ℓ3 +ℓ4 +¯ℓ4 +¯ℓ1 +ℓ1 +¯¯ℓ2 +¯ℓ2 +s +ℓ∗ +1 +ℓ2 +ℓ3 +ℓ4 +a +b +c +0 +� += δss′ +� +α +F ℓ1ℓ∗ +10 +ss∗α +����� +¯¯ℓ3 +¯ℓ3 +ℓ4 +¯ℓ4 +s +α +ℓ∗ +1 +¯ℓ1 +ℓ1 +¯¯ℓ2 +¯ℓ2 +ℓ2 +ℓ3 +ℓ4 +a +b +c +� +=δss′ +� +α,β +F ℓ1ℓ∗ +10 +ss∗α F +¯ℓ1aℓ1 +sα∗β +����� +¯¯ℓ3 +¯ℓ3 +ℓ4 +¯ℓ4 +α +¯ℓ1 +ℓ1 +β +a +¯¯ℓ2 +¯ℓ2 +s +ℓ2 +ℓ3 +ℓ4 +b +c +� += δss′ +� +α,β,γ +F ℓ1ℓ∗ +10 +ss∗α F +¯ℓ1aℓ1 +sα∗β F +¯ℓ2ℓ2a∗ +sβ∗γ +����� +¯¯ℓ3 +¯ℓ3 +ℓ4 +¯ℓ4 +α +¯ℓ1 +ℓ1 +γℓ2 +¯¯ℓ2 +¯ℓ2 +s +ℓ3 +ℓ4 +β +b +c +� += · · · (repeat similar steps across the subsequent vertices around the plaquette) +=δss′ +� +α,β,γ, +δ,ε,η,τ +F ℓ1ℓ∗ +10 +ss∗α F +¯ℓ1aℓ1 +sα∗β F +¯ℓ2ℓ2a∗ +sβ∗γ F +¯¯ℓ2bℓ∗ +2 +sγ∗δ F +¯ℓ3ℓ3b∗ +sδ∗ε +F +¯¯ℓ3cℓ∗ +3 +sε∗η F +¯ℓ4ℓ4c∗ +sη∗τ +����� +¯¯ℓ3 +¯ℓ3 +ℓ4 +¯ℓ4 +¯ℓ1 +ℓ1 +¯¯ℓ2 +¯ℓ2 +s +α +γ +ε +τ +β +δ +η +� +=δss′ +� +α,β,γ, +δ,ε,η,τ +F ℓ1ℓ∗ +10 +ss∗α F +¯ℓ1aℓ1 +sα∗β F +¯ℓ2ℓ2a∗ +sβ∗γ F +¯¯ℓ2bℓ∗ +2 +sγ∗δ F +¯ℓ3ℓ3b∗ +sδ∗ε +F +¯¯ℓ3cℓ∗ +3 +sε∗η F +¯ℓ4ℓ4c∗ +sη∗τ +����� +¯¯ℓ3 +¯ℓ3 +ℓ4 +¯ℓ4 +¯ℓ1 +ℓ1 +¯¯ℓ2 +¯ℓ2 +α +γ +ε +τ +β +δ +η +s +� +. +(D1) +2. +Isometric property +We now show that Gs +p is an isometry. That is, Gs +p +†Gs +p +equals identity on the input space, which we called VSN +p,s +in section VI A 1. +Graphically, Gs +p +† removes a string of s from the edges +of the plaquette when the controlled edge is in the state + +26 +|s⟩. Let us denote the matrix elements of Gs +p by +[Gs +p](s,α,β,γ,δ,ε,η,τ) +(s′,ℓ∗ +1,a,ℓ2,b,ℓ3,c,ℓ4)(ℓ1, ¯ℓ1, ¯ℓ2, ¯¯ℓ2, ¯ℓ3, ¯¯ℓ3, ¯ℓ4, ℓ4) += δss′F ℓ1ℓ∗ +10 +ss∗α F +¯ℓ1aℓ1 +sα∗β F +¯ℓ2ℓ2a∗ +sβ∗γ F +¯¯ℓ2bℓ∗ +2 +sγ∗δ F +¯ℓ3ℓ3b∗ +sδ∗ε +F +¯¯ℓ3cℓ∗ +3 +sε∗η F +¯ℓ4ℓ4c∗ +sη∗τ +. +(D2) +Then, Gs +p +† has an algebraic expression of +Gs +p +† +����� +¯¯ℓ3 +¯ℓ3 +ℓ4 +¯ℓ4 +¯ℓ1 +ℓ1 +¯¯ℓ2 +¯ℓ2 +α′ +γ′ +ε′ +τ ′ +β′ +δ′ +η′ +s′ +� += +� +t1,t2,t3, +t4,t5 +� +[Gs +p](s′,α′,β′,γ′,δ′,ε′,η′,τ ′) +(s,ℓ∗ +1,t5,t4,t3,t2,t1,ℓ4) (ℓ1, ¯ℓ1, ..., ℓ4) +�∗ +× +����� +¯¯ℓ3 +¯ℓ3 +ℓ4 +¯ℓ4 +¯ℓ1 +ℓ1 +¯¯ℓ2 +¯ℓ2 +ℓ∗ +1 +t4 +t2 +ℓ4 +t5 +t3 +t1 +s +� +. +(D3) +Using the orthogonality relation Eq. (B9), we find +Gs +p +†Gs +p +����� +¯¯ℓ3 +¯ℓ3 +ℓ4 +¯ℓ4 +¯ℓ1 +ℓ1 +¯¯ℓ2 +¯ℓ2 +ℓ∗ +1 +ℓ2 +ℓ3 +ℓ4 +a +b +c +s′ +� += δss′ +� +t1,t2,t3, +t4,t5 +δt1cδt2ℓ3δt3bδt4ℓ2δt5a +����� +¯¯ℓ3 +¯ℓ3 +ℓ4 +¯ℓ4 +¯ℓ1 +ℓ1 +¯¯ℓ2 +¯ℓ2 +ℓ∗ +1 +t4 +t2 +ℓ4 +t5 +t3 +t1 +s +� +, +(D4) +thereby establishing Gs +p as an isometry. +Hence, Gp = +� +s Gs +p is also an isometry. +3. +Commutation relations +The +commutation +relations +of +the +Gs +p +operators +immediately follow from the graphical definition. +Any +two Gs +p and Gs′ +p′ commute, provided that they do not act +on each other’s controlled edge. When p and p′ are the +same plaquette, Gs +p and Gs′ +p commute trivially, because +they act on orthogonal spaces with the control edge in +|s⟩ and |s′⟩ respectively. If the plaquettes p and p′ are +not next to each other, Gs +p and Gs′ +p′ obviously commute. +When p and p′ are adjacent, the proof of commutation +amounts to showing the order, in which the string s and +s′ are fused into the bordering edges, does not matter. +Consider Gs +p and Gs′ +p′ acting on two adjacent plaquettes +p and p′. We focus on the bordering edges of these two +plaquettes. We will show that the F-symbols associated +with the two diagrams (the thickened red arrows indicate +the direction of motion of the s- and s′-strings), +s′ +s +1st +2nd +p′ +p +and +s′ +s +2nd +1st +p′ +p +are equal. The left diagram corresponds to computing +Gs +pGs′ +p′ on a reference ket-vector, and the right diagram +corresponds to Gs′ +p′Gs +p. +In the case where s′ moves first, we find +����� +s′ +s +ε′ +d +α +b +c +a +e +� += +� +ρ +F b∗ca +s′ε′∗ρ +����� +s′ +s +ε′ +d +α +b +c +ρ +e +� += · · · += +� +q +�� +ρ +F b∗ca +s′ε′∗ρF e∗dc∗ +s′ρ∗τ ′F τ ′ρ∗e∗ +sα∗q∗ F ε′∗b∗ρ +sqγ +� ����� +s′ +s +ε′ +d +α +b +q∗ +γ +τ ′ +� +. +(D5) +In the case where s moves first, we have +����� +s′ +s +ε′ +d +α +b +c +a +e +� += +� +ρ +F dc∗e∗ +sα∗ρ +����� +s′ +s +ε′ +d +α +b +ρ +c +a +� += · · · += +� +q +�� +ρ +F dc∗e∗ +sα∗ρ F ab∗c +sρ∗γF γρ∗a +s′ε′∗qF α∗dρ +s′q∗τ ′ +� ����� +s′ +s +ε′ +d +α +b +q +γ +τ ′ +� +. +(D6) +For each fixed q, we want to show that the coefficients, +i.e. free sums over ρ, in Eq. (D5) and Eq. (D6) are equal. +To do this, we consider an alternative way of moving +the strings s and s′. We will show that this alternative +expression can be simplified, via the pentagon equation, + +27 +to produce either Eq. (D5) or Eq. (D6). The alternative +expression is obtained by moving both s and s′ to the +central edge, +����� +s′ +s +ε′ +d +α +b +c +a +e +� += +� +η′,β +F b∗ca +s′ε′∗η′F dc∗e∗ +sα∗β +����� +s′ +s +ε′ +d +α +b +c +η′ +β +� += +� +η′,β,q +F b∗ca +s′ε′∗η′F dc∗e∗ +sα∗β F sβ∗c∗ +s′η′∗q +����� s′ +s +ε′ +d +α +b +q +η′ +β +� += · · · += +� +q +Ccoef.(q) +����� +s′ +s +ε′ +d +α +b +q +γ +τ ′ +� +, +where, for each q, the coefficient of the alternative +expression is +Ccoef.(q) = +� +η′,β +F b∗ca +s′ε′∗η′F dc∗e∗ +sα∗β F sβ∗c∗ +s′η′∗q F α∗dβ +s′q∗τ ′F ε′∗b∗η′ +sqγ +. +(D7) +Manipulating the F-symbols via the tetrahedral symme- +try Eq. (B7) and performing the above sum over either +η′ or β via the pentagon equation Eq. (B8), we obtain +� +� +� +� +� +� +� +� +ρ +F b∗ca +s′ε′∗ρF e∗dc∗ +s′ρ∗τ ′F τ ′ρ∗e∗ +sα∗q∗ F ε′∗b∗ρ +sqγ +if sum over β, +� +ρ +F dc∗e∗ +sα∗ρ F ab∗c +sρ∗γF γρ∗a +s′ε′∗qF α∗dρ +s′q∗τ ′ +if sum over η′, +(D8) +which are exactly the coefficients in Eq. (D5) and +Eq. (D6) for fixed q. +Hence, the order, in which the +strings s and s′ are fused into the bordering edges does +not matter. That is, [Gs +p, Gs′ +p′] = 0, as long as they do not +act on each other’s controlled edge. The proofs for the +remaining commutation relations in Eq. (22) and Eq. (23) +are similar. +4. +The central equation +Let us prove the central equation on a triangular +plaquette. The proof on any polygon-shaped plaquette +is similar. With the graphical definitions, we find +Gγ +p (|γ⟩ ⟨c|)ct Gc +p +† +����� +ℓ2 +ℓ3 +ℓ1 +c′ +a +b +� +=Gγ +p (|γ⟩ ⟨c|)ct δcc′ +� +t1 +F ℓ2a∗b +cℓ3t∗ +1 +����� +ℓ2 +ℓ3 +ℓ1 +a +t1 +ℓ3 +c +� +=Gγ +p (|γ⟩ ⟨c|)ct δcc′ +� +t1 +F ℓ2a∗b +cℓ3t∗ +1 F ℓ1c∗a +ct10 δt1ℓ∗ +1 +����� +ℓ2 +ℓ3 +ℓ1 +t1 +ℓ3 +c +� +=δcc′ +� +F ℓ2a∗b +cℓ3ℓ1 F ℓ1c∗a +cℓ∗ +10 +� ����� +ℓ2 +ℓ3 +ℓ1 +ℓ∗ +1 +ℓ3 +γ +� +=δcc′ +� +F ℓ2a∗b +cℓ3ℓ1 F ℓ1c∗a +cℓ∗ +10 +� � +α +F ℓ∗ +1ℓ10 +γ∗γα∗ +����� +ℓ2 +ℓ3 +ℓ1 +α +ℓ∗ +1 +ℓ3 +γ +� +=δcc′ +� +α,β +F ℓ2a∗b +cℓ3ℓ1 F ℓ1c∗a +cℓ∗ +10 F ℓ∗ +1ℓ10 +γ∗γα∗F ℓ2ℓ3ℓ1 +γα∗β +����� +ℓ2 +ℓ3 +ℓ1 +γ +α +β +� +, +(D9) +Now, it remains to show that the coefficient for every +basis ket-vector (i.e. fixing α and β) is the same as that +of � +k P γ +ct +� +dk +dγdc Bk +p +� +P c +ct. To do this, first, let us write the +pentagon equation Eq. (B8) in a different form via the +tetrahedral symmetry Eq. (B7) +F ijp +kq∗rF iq∗r +ℓ∗sm = +� +n +F rj∗k∗ +nℓs∗ F ipj +nsmF q∗kp∗ +nm∗ℓ∗ +dndr +� +dkdsdjdℓ +. +(D10) +Focusing on the F-symbols of Eq. (D9), we find +F ℓ2a∗b +cℓ3ℓ1 F ℓ1c∗a +cℓ∗ +10 F ℓ∗ +1ℓ10 +γ∗γα∗F ℓ2ℓ3ℓ1 +γα∗β += +� +dα +dℓ1dγ +� +da +dℓ1dc +F ℓ2a∗b +cℓ3ℓ1 F ℓ2ℓ3ℓ1 +γα∗β += +� +k +dk +dγdc +F ℓ1ac∗ +kγ∗α F ℓ2ba∗ +kα∗β F ℓ3cb∗ +kβ∗γ , +(D11) +The first equality follows from the normalization of F- +symbols in Eq. (B6), and we have used Eq. (D10) in + +28 +the second equality to get the last line. +The last line +is exactly the coefficient of � +k P γ +ct +� +dk +dγdc Bk +p +� +P c +ct on the +same ket-vector. +Hence, we have proven the central +equation on the triangular plaquette. + diff --git a/ItAyT4oBgHgl3EQfTPfP/content/tmp_files/load_file.txt b/ItAyT4oBgHgl3EQfTPfP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..351366ffee26452d2d112a998336c597f976f5e1 --- /dev/null +++ b/ItAyT4oBgHgl3EQfTPfP/content/tmp_files/load_file.txt @@ -0,0 +1,1450 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf,len=1449 +page_content='Renormalization of Ising cage-net model and generalized foliation Zongyuan Wang,1 Xiuqi Ma,1 David T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Stephen,2, 1 Michael Hermele,2 and Xie Chen1 1Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA 2Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA (Dated: January 3, 2023) A large class of type-I fracton models, including the X-cube model, have been found to be fixed points of the foliated renormalization group (RG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The system size of such foliated models can be changed by adding or removing decoupled layers of 2D topological states and continuous deformation of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In this paper, we study a closely related model – the Ising cage-net model – and find that this model is not foliated in the same sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In fact, we point out certain unnatural restrictions in the foliated RG, and find that removing these restrictions leads to a generalized foliated RG under which the Ising cage-net model is a fixed point, and which includes the original foliated RG as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The Ising cage-net model thus gives a prototypical example of the generalized foliated RG, and its system size can be changed either by condensing / uncondensing bosonic planon excitations near a 2D plane or through a linear depth quantum circuit in the same plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We show that these two apparently different RG procedures are closely related, as they lead to the same gapped boundary when implemented in part of a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Finally, we briefly discuss the implications for foliated fracton phases, whose universal properties will need to be reexamined in light of the generalized foliated RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' INTRODUCTION The renormalization group (RG) plays a fundamental role in the characterization and classification of quantum phases of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='1–3 It is a piece of conventional wisdom that each phase – defined as a deformation class of quantum systems – is characterized by a unique RG fixed point, which encodes the universal long-distance and low- energy properties of the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Moreover, the existence of such a fixed point underlies the key role played by continuum quantum field theory as a tool to describe universal properties of phases (and phase transitions) while discarding extraneous non-universal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Fracton models in three spatial dimensions (3D)4,5 pro- vide exceptions to this conventional wisdom, and accord- ingly challenge our understanding of the relationships among quantum phases of matter, the renormalization group, and quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This is nicely illus- trated in the X-cube model,6 perhaps the simplest frac- ton model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The defining characteristic of a fracton model is the presence of excitations of restricted mobility, and the X-cube model supports point-like excitations mobile in planes (planons), along lines (lineons), and for which an isolated excitation is fully immobile (fractons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The model is exactly solvable and has zero correlation length, so we might expect it to be a fixed point of the RG, as is the case for toric code and string-net models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='7,8 However, the X-cube model on a lattice of linear size L is equivalent (under the application of a finite-depth circuit) to an X-cube model on a smaller lattice stacked with 2D toric code layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='9 Therefore, when trying to coarse-grain the X-cube model, non-trivial 2D layers are left behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' These layers cannot be integrated out or otherwise removed, thus preventing the model from being a fixed point of any conventional RG procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This behavior is closely related to the striking system-size dependence of certain properties, such as the ground state degeneracy (GSD) and the number of types of fractional excitations, both of which grow exponentially in the linear system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='9,10 Similar phenomena occur in other fracton models, including Haah’s cubic code11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' It is interesting to ask whether some fracton models are fixed points of a suitably generalized RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' While there are many schemes and procedures for carrying out RG in different settings, it is important to emphasize that simply finding a new RG scheme is not enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Instead, a more radical generalization of what we usually mean by RG is needed, because, for instance, any RG procedure that can have the fracton models as fixed points must allow for the increase / decrease in GSD and the addition / removal of fractional excitations in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Along these lines, it was found the X-cube model is a fixed point of a foliated RG procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='9,12–14 It is helpful to recall the conventional RG procedure for gapped phases2,3, which allows, in each RG step, for continuous deformations of the Hamiltonian that keep the gap open, and for the addition/removal of trivial gapped systems (those whose ground state is a product state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In the foliated RG, one also allows addition or removal of decoupled, gapped 2D systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Such 2D systems can be topologically ordered and thus carry non-trivial GSD and fractional excitation types, hence allowing for these properties to change under RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In the case of the X-cube model, we can remove 2D toric code layers under the foliated RG, thus making the model into a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' More generally, a large class of type- I fracton models6 – those where some of the fractional excitations are mobile – are fixed points of the foliated RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The foliated RG leads to the closely related notion arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='00103v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='str-el] 31 Dec 2022 2 of foliated fracton phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='10,15 Foliated fracton phases, which we define in Appendix A, are a coarser equivalence relation on ground states than ordinary phases, and each foliated fracton phase contains a fixed point of the foliated RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This fixed point captures certain universal properties that are the same everywhere in the foliated phase, and these properties are referred to as foliated fracton order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' When a model belongs to a foliated fracton phase, it is a convenient shorthand terminology to refer to the model as being foliated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' An interesting type-I fracton model that has not been investigated from this perspective is the Ising cage-net model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='16 The Ising cage-net model is very similar to the X-cube model in many ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Both are exactly solvable models that can be obtained from a coupled layer construction, based on toric code layers in the X-cube case,17,18 and doubled-Ising string-net layers in the cage- net case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='16 Both have fracton excitations that are created at the corners of a rectangular membrane operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Both have lineon excitations (abelian in the X-cube model and non-abelian in the cage-net model) that move in the x, y and z directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Both have other planon excitations that move in xy, yz or zx planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Despite these similarities, it has not been clear whether the Ising cage-net model is foliated in the sense defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' It is important to emphasize that, while both involve a layer structure, the coupled-layer constructions of X-cube and cage-net models are very different from foliated RG and from the notion of foliated fracton phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In particular, there is no obvious relationship between whether a model can be obtained by a coupled- layer construction and whether it is foliated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' By analogy with the X-cube model, it is natural to guess that the Ising cage-net model is a foliated RG fixed point upon adding/removing doubled-Ising string-net layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' However, this cannot be the case, because the doubled- Ising string-net model contains non-abelian excitations with quantum dimension √ 2, while the cage-net model has excitations with integer quantum dimension only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='16 While this argument does not rule out the possibility of a foliated RG fixed point with other 2D topological states as resources, in fact the Ising cage-net model is not foliated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This can be seen by studying the model’s GSD, which has been computed by some of the authors in a separate paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='19 It is found that the GSD does not grow by integer multiples when the system size grows by unity in the x, y or z directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The question is then open again: can we think of the Ising cage-net model as a fixed point of a suitably generalized RG?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' More specifically, can the foliated RG be generalized somehow to include the Ising cage-net model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In fact, we argue in this paper that the foliated RG should be extended, independent of the Ising cage-net example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We do this by re-examining foliated RG from two complementary perspectives, one based on planon condensation, and the other based on quantum circuits, and point out that in both these pictures, the foliated RG has unnatural restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' These observations lead us to a generalized foliated RG under which, remarkably, the Ising cage-net model is a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' finite depth circuit condensation/ linear depth circuit foliated fracton model 2D topological order Lx × Ly × Lz Lx × Ly × (Lz − 1) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 1: Top: the foliated RG scheme, where a layer of topologically ordered state (shown in orange) can be added into or removed from a foliated fracton model via a finite depth circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Bottom: generalized foliated RG scheme realized by condensation of bosonic planons or a sequential linear depth circuit around the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The generalized foliated RG can be carried out either by condensing or uncondensing bosonic planon excitations supported near a 2D plane, or by acting with a quantum circuit, supported near a 2D plane, whose depth scales with the linear size of the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We show that either of these operations can be used to decrease or increase the system size of the Ising cage- net model, which is thus a generalized foliated RG fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The two apparently different ways of carrying out the generalized foliated RG are closely related, through a connection that we explain between anyon condensation and a class of linear depth circuits that we refer to as sequential circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We note that the original foliated RG arises as a special case of the generalized procedure introduced here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In particular, for the X-cube model, instead of decoupling a toric code layer and removing it to decrease system size, we can condense the bosonic planon that effectively comes from the toric code layer (either e or m), which has the same effect as removing the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Alternatively, we can act with a certain linear-depth circuit (more specifically, a sequential circuit) whose effect is to condense the same bosonic planon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Therefore, we can use generalized foliation to study the X-cube model, the Ising cage-net model and many other type-I fracton models within a single framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Just as foliated RG comes with the notion of foliated fracton phases and foliated fracton order, we expect that the generalized foliated RG comes with corresponding notions of generalized foliated 3 fracton phases and generalized foliated fracton order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' It will be interesting to study these notions in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The paper is structured as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' II, we review the original foliated RG by focusing on the X-cube model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' III, we review the Ising cage-net model, which is not foliated according to the original scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Section IV then briefly points out some unnatural restrictions within the original foliated RG, and proposes a generalized foliated RG where these restrictions are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' V, we show that the Ising cage-net model is foliated in terms of a generalized foliated RG defined by planon condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Then, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' VI, we demonstrate that the generalized foliated RG can also be implemented by a planar linear depth circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The linear depth circuit has a special structure, and we dub it a sequential circuit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' VII we show how the sequential circuit we use is closely related to the condensation of planons via gapped boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' VIII, we conclude with a brief discussion on the implications of and outlook for the generalized foliated RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' FOLIATION IN X-CUBE Before our discussion of the ‘generalized foliation’, it is instructive to review the original notion of foliation and see how the corresponding RG procedure is carried out for the X-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The X-cube model has a foliated structure, where layers of the toric code can be added to or removed from the X-cube via a finite depth circuit S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='9 Given an X-cube ground state |ΨX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='⟩ of the system size Lx × Ly × Lz and a toric code ground state |ΨT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='⟩, S yields a |ΨX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='⟩ of the size Lx × Ly × (Lz + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In rest of this section, we review the finite depth circuit S on the three-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 2: (a) The three types of vertex terms in the X-cube Hamiltonian Ax v, Ay v, and Az v, which are tensor products of Pauli-Z operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (b) The cube term Bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Let us consider the X-cube Hamiltonian defined on a cubic lattice on the three-torus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' and one copy of the toric code Hamiltonian defined on a square lattice on the two- torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For both models, the local qubit DOFs are placed on the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The X-cube Hamiltonian6 HX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' = − � v (Ax v + Ay v + Az v) − � c Bc (1) contains three types of vertex terms Ax v, Ay v, and Az v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' and one type of cube term Bc, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The toric (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 3: (a) The vertex term Qv in the toric code Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (b) The plaquette term Bp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' code Hamiltonian20 HT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' = − � v Qv − � p Bp (2) is a sum of local terms as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 4: The insertion of a layer of toric code living on an xy-plane (blue colored square lattice) into a cubic lattice, which hosts the X-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The inserted layer bisects an edge i near the inserted plane into edges labeled by i′ and k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For every bisected edge, the X-cube Hamiltonian is modified by replacing Zi → Zi′ and Xi → Xi′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The new edges k′ are product states with the Hamiltonian of H0 = − � {k′} Zk′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' To construct the circuit, we first insert a decoupled toric code into the X-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 4, when the inserted toric code lies in the xy-plane, it bisects the z-direction edges in the X-cube model, thus creating new qubit edges k′ colored in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' These new k′ edges are added to the system as product states whose Hamiltonian is chosen to be H0 = − � {k′} Zk′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For each bisected edge i in the X-cube Hamiltonian, we substitute Zi → Zi′ and Xi → Xi′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The circuit S is a product of two finite depth circuits S2 and S1, S = S2S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Each is a product of the controlled- NOT (CNOT) gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The circuit S1 acts on the edges of the modified X-cube Hamiltonian, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Every CNOT gate in S1 has an i′ edge serving as the controlled qubit and the corresponding k′ edge as the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' On the other hand, S2 acts on both edges of the X-cube and those of the toric code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=" Every edge of the X X X X X X X X X X X2 ZX X X XZ x k'4 (a) (b) FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 5: An illustration of the finite depth circuit S = S2S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (a) The action of the circuit S1 when focus on an elementary cube of the original cubic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The arrows, representing the CNOT gates, point from the controlled qubits to the targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (b) S2’s action viewed at a cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' toric code serves as the controlled qubit for the CNOT gates whose targets are edges in the modified X-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' An illustration of S2 is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The CNOT gate, acting by conjugation, has the actions of ZI �→ ZI, IZ ↔ ZZ, XI ↔ XX, IX �→ IX, (3) where the first qubit is the control and the second is the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' All the CNOT gates in S1 or S2 commute with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Therefore, S is a finite depth circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' By direct computation, we see that S � ˜H(Lx,Ly,Lz) X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' + HT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' + H0 � S† ∼= H(Lx,Ly,Lz+1) X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' , (4) where ˜HX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' is the modified X-cube Hamiltonian, and the symbol ∼= denotes that the L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' and the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' share the same ground space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' ISING CAGE-NET In this section, we review the basic definition and properties of the Ising cage-net model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The Ising cage-net is an exactly solvable model obtained from the coupled layer construction16, in which decoupled layers of the doubled-Ising string-net21–25 are coupled together through the particle-loop (p-loop) condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Specifically, we take three stacks of the doubled-Ising string-net defined on a square-octagon lattice (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 6), and stack them together to form a truncated cubic lattice, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Each of the six faces of a cube is an octagonal plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We call an edge l, parallel to the µ- direction for µ ∈ {x, y, z}, a µ-principal edge, and denote it by lµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' As a 2D lattice model, the doubled-Ising string-net is built from the Ising unitary modular tensor category26,27, which consists of an index set {0, 1, 2} and a set of symbols (δijk, ds, F ijm kln , Rij k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The model has a three- dimensional local Hilbert space of spanC{|0⟩ , |1⟩ , |2⟩} for each edge of the square-octagon lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The states |0⟩, Qv Bs p W fluxon l FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 6: A square-octagon lattice, where the doubled- Ising string-net is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Each edge has a local Hilbert space with a basis {|0⟩ , |1⟩ , |2⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Qv is defined for every trivalent vertex, and Bp = �2 s=0(ds/D)Bs p is defined for each square and octagonal plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The string- operator for a fluxon excitation W fluxon l violates the two Bp terms containing the edge l and no Qv term anywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 7: A truncated cubic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' It is formed by intersecting layers of the square-octagon lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Every cube has six octagonal faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' At the corners of each cube are octahedrons (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The edges l, parallel to µ direction for µ ∈ {x, y, z}, are called the µ- principal edges, which are denoted by lµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For the system of decoupled layers, a µ-principal edge has a nine- dimensional local space given by the tensor product of ( spanC{|0⟩ , |1⟩ , |2⟩} )⊗2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' |1⟩, |2⟩ are dubbed as 0-string, 1-string, and 2-string respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The commuting projector Hamiltonian HD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' = − � v Qv − � p Bp (5) consists of the vertex projector Qv and the plaquette projector Bp = �2 s=0(ds/D)Bs p (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The symbol ds takes values in d0 = d2 = 1, and d1 = √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' D = � s(ds)2 is the total quantum dimension of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Z5 Qv’s action is defined by Qv ����� j i k � = δijk ����� j i k � , (6) where the symbol δijk is symmetric under permutation of its indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The non-zero elements are δ000 = δ011 = δ211 = δ022 = 1, up to permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The subspace where all the vertex terms Qv are satisfied is called the stable vertex subspace HD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Qv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='25 The plaquette operator Bs p’s action are evaluated by the graphical rules, which are defined via the d- and F-symbols (Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Bs p acts on a plaquette by fusing a loop of s into the edges as, for example, Bs p ����� ℓ6 ℓ5 ℓ8 ℓ7 ℓ2 ℓ1 ℓ4 ℓ3 e2 e4 e6 e8 e3 e5 e7 e1 � = ����� ℓ6 ℓ5 ℓ8 ℓ7 ℓ2 ℓ1 ℓ4 ℓ3 e2 e4 e6 e8 e3 e5 e7 e1 s � = � e′ 1,e′ 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=',e′ 8 � 8 � i=1 F ℓi ei+1 ei s e′ i e′ i+1 � ����� ℓ6 ℓ5 ℓ8 ℓ7 ℓ2 ℓ1 ℓ4 ℓ3 e′ 2 e′ 4 e′ 6 e′ 8 e′ 3 e′ 5 e′ 7 e′ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (7) For every ground state |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='⟩, which is a superposition of different configurations of closed loops satisfying Qv at each vertex, Bs p acts as Bs p |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='⟩ = ds |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (8) Moreover, the Bs p operators form a commutative fusion algebra of Bi pBj p = 2 � k=0 δijkBk p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (9) The doubled-Ising string-net has nine topological excitations {1, ψ, ¯ψ, σ, ¯σ, σ ¯ψ, ψ¯σ, σ¯σ, ψ ¯ψ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In terms of the theory of anyons, these excitations come from a copy of the chiral Ising anyon {1, σ, ψ}, and an anti-chiral copy {1, ¯σ, ¯ψ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The fusion rules for the chiral Ising anyon are × 1 σ ψ 1 1 σ ψ σ σ 1 + ψ σ ψ ψ σ 1 (10) The anti-chiral Ising anyon obeys the same fusion rules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' we simply replace the anyon labels above with the barred version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Among the nine excitations, the non-abelian σ¯σ and the abelian ψ ¯ψ are bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' They are also the only non-trivial pure fluxon excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A fluxon excitation violates exactly one Bp term and none of the Qv terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A fluxon string-operator W fluxon l creates the fluxon and its anti-particle on the two adjacent plaquettes sharing the edge l (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In particular, the ψ ¯ψ has a string- operator W ψ ¯ ψ l = (−1)n1(l), (11) where n1(l) = 1 if the edge l is in the state |1⟩, and n1(l) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 8: An elementary ψ ¯ψ particle-loop (p-loop), the red loop, created by the coupling operator Vlµ shown by the green tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We represent a flux by a line segment normal to the hosting plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Joining the segments together, we have the red loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' To couple the stacks of the doubled-Ising string-net layers together, we condense the ψ ¯ψ p-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 8 is the smallest ψ ¯ψ p-loop created by the coupling operator Vlµ = W (ψ ¯ ψ)µν lµ W (ψ ¯ ψ)µρ lµ , (12) which is a product of ψ ¯ψ string-operators, from the µν- and µρ-planes, acting on the edge lµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We add −Vlµ for every principal edge to the Hamiltonian of the decoupled layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' −Vlµ penalizes the presence of the states |01⟩, |10⟩, |21⟩, and |12⟩ on lµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Using the Brillouin-Wigner degenerate perturbation theory and treating doubled-Ising string-nets as perturbations, we arrive at the Ising cage-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Hence, on a principal edge, the Ising cage-net has a five-dimensional local Hilbert space of spanC{|00⟩ , |11⟩ , |02⟩ , |20⟩ , |22⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Other edges are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The Ising cage-net has a commuting Hamiltonian of HI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' = − � µν,v Aµν v − � ps Bps− � po 1 2 � B0 po + B2 po � − � c Bc, (13) where Aµν v is the vertex projector in a µν-plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Bps is the doubled-Ising string-net plaquette projector for a square plaquette;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 1 2 � B0 po + B2 po � is a plaquette term associated with each octagonal plaquette po;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' and Bc = � po∈c √ 2 2 B1 po (14) 少小 少小6 is the cube term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The vertex term acts as Aµν v ����� j ℓ i k � = δijkδ(j,ℓ) ����� j ℓ i k � , (15) where we have used the doubled line to represent a principal edge in the state |jℓ⟩, and δ(j,ℓ) = � 1 for (j, ℓ) ∈ I 0 otherwise (16) with the index set I = {(0, 0), (1, 1), (0, 2), (2, 0), (2, 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Mobility Type Excitations Planons Abelian (ψ ¯ψ)µν ψµν ¯ψµν Non-abelian (σ¯σ)µν Lineons Abelian — Non-abelian σµνσµρ ¯σµνσµρ σµν ¯σµρ ¯σµν ¯σµρ TABLE I: Excitations in the Ising cage-net for each µν- plane, written in terms of the doubled-Ising excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Amongst, the only composite excitation, (ψ ¯ψ)µν, is a fracton dipole, a planon in the µν-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A lineon can only move along the line specified by the repeated position index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For example, σµνσµρ is mobile along a line in the µ direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Moreover, pairs of lineons from different planes can form a lineon dipole, which is a planon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 9: The vacuum fusion channel of three different lineons at an octahedron (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The colored solid lines represent the labeled string-operators in the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The dashed lines represent edges without string- operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The quasi-particle excitations of the Ising cage-net follow directly from the constituent doubled-Ising layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Excitations that survive the condensation must have string-operators that commute with Vlµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Thus, some of the doubled-Ising planons must now exist together with some other doubled-Ising planons from a perpendicular plane, hence the emergence of lineons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A lineon can turn at a corner and become another lineon at the cost of emitting a third one (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The ψ ¯ψ, on the other hand, splits into two fractons, where each fracton is immobile as there is no operator that can annihilate an individual fracton and create it at a different location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We summarize the excitations in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 10: A cage configuration, as dictated by Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The orange colored cage is formed by a loop of 1 on each octagonal face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The purple lines represent strings of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A ground state of the Ising cage-net is a superposition of different configurations of cages, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Bps, B0 po, B2 po, and Bc all have the eigenvalue of 1 on the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In a separate paper19, we find the GSD of a Lx × Ly × Lz Ising cage-net to be GSD(Lx, Ly, Lz) = 1 8 � A + B + 5C + 45 � , (17) where A = 9Lx+Ly+Lz, B = 9Lx+Ly + 9Ly+Lz + 9Lz+Lx, and C = 9Lx + 9Ly + 9Lz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We immediately see that GSD(Lx, Ly, Lz + 1)/GSD(Lx, Ly, Lz) is not an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Thus, the Ising cage-net is ‘not’ foliated according to the foliation10,15 introduced previously for the X-cube and other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Nevertheless, as we will see, the Ising cage- net is foliated in a generalized sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' GENERALIZING THE NOTION OF FOLIATION The calculation of the GSD for Ising cage-net model shows that it is not foliated in the usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' However, from its construction in terms of stacks of 2D topological orders, it is reasonable to expect that it may be foliated in some generalized sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Indeed, once we examine the original defnition of foliation in more detail, we can uncover two parallel ways in which it is unnaturally restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' First, let us formulate the original foliated RG process purely in terms of quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Recall that foliated RG in the X-cube model involves adding a topologically ordered layer and then coupling it to the X-cube bulk Z OxzO yz X Oyz: xy2-string L-string7 with a finite-depth quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The topological layer cannot itself be created with a finite-depth circuit from a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' However, it is now well-understood that it can be created with a linear-depth circuit28,29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Therefore, if we view foliated RG as a generalization of usual entanglement RG2,3, in which one is allowed to add ancillary degrees of freedom in a product state and then apply finite-depth circuits, moving to foliated RG corresponds to additionally allowing linear-depth circuits within a 2D subsystem of the 3D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' However, from this perspective, the current definition of foliated RG is restricted, in that we only allow the linear-depth circuit to act on the ancillae qubits and not on the 3D bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A more natural definition would be to allow the linear- depth circuit to act arbitrarily within a 2D layer on both the ancillae and the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We remark that the kinds of linear-depth circuits involved here have a special structure that preserves the area law of entanglement, as discussed in more detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Second, we can also view foliated RG in terms of condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Namely, suppose we want to implement the inverse process of removing a single layer from the X- cube model, reducing its size in one direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This can be achieved by condensing a planon within a single layer, corresponding to disentangling the toric code layer and then trivializing that layer by condensing a boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In this case, the planon which we condense is very special: it can be viewed as being part of a 2D theory that is decoupled from the rest of the excitation spectrum of the 3D bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' To be more general, if we allow condensation of planons in RG, we should allow condensation of arbitrary planons, not only those that are part of decoupled 2D theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In light of the above, there are two natural ways to extend the notion of foliated RG: linear-depth circuits and planon condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In what follows, we will show that both approaches lead to a generalized foliated RG that is applicable to the Ising cage-net model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Then, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' VII, we argue that these two approaches, while seemingly distinct, are in fact very closely related to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' RG VIA CONDENSATION How can the system size of the Ising cage-net model be increased / decreased?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In this section, we show that it can be changed through condensation and un- condensation of bosonic planons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This is closely tied to the topic of anyon condensation in 2D systems, and we refer the reader to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 30 and references therein for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Let us begin by considering the process of condensing planons in an xy-plane to decrease the system size in the z direction by one (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Recall from the last section that for each xy-plane there is a bosonic planon ψ ¯ψ which can be condensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' When ψ ¯ψ in plane z = 0 is condensed, the quasi-particle content of the model changes as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Since ψ ¯ψ is the fracton dipole, fractons between z = −1 z = 0 z = 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 11: An illustration of the relevant xy-planes of a Lx×Ly×Lz Ising cage-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Via the condensation process described in the text, we remove the z = 0 plane and obtain a Lx × Ly × (Lz − 1) Ising cage-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' planes z = 0 and z = 1 are identified with the corresponding fracton between planes z = −1 and z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The planons ψ and ¯ψ on the z = 0 plane are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The σ¯σ planon on the z = 0 plane splits into two abelian bosonic planons e and m with a mutual −1 braiding statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The lineons in the z = 0 plane composed of σxyσxz, ¯σxyσxz, σxy¯σxz, and ¯σxy¯σxz are all confined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Planons and lineons on other planes are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' After this step, we can further condense either e or m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This gets rid of the remaining planons on the z = 0 plane without affecting other quasi-particle excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Now, we see that the quasi-particle content of the model is the same as that of an Ising cage-net model with the z = 0 plane removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The planons and lineons on planes other than z = 0 are left intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Moreover, the fracton between z = 0 and z = 1, which is now identified with the fracton between z = −1 and z = 0, becomes the new fracton between z = −1 and z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Therefore, the size of the Ising cage-net model can be decreased by one in the z direction by first condensing the ψ ¯ψ planon in a plane, and then by condensing one of the split channels of the σ¯σ planon on the same plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We see that if we allow condensation of bosonic planons as a RG operation, we obtain a generalized foliated RG under which the Ising cage-net model is a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' As noted in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' IV, the original foliated RG for the X-cube model can also be viewed in terms of such condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The condensation of planons is, of course, a singular process where the bulk gap needs to close and then reopen, corresponding to a phase transition between z X8 different standard phases (see Appendix A for the definition of standard phases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This means that, similar to the original foliated RG, the generalized foliated RG operations can move across certain phase boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' However, only certain phase boundaries can be crossed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' the singularity involved in planon condensation is localized to a selected plane and is hence a “subsystem” singularity, not one in the full 3D bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A useful way to think about the condensation process is to use the fact that the Ising cage-net model can be obtained by gauging the planar Z2 symmetries of a subsystem symmetry protected topological (SSPT) model protected by such symmetries31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The planons being condensed correspond to the symmetry charges of the planar symmetries in the SSPT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Hence the condensation of the planons in a given plane corresponds to breaking / removing that planar symmetry and reducing the size of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' On the other hand, if we want to increase the size of the system by adding a plane at z = 0, we need to add the planar symmetry and the corresponding planar state back to the SSPT model and ‘re-gauge’ the planar symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' RG VIA PLANAR LINEAR DEPTH CIRCUIT The planar linear depth circuit we construct for the Ising cage-net model is a direct generalization of a RG scheme that maps product states to ground states of a string-net model, introduced by Liu Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' VI A, we review this RG procedure for the string-net models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We describe carefully an initialization step that is nontrivial for non-abelian string-net models, which was not discussed in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' VI B, we describe the RG scheme as a linear depth circuit for the Ising cage-net model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We will see that the initialization step is also important and nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' String-net RG In this section, we will first describe an important step in the RG procedure – the ‘controlled gate’ which adds a plaquette to the string-net wave-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' After that, we will describe the full RG procedure starting from the string-net wave-function on the minimal lattice on a torus and then adding plaquettes row by row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A brief review of the string-net models is given in Appendix B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Adding plaquettes via the controlled gate The controlled gate can be used to add a plaquette to the string-net wave-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We present the definition and properties of the gate in this sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Computational details of the results discussed here can be found in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Suppose that on a trivalent lattice, a plaquette is added by adding an edge (the red edge in the diagrams below), and we want to extend the string-net wave-function from the original lattice to that including this new plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' When the edge is added, it is not entangled with the rest of the lattice and is in the state |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' To merge the added edge into the lattice, first, map it to � s ds √ D|s⟩ where D is the total quantum dimension of the string-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' |0⟩ �→ � s ds √ D |s⟩ (18) Then, we use this edge as the control to draw loops around the added plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' More specifically, we can represent the controlled gate Gp = � s Gs p graphically as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The action of Gs p is similar to the action of Bs p which adds a loop s to a plaquette, but for the graphical evaluation of Gs p, we treat the control edge as if it is in the state |0⟩, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Gs p ����� ¯¯ℓ3 ¯ℓ3 ℓ4 ¯ℓ4 ¯ℓ1 ℓ1 ¯¯ℓ2 ¯ℓ2 ℓ∗ 1 ℓ2 ℓ3 ℓ4 a b c s′ � = δss′ ����� ¯¯ℓ3 ¯ℓ3 ℓ4 ¯ℓ4 ¯ℓ1 ℓ1 ¯¯ℓ2 ¯ℓ2 s ℓ∗ 1 ℓ2 ℓ3 ℓ4 a b c 0 � = δss′ � α,β,γ, δ,ε,η,τ F ℓ1ℓ∗ 10 ss∗α F ¯ℓ1aℓ1 sα∗β F ¯ℓ2ℓ2a∗ sβ∗γ F ¯¯ℓ2bℓ∗ 2 sγ∗δ F ¯ℓ3ℓ3b∗ sδ∗ε F ¯¯ℓ3cℓ∗ 3 sε∗η F ¯ℓ4ℓ4c∗ sη∗τ ����� ¯¯ℓ3 ¯ℓ3 ℓ4 ¯ℓ4 ¯ℓ1 ℓ1 ¯¯ℓ2 ¯ℓ2 α γ ε τ β δ η s � , (19) 9 where the red line with an arrow marks the control edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We carry out the explicit graphical evaluation in Appendix D 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Note that Gs p can be defined on any polygonal plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Gs p is not a unitary on the full Hilbert space, but only between subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' More specifically, it is an isometry from VSN p,s to HSN p,s, both of which involve the DOF around a plaquette p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In VSN p,s, the control edge is set to |s⟩ while the other edges come from the string-net wave-function on the lattice with the control edge missing (pretending that it is set to |0⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The vertices containing the control edge, then, involve configurations like ℓ∗ ℓ s (20) In HSN p,s, all edges, including the control edge, come from the string-net wave-function with the control edge set to |s⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In Appendix D 2, we prove that Gs p is an isometry from VSN p,s to HSN p,s by demonstrating Gs p †Gs p ����� ¯¯ℓ3 ¯ℓ3 ℓ4 ¯ℓ4 ¯ℓ1 ℓ1 ¯¯ℓ2 ¯ℓ2 ℓ∗ 1 ℓ2 ℓ3 ℓ4 a b c s′ � = δss′ ����� ¯¯ℓ3 ¯ℓ3 ℓ4 ¯ℓ4 ¯ℓ1 ℓ1 ¯¯ℓ2 ¯ℓ2 ℓ∗ 1 ℓ2 ℓ3 ℓ4 a b c s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (21) The controlled gates commute with each other � Gs p, Gs′ p′ � =0 = � Gs p †, Gs′ p′ � , (22) as long as they do not act on each other’s controlled edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Moreover, we can show � Gs p, Bs′ p′ � = 0 = � Gs p †, Bs′ p′ � , (23) provided that Bs′ p′ does not act on the control edge of Gs p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We prove these commutation relations in Appendix D 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In Appendix D 4, we prove a useful equation, which we call the central equation Gs p (|s⟩ ⟨s′|)ct Gs′ p † = P s ct �� k dk dsds′ Bk p � P s′ ct , (24) where (|s⟩ ⟨s′|)ct acts on the control edge and P s ct = |s⟩ ⟨s| is a projector on the control edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' With the central equation, we can show that the controlled gate does what we claimed – it adds a plaquette to the string-net wave-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In particular, we show below that under conjugation by Gp = � s Gs p, the projector on the control edge Pct = � s,s′ dsds′ D |s⟩⟨s′| is mapped to the plaquette projector Bp = � s ds D Bs p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' GpPctG† p = � s,s′ dsds′ D Gs p (|s⟩⟨s′|)ct Gs′ p † = � s,s′,k dk D P s ctBk pP s′ ct = � k dk D Bk p = Bp (25) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The RG circuit Using the controlled gate as a building block, we can construct the full linear depth circuit that maps a product state to the string-net wave-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We present the linear depth circuit in two steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' from a product state to a string-net wave-function on the minimal lattice on torus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' from the string-net wave- function on the minimal lattice to the full lattice by adding plaquettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We are going to focus on the trivalent square-octagon lattice, although the general procedure applies to other trivalent graphs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The minimal lattice on the torus consists of three edges, two vertices, and one plaquette, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' On the square-octagon lattice, we start from the product state ⊗l|0⟩l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Pick three edges around a vertex as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Apply a local unitary transformation on the three edges so that they become one of the ground states on the minimal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Note that for abelian string-net states, the ground states can be chosen to be a product state of the three edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In fact, the ⊗l|0⟩l state is a legitimate state already, because it satisfies the vertex term while the plaquette term is trivial for abelian strings on the minimal lattice (for proof see Appendix B 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' However, for non-abelian string- nets, the Bs p term for a non-abelian s-string acts non- trivially in the stable vertex subspace, and the ground states generally become entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In the case of the doubled-Ising on the minimal lattice, ten configurations satisfy the vertex constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Of this ten-dimensional space, only nine dimensions belong to the ground space, where B0 p = 1, B1 p = √ 2, and B2 p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The remaining one dimension carries a ψ ¯ψ fluxon excitation such that B0 p = 1, B1 p = − √ 2, and B2 p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' One possible choice of the nine doubled-Ising ground states on the minimal lattice is given in Appendix C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Now, we need to grow this minimal structure so that it reaches the full extent of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' To do this, we ‘copy’ the states on the i and j edges along the non- contractible loops in the y and x directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' To achieve this, we use controlled gates of the form � i |i⟩|i⟩⟨i|⟨0|, and apply them sequentially along the non-contractible loops, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' As this step has to be done sequentially along the loop, its depth increases linearly with the size of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This completes step 1 of the linear depth circuit, which we call initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 10 i j k i j k y x FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 12: The initialization step in the RG circuit for generating the string-net wave-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Left: pick three edges around a vertex and map them into one of the ground states of the string-net on the minimal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Right: grow the minimal structure by copying the string states |i⟩ and |j⟩ along non-contractible loops so that they reach the full extent of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (2) (1) (2) (Ly − 1) (1) {1} (Lx − 1) ˜p y x FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 13: Adding loops to plaquettes in step 2 of the RG circuit for generating the string-net wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The state has been initialized into one of the ground states on the minimal lattice (black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' First, loops are added to the square plaquettes (shown in red) in a single step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Then, loops are added to octagon plaquettes in row (1), (2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (ly − 1) sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For the last row, loops are added to octagon plaquette in column (1), (2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='., (Lx − 1) sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' No action is needed in the last plaquette ˜p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Step 2 is also of linear depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The minimal lattice has only one plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In step 2, we add more plaquettes to the lattice using the controlled gate introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' VI A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The plaquettes cannot be added all at once, because the controlled gates commute only when they do not act on each other’s control edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A linear depth circuit is hence needed to add all the plaquettes to the square-octagon lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A particular sequence for adding these plaquettes is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Firstly, all the square plaquettes (red circles) can be added at the same time because they do not overlap with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The small circle indicates the control edge while the big circle indicates the action of Gs p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Secondly, we add the square- octagon lattice in row one (labeled (1) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' All controlled gates in row one commute with each other so they can be added in one step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Then we add row two, row three, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=', until the next to last row (labeled (Ly − 1) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For the last row, we need to choose the control edges side ways because we need un-entangled edges to be used as control edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Due to this change, the plaquettes in the last row need to be added sequentially as the controlled gates do not commute any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' As shown in the figure, we can add them in the order of (green labels) (1), (2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=', (Lx − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We do not need to act in the last plaquette (labeled ˜p) as the constraint due to the last plaquette is already implied by that of the largest plaquette that we started from combined with all the small plaquettes added so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Therefore, at this point, we have finished the linear depth RG procedure that starts from a product state and maps it to the the string-net wave-function on the square-octagon lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Ising cage-net In this section, we use the controlled gate of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (19) to build up the RG circuit to enlarge an Ising cage-net ground state on the three-torus by one layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We will start, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' VI B 1, by introducing finite depth circuits that grow cages on the cage-net ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' They serve as the building blocks of the full planar linear depth RG circuit, which we discuss in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' VI B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Adding cages via the controlled gate In 2D, we have seen that a plaquette can be added to the string-net wave function, via the controlled gates, after an edge is added to the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We can extend this procedure to 3D cage-net states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Suppose that we start with the Ising cage-net ground state on the truncated cubic lattice (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 7) and add a plane in the xy direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' At each point where the added plane bisects the z direction edges, an octahedron is added, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 14, to ensure the trivalent structure in each of the coupled planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In the added plane, octagonal plaquettes fill in the space between the octahedrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Every edge of the added octahedrons carries a three dimensional Hilbert space spanned by {|0⟩, |1⟩, |2⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We start with these edges all set to the state |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The principal edges on the octagons each carry a five dimensional Hilbert space spanned by {|00⟩, |02⟩, |20⟩, |22⟩, |11⟩}, which is a subspace of the tensor product Hilbert space of two three dimensional DOFs {|0⟩, |1⟩, |2⟩} ⊗ {|0⟩, |1⟩, |2⟩} that come from the two intersecting planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We start with these principal edges in the state |00⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 14: Insertion of an xy-plane bisects a cube in the original cage-net lattice into two cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Each intersection point between the xy-plane and the z-principal edges is expanded into an octahedron to preserve the trivalent structure in the xy, yz and zx planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 15: ‘Copying’ the states on the bisected z-principal edges onto edges of the added octahedron to satisfy vertex rules in the xz and yz planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The copying process can be performed by controlled gates of the form � xy |xy⟩⟨xy|⊗|x⟩⟨0| and � xy |xy⟩⟨xy|⊗|y⟩⟨0|, indicated by the arrows pointing from the control to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We describe first the process to add one cube into the new layer, which consists of two steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' add the octahedrons to the cage-net wave-function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' grow a cage structure in the upper truncated cube of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In step one, we first need to copy the state of the bisected z-principal edge onto some of the octahedron edges so that the vertex rules are satisfied at the octahedrons’ vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Suppose the bisected edge is in the state |xy⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The copying process can be achieved with the controlled gates � xy |xy⟩⟨xy| ⊗ |x⟩⟨0| and � xy |xy⟩⟨xy| ⊗ |y⟩⟨0| as indicated by the blue and green arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Then, we add the square plaquettes to the cage-net wave- function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This can be done as described in the previous section on how to add a square plaquette to the doubled- Ising string-net wave function, as the square plaquettes remain unaffected when the doubled-Ising layers are coupled into Ising cage-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' More specifically, for each square plaquette, we pick an edge in the state |0⟩ as the control edge, map it to � s ds √ D|s⟩, and use it as the control in the controlled gate Gp that adds loops into the plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 16: Growing a cage structure in an added cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (a) First, using an edge from the bottom face (colored green) as control, add loops to the bottom and top faces, (b) then use the edges on the side faces (colored green) as control to add loops to the side face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Step 2, which adds a cage structure to the cube, is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 16, first we add loops to the bottom and top faces and then to the side faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' More specifically, first we pick a principal edge on the bottom face in the state |00⟩ as the control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We will use the convention where the first |0⟩ comes from the xy plane while the second |0⟩ comes from the vertical xz and yz planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Map the control edge as |00⟩ �→ � s ds √ D |s0⟩ , (26) Note that this takes the controlled edge out of the five dimensional subspace of {|00⟩, |02⟩, |20⟩, |22⟩, |11⟩} but keeps it in the nine dimensional space of {|0⟩, |1⟩, |2⟩}⊗2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This will also happen to other principal edges as we implement the procedure, but at the end of the process of growing a cube, all principal edges will be back to the five dimensional subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Now, using the |s⟩ state as the control, apply the controlled gate to the bottom face pb and top face pt as G0 pb + G2 pb + 1 √ 2G1 pbB1 pt (27) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 16 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Note that Gs pb and Bs pt act on the first part of the principal edges (the part that comes from horizontal planes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' After these controlled gates, the projector on the control edge |0⟩⟨0| (the first part) gets mapped to (|0⟩⟨0|)ct �→ � ss′ dsds′ D (|s⟩⟨s′|)ct �→ B0 pb + B2 pb + B1 pbB1 pt, (28) where in deriving the last line, we used the fact that the top face is part of the original cage-net wave-function + ZZ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' ■ ■ ■.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='12 and B0 pt = B2 pt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Note that it might seem that the operator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (27) is not unitary as B1 p is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' But since B1 ptB1† pt = B0 pt + B2 pt = 2, the action of the operator restricted to the ground space of the original cage-net model is indeed unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Next, we need to add loops to the side faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' To do this, we take the principal edges on the bottom face, which are now in the states |s0⟩ and send them to |sαs⟩, where αs comes from the xz or yz planes and αs = 0 if s is even, αs = 1 if s is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This brings the principal edges on the bottom face back to the five dimensional Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Then map the |αs⟩ states to |0⟩ �→ 1 √ 2 (|0⟩ + |2⟩) , |1⟩ �→ |1⟩ (29) Use the |αs⟩ states as the control to draw loop on the side faces by applying � αs Gαs ps as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 16 (b) to each side face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Let us see how the Hamiltonian terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (28) transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We show the step by step calculation for the third term B1 pbB1 pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The B1 pt part is not affected by the transformation and will be omitted from the following equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Let us focus on the transformation induced by on principal edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We label the two three-dimensional DOFs on the principal edge as 1 and 2 respectively, where 1 comes from the bottom face whose state is labeled by s and 2 comes from the side face whose state is labeled by αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' � (P 0 1 + P 2 1 )B1 pbP 1 1 + P 1 1 B1 pb(P 0 1 + P 2 1 ) � ⊗ (|0⟩⟨0|)2 �→ 1 √ 2(P 0 1 + P 2 1 )B1 pbP 1 1 ⊗ (|0⟩2 + |2⟩2) 2⟨1| + 1 √ 2P 1 1 B1 pb(P 0 1 + P 2 1 ) ⊗ |1⟩2 (2⟨0| + 2⟨2|) �→ 1 √ 2(P 0 1 + P 2 1 )B1 pbP 1 1 ⊗ � P 0 2 + P 2 2 � B1 psP 1 2 + 1 √ 2P 1 1 B1 pb(P 0 1 + P 2 1 ) ⊗ P 1 2 B1 ps � P 0 2 + P 2 2 � (30) The result is the product of B1 pb and B1 ps projected onto the five dimensional subspace of the principal edge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' as promised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This works for all side faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Similar calculations can be carried out for the first two terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' If we put everything together and omit the projection onto the five dimensional subspace of the principal edges, we see the Hamiltonian terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (28) becomes � B0 pb + B2 pb � � ps � B0 ps + B2 ps � + B1 pbB1 pt � ps B1 ps, (31) which is a sum over the desired plaquette terms on the bottom and side faces as well as the cube term on the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In the RG circuit to be discussed in the next section, we need to grow cubes in the same row at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This works in a similar way as growing a single cube and we describe the procedure here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' First, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 17 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 17: Adding a row of cubes to the cage-net state, step 1: the inserted xy-plane bisects the cubes into two;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' octahedrons are added at the intersection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 18: Adding a row of cubes to the cage-net state, step 2: (a) first, we simultaneously add loops to the bottom and the top faces of all cubes in the row;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (b), use the edges on the side face (colored green) as control to add loops to all the side faces at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' which illustrates the situation with two cubes in the row, a new plane is added which bisects the row of cubes into two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Octahedrons are added to the intersection points to preserve the trivalent structure in the coupled xy, yz and zx planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The ‘copying’ process illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 15 is then used to restore vertex rules at the vertices of the octahedrons and then the square plaquettes in the octahedrons are added to the cage-net wave-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 区 区区 Z13 The next step is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 18, which adds cage structures to a whole row of cubes at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This is done by first picking the principal edge in, for example, the x direction and use them as controls to add loops in the bottom and top faces as described above for each cube in the row (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 18 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The operations in each cube commute with that in another cube, and hence they can be done all at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Next, loops are added to the side faces using the principal edges on the bottom face as control, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 18 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Again, the operations on each side face commute with each other, so they can be done at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' As a result of this process, all the cubes in the row are now added to the cage-net wave- function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Note that the process illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 18 applies to the first row in the added plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' When we try to add subsequent rows, some of the side faces would have been added to the cage-net state already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Those side faces can be treated in the same way as the top face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' That is, apply B1 ps in step Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 18 (a) when the x-principal edge is in the state |10⟩, instead of applying � αs Gαs ps controlled by the bottom principal edge of the side face in the state |sαs⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A similar procedure applies to the cubes in the last row of the added plane as well, which have to be added one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' RG circuit – Ising cage-net The processes for adding single cubes and a row of cubes are building blocks for the full RG circuit that adds a full plane to the cage-net state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Similar to the case of the doubled-Ising, we first need to initialize the added plane into proper eigenstates of the non-local logical operators before adding the local structures of cubic cages (plaquettes in the case of doubled-Ising).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A commuting set of logical operators of the Ising cage- net ground space can be chosen to be generated by the string-operators of ψ, ¯ψ planons in each µν plane along the µ and ν directions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We can choose the original cage-net state (before adding the plane) to be an eigenstate of all such logical operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The added xy plane can be initialized into an eigenstate of ψx, ψy, ¯ψx and ¯ψy on that plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The circuit described in the last section on how to add cubic cages and plaquette terms to the wave-function does not affect these nonlocal logical operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Therefore, the resulting cage-net state after the RG circuit remains an eigenstate of all the ψ, ¯ψ logical operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' But the choice of the eigenvalue for the ψ, ¯ψ logical operators is not arbitrary as the operators are related to each other and hence their eigenvalues are constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 19, we study carefully the relations among these operators, which allowed us to derive the ground state degeneracy of the Ising cage-net model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The relations are listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For derivation, see the discussion in section VII of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For {µ, ν, λ} = {x, y, z} � i � ψ ¯ψ �µ µλ (ν = i) � j � ψ ¯ψ �ν νλ (µ = i) = 1 rµν(λ = i)¯rµν(λ = i) = 1, ∀i, ∀{µ, ν} rµν(λ = i)rµν(λ = i + 1) = 1, ∀i, ∀{µ, ν} (32) where rµν = 1 2 � 1 + ψµ µν + ψν µν − ψµ µνψν µν � , ¯rµν = 1 2 � 1 + ¯ψµ µν + ¯ψν µν − ¯ψµ µν ¯ψν µν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' As we started from a ground state of the cage-net model, the original set of ψ, ¯ψ operators satisfy the relations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' When we add a new xy-plane, we need to make sure that after the new ψx xy, ψy xy, ¯ψx xy, ¯ψy xy operators are added to the original set, the total set still satisfy the relations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This can be guaranteed when the added string- operators satisfy ψx xy ¯ψx xy = 1, ψy xy ¯ψy xy = 1 (33) rxy = ¯rxy = ±1 (34) The choice of ±1 in the last relation depends on whether rxy(z = i) = 1 or −1 in the original set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Compared to the eigenstates listed in Appendix C 1, |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩1, |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩5, |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩9 satisfy the relations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (33) and rxy = 1 while |ψ ¯ψ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩ satisfies the relations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (33) and rxy = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Therefore, we can initialize the added layer into one of these states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 19: Inserting an xy-plane into the original cage-net lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Each red ball represents an octahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The new principal edges are shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In particular, consider the added xy-plane in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Each red ball represents an octahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The added DOF are initially set to be either in state |0⟩ (on edges of the octahedron) or |00⟩ (on principal edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Now initialize the trivalent lattice in the xy-plane into one of |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩1, |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩5, |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩9 and |ψ ¯ψ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩ following the procedure described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This linear depth process set up the stage for the next step of the RG circuit: adding cage structures to the cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Z x 1 114 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 20: Adding cage structures to the cubes in step 2 of the RG circuit for the cage-net state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The red lines indicate the minimal lattice state determined by the initialization step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Cage structures are added to the cubes in the 1st row, the 2nd row, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' the (Ly − 1)th row in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In the last row, cage structures are added to the cube in the 1st column, 2nd column, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=', (Lx − 1)th column in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' No action is required in the last cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Now we can use the procedure described in the last section to add cage structures to the cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 20, on top of the minimal structure set up in the initialization step (red lines), cage structures are added to the cubes in the 1st row, the 2nd row, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' the (Ly − 1)th row in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In the last row, cage structures are added to the cube in the 1st column, 2nd column, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=', (Lx − 1)th column in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' No action is required in the last cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This process has depth ∼ (Lx + Ly) and completes the addition of a new layer into the cage-net wave-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' RELATING CONDENSATION AND LINEAR-DEPTH CIRCUITS VIA GAPPED BOUNDARIES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' General discussion In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' V, we discussed the RG process in terms of condensation of planons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' VI, we discussed the RG process in terms of a linear depth circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In this section, we show that these two are closely related to each other by understanding each in terms of gapped boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We first consider a gapped boundary between a 2D topological order and vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' If an excitation moves from the bulk to the boundary, it may become trivial in the sense that it can be destroyed by a local operator on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This phenomenon is referred to as condensation at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' On the other hand, some excitations remain non-trivial as they approach the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' These phenomena can be characterized precisely in a category-theoretic language32–35;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' in the abelian case, this amounts to specifying a maximal subset of bosons that can simultaneously condense at the boundary36–39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' It is believed the universality class of a gapped boundary is fully determined by its category- theoretic characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The above discussion allows us to define distinct types of anyon condensation (to vacuum) in a precise way, as distinct types of gapped boundaries (to vacuum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Such a definition is natural if we view the vacuum as a condensate of certain anyons in the 2D topological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For instance, creating a puddle of anyon condensate within the bulk 2D topological order amounts to creating a puddle of trivial state (vacuum) separated from the bulk by a gapped boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This discussion, and the definition of anyon condensation in terms of gapped boundaries, can be generalized to gapped boundaries between arbitrary 2D topological orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In the context of generalized foliated RG, we consider condensation of planons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Condensation of a single planon can similarly be associated with – and defined in terms of – certain gapped boundaries between two fracton orders, with the property that the boundary should be transparent to mobile excitations away from the selected plane where the condensation occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' It will be an interesting problem for future work to fully characterize those boundaries between fracton phases that correspond to planon condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We note that there has been some related prior work discussing gapped boundaries of fracton models in terms of condensation40,41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' It turns out that the kind of linear-depth circuits considered here can also be associated with a type of gapped boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A linear depth circuit has the general form U = �K ℓ=1 Uℓ where each layer Uℓ consists of a number of local unitary gates with non-overlapping support, and the number of layers K is proportional to the linear system size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In general, Uℓ can contain gates acting across the entire system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' However, for the circuits we employed for RG, each layer Uℓ only contains gates acting in a lower dimensional subsystem of the entire system, such as the rows in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 13 and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Such circuits are much more restrictive than generic dense linear-depth circuits, particularly because they preserve the area law when acting on a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We call this class of circuits sequential circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Again we first focus on the 2D case, where as we have discussed, sequential circuits can be used to generate topologically ordered ground states from an initial product state (the topological “vacuum”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In order to avoid complications associated with periodic boundary conditions, we make a simplification as compared to the circuits discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' VI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' namely, we work with an infinite system and consider circuits that generate a disc Z MOL 1 1st col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='15 of 2D topological order from vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' If desired, the size of the disc can later be taken to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This allows us to drop the initialization step, whose role is to take care of the non-trivial ground state degeneracy on a 2-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We can also drop the final linear-depth sequence of gates needed to stitch two gapped boundaries together in a manner consistent with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' With these simplifications, the circuits operate in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We slice the 2D space into 1D concentric circles surrounding the center of the disc, and order these subspaces according to their radial coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The ℓth layer of the circuit is assumed to be supported near (but not entirely within) the ℓth circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' After applying some number of layers of the circuit, one is left with a disc of topological order which has a gapped boundary to the vacuum region which has not yet been acted on by the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Then, the next layer in the circuit acts only within the vicinity of the one-dimensional gapped boundary between the topological order and the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The action of the unitary in this layer is to “grow” the topological order by a small amount, pushing the gapped boundary further into the vacuum region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Continuing in this way allows one to grow the topologically ordered region arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Based on the above, given a sequential circuit, we can associate the universality class of the gapped boundary to vacuum which emerges when the circuit is truncated at some radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This association is well-defined in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We can define a truncation of the circuit ¯U = �K0 ℓ=1 Uℓ where K0 < K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This will create a disc of topological order with a particular gapped boundary to vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Now, consider a different truncation ¯U′ = �K0 ℓ=1 Vℓ where each Vℓ again consists of non-overlapping gates such that Vℓ = Uℓ for ℓ sufficiently less than K0, but the layers near the boundary may differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' By definition, the two truncated circuits differ only by a finite-depth circuit near the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' But a 1D finite depth circuit cannot change the universality class of the gapped boundary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' it cannot change the set of anyons which can condense on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' So the gapped boundary type is independent of how the sequential circuit is truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We note this conclusion only holds for truncations that are compatible with the 1D layer structure of concentric circles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' the key property is that the truncation only cuts through a finite number of 1D layers, which is bounded above as the size of the disc increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We emphasize that this discussion can be generalized to gapped boundaries between two different 2D topologi- cal orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' That is, given two topological orders referred to as A and B that admit a gapped boundary, an A- ground-state can be converted into a B-ground-state by applying a sequential circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Or, if we apply a trun- cated version of the same sequential circuit, we can cre- ate a puddle of B within the bulk topological order A, separated by a gapped boundary whose universality class does not depend on how the circuit is truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In formulating the generalized foliated RG in terms of quantum circuits, we apply sequential circuits within 2D layers of a 3D fracton model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Truncating such a sequential circuit (along its 1D layer structure) results in a gapped boundary between two different fracton orders, where some of the mobile excitations may condense along the layer where the circuit is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This is how we described planon condensation above, and thus we propose that planon condensation and applying 2D sequential circuits are different ways to realize the same operation in generalized foliated RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Condensation in the Ising cage-net circuit In accordance with the above discussion, we now identify the type of gapped boundary that is associated with the sequential circuits used to create Ising cage- net model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' To accomplish this, we are going to apply the circuit only to a finite disc-shaped region within a plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' we will not take the limit that the size of the disc goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Inside the region, we get the fracton order as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Outside of the region, the added degrees of freedom remain unentangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' There is a gapped boundary between the two sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We show that the gapped boundary and the region outside can be obtained by condensing bosonic planons starting from a complete fractonic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' First, let’s see how a similar relation works in the doubled-Ising string-net state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We imagine a very large disc of string-net state, and we ignore the curvature of the disc’s boundary to simplify the following discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Recall that in the RG circuit, the plaquettes are added row by row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Suppose that we stop the process at row i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The boundary between row i and row i + 1 is a smooth boundary on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' As the Hamiltonian terms remain commuting throughout the process, the boundary is gapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The gapped boundary can be induced by the condensa- tion of ‘fluxon excitations’22 ψ ¯ψ and σ¯σ on the boundary and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' To see that, consider a string-operator of the form shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 21, which consists of a string segment above the lattice, a parallel segment under the lattice and the two are connected by segments that vertically go through the lattice plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Note that, while embed- ded in the 3D space, the string-operator is a closed loop, from the 2D perspective, it ends at the locations where the string goes through the lattice plane and can create excitations at those points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In particular, such string- operators in general violate the plaquette term at their ends, as the plaquette terms correspond to a loop oper- ator that links with the string-operator and the linking generates nontrivial action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Therefore, in the bulk of the string-net state, the string-operator generates ‘fluxon ex- citations’ at its ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In the doubled-Ising model, there are two string-operators of this type, corresponding re- spectively to a loop of string type 1 and a loop of string type 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The two string-operators generate the ψ ¯ψ and σ¯σ excitations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' If the string-operator ends 16 s-loop i i − 1 i + 1 p p′ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 21: Condensation of the ψ ¯ψ and the σ¯σ fluxons on the smooth boundary of the doubled-Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The vertex details are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The dashed lines represent the unentangled edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' An open ended fluxon string- operator is constructed from a loop of s-string that passes through the lattice plane vertically at a plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' If the plaquette (for example, the one labeled p) lies within the doubled-Ising region, it creates a fluxon excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' If the plaquette (for example, the one labeled p′) falls outside the string-net region, then no excitation is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Thus, all fluxons condense on the smooth boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For computational details on the condensation, see Appendix C 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (goes vertically through the lattice plane) outside of the smooth boundary (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 21), there are no more plaquette terms to violate and the string-operator does not gener- ate any excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Detailed calculations can be found in Appendix C 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Therefore, the ψ ¯ψ and σ¯σ excitations condense on the boundary and beyond, thus demonstrat- ing the connection between anyon condensation and the linear depth circuit for the doubled-Ising string-net state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The situation is very similar in the Ising cage-net model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The RG circuit is again implemented row by row in a sequential manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Suppose that we stop the process at row i, there will be a gapped boundary between row i and row i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 22, like for the string-nets, a vertical loop operator that goes through the lattice plane at two points generates planon excitations ψ ¯ψ and σ¯σ in the bulk of the cage-net state (in rows j ≤ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Beyond row i, however, it does not generate any excitations and hence the ψ ¯ψ and σ¯σ are condensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This agrees with the RG procedure driven by condensation described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Therefore, the process of sequential application in the linear depth circuit can be interpreted as moving the boundary between the cage-net state and the condensed state, hence enlarging or shrinking the fracton order in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (s-loop)xy i i + 1 p p′ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 22: Condensation of the ψ ¯ψ and the σ¯σ fluxon excitations in the half xy-plane (shown in blue) in the Ising cage-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' If the end of the the fluxon string operator falls within the Ising cage-net region (for example at the plaquette p), a fluxon excitation is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' If the end falls outside of the Ising cage-net region (for example at the plaquette p′), then no excitation is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Therefore, both ψ ¯ψ and σ¯σ planons condense on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' SUMMARY AND DISCUSSION In this paper, we studied the renormalization group transformation for the Ising cage-net model and found that the system size of the Ising cage-net model can be decreased / increased by condensing / uncondensing planon excitations near a 2D plane, or correspondingly through a so-called sequential circuit which preserves the area law and whose depth scales with the linear size of the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We argued that these two ways of carrying out the RG are closely related through gapped boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We call this procedure the generalized foliated RG, because the previously defined foliated RG, under which the X-cube and related models are fixed points,9 fits into this new definition as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' On the one hand, the system size of the X-cube can be decreased / increased by condensing / uncondensing a lineon dipole or fracton dipole on a given plane (both these excitations are planons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Or, the RG procedure can be carried out with a linear depth circuit in the same plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' One way to construct the linear depth circuit is to use the finite depth circuit discussed for the original foliation scheme9 to decouple a layer of toric code out of the X-cube model, and then disentangled the toric code into product state Z1 1 1 1 1 1 1 1 1 1- 1 1 1 1 1 1 1 1 117 with a linear depth circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Altogether this is a linear depth circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Alternatively, we can use a circuit similar to that discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' VI to remove cage structures in a plane row by row and hence removing a plane from the X-cube model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' On the other hand, the generalized foliated RG allows a broader class of RG operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Indeed, the Ising cage- net model is not a fixed point of the original foliated RG as can be seen from its ground state degeneracy calculation19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We recall that the original foliated RG led to an associated notion of foliated fracton phases (see Appendix A for a definition), with the key property that two systems related by a foliated RG operation lie within the same foliated fracton phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Similarly, we expect that there exists a notion of generalized foliated fracton phase (GFF phase), again with the key property that two systems related by a generalized foliated RG operation lie in the same GFF phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' GFF phases should be a coarser equivalence relation on quantum systems than foliated fracton phases, because a broader class of RG operations are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We do not currently know how to give a definition of GFF phases along the lines of those in Appendix A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' however, one possibility is to give a definition based on circuit equivalence of ground states, where one allows certain linear depth circuits supported on planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' IV, we pointed out that the original foliated RG contains certain unnatural restrictions, while the gener- alized foliated RG seems to be more natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' There- fore, we expect that GFF phases are correspondingly a more natural concept than foliated fracton phases as orig- inally defined, so it will be important to revisit what we have learned about foliated fracton phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In particu- lar, several invariants have been devised for foliated frac- ton phases as originally defined, including those based on fractional excitations and entanglement entropy10,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Now, with a new notion of GFF phases, we need to re- consider the question of what quantities remain invariant under the new equivalence relation, and which models belong to the same GFF phase and which do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For example, we can ask whether the twisted foliated fracton model proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 13 is still in a different phase than the X-cube model or not under the new definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Finally, we want to comment that the generalized foliation defined in this paper makes the discussion of type I fracton models more in-line with that of Subsystem Symmetry Protected Topological (SSPT) phases with planar symmetry in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 42–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In the definition of ‘strong SSPT’ in these papers, when a decoupled layer with planar symmetry is added to the bulk of the system, the planar symmetry can be combined with an existing planar symmetry in the system, which corresponds to the condensation of the composite of the symmetry charges from the decoupled plane and a planar symmetry charge in the bulk of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The ‘strong SSPT’ orders discussed in these papers hence may become nontrivial (twisted) foliated fracton orders when the planar symmetries are gauged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' ACKNOWLEDGMENTS We are indebted to inspiring discussions with Dave Aasen, Kevin Slagle, Nathan Seiberg, and Dominic Williamson, and helpful correspondence with Fiona Burnell and Michael Levin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=', X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' are supported by the National Science Foundation under award number DMR-1654340, the Simons Investigator Award (award ID 828078) and the Institute for Quantum Information and Matter at Caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' is also supported by the Walter Burke Institute for Theoretical Physics at Caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The research of MH is supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) under Award number DE- SC0014415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This work is also partly supported by the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (651438, XC and ZW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 651440, MH and DTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The work of MH on general aspects of the generalized foliated RG 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Levin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' X 3, 021009 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 40 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Bulmash and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Iadecola, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' B 99, 125132 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 41 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Luo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Spieler, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Sun, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Karch, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' B 106, 195102 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 42 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' You, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Devakul, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Burnell, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Sondhi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' B 98, 035112 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 43 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Devakul, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Williamson, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' You, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' B 98, 235121 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 44 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Devakul, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Shirley, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Research 2, 012059 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 45 While it may seem the latter condition is redundant, it is needed if one defines the energy gap as the gap to local excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 46 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Wang, Topological quantum computation, 112 (American Mathematical Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Appendix A: Definition of foliated fracton phases Here we give a definition of foliated fracton phases, and in the process provide a framework for thinking about the relationship among fracton phases, universality and RG fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Here we focus on foliated fracton phases as introduced in earlier works;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' VIII, we briefly comment on a possible definition of generalized foliated fracton phases associated with the generalized foliated RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' An important point is that notions of standard and foliated phases both play important, but different, roles in fracton physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For ease of presentation, we do not consider symmetry in this discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' First we recall the definition of standard gapped phases and make some comments on the physical basis of this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Phases are equivalence classes of systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' by a system we mean a specification of the degrees of freedom on some d-dimensional spatial lattice, together with a local Hamiltonian H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In a slight abuse of notation we use H to denote the system and not just its Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Two systems H and H′ are in the same phase (considered equivalent) if there exist resource systems R and R′ so that there is a continuous path between the Hamiltonians for the systems H ⊗ R and H′ ⊗ R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Here “⊗” denotes the operation of stacking two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Each “trivial” resource system is a collection of gapped, decoupled zero- dimensional systems arranged in d-dimensional space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' R and R′ have product ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The energy gap is required to remain open along the continuous path, which must also avoid first-order phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='45 A special case of the above definition is that H and H′ are equivalent if there is a continuous path between their Hamiltonians (without stacking with resource systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Typically we are interested in the universal properties of phases, which we define simply as those properties that are the same everywhere within a phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In many cases, a standard phase contains within it a representative system that is a RG fixed point under some (conventional) scheme for carrying out the RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' When this occurs, the universal properties of a phase are encapsulated in the properties of the RG fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This holds because two (infinite) systems related by a RG step are in the same phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' for instance, this property is clear in “entanglement RG” schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Why do we consider stacking with trivial resource 19 systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This certainly leads to nice mathematical properties, and has the advantage of allowing for comparison between systems with different local degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' However, there is a more fundamental reason, namely to obtain a definition of phases that can be distinguished in experiments, at least in principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The key point is that lattice models are always idealizations of continuum systems, where some degrees of freedom are deemed unimportant and left out of the model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' atomic core levels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Any physically measurable notion of phases cannot depend on which degrees of freedom we choose to include or ignore in a theoretical model, and this issue is addressed by including stacking with trivial resource systems in the equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' More generally, we emphasize that the equivalence relation used to define standard phases is not arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Given the idea that gapped phases should be connected components of parameter space where the gap remains open, standard phases are the finest equivalence classes that can be distinguished (in principle) by experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Therefore, it is always physically relevant to consider standard phases, even when the universal properties of a standard phase are not captured in a RG fixed point, as occurs in fracton models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Put another way, we cannot achieve a complete understanding of fracton physics if we ignore standard phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' However, this does not preclude the relevance of other notions of phases to fracton physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Before defining foliated fracton phases, we first introduce the closely related notion of F-phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In order to talk about F-phases (and foliated fracton phases), we need to introduce a foliation of 3D space, which is a certain geometrical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In particular, a foliation consists of one or more decompositions of space into parallel 2D layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' If we have k separate decompositions, we sometimes speak more specifically about a k-foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' An important example of a 3-foliation is given by the sets of all xy, yz and xz planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Similar to standard phases, two systems H and H′ are considered to be in the same F-phase (or to be F- equivalent) if there is a continuous path between the Hamiltonians for H ⊗ R and H′ ⊗ R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The difference from standard phases is that the resource systems R and R′ are allowed to consist of decoupled gapped 2D systems on any layers of the foliation structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For instance, R can consist of decoupled 2D toric codes lying on a set of xy planes (as long as the foliation structure includes xy planes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Here, as in the foliated RG, we view 2D layers as a free resource, analogous to product states in the definition of standard phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The relation of F-equivalence is obviously coarser than standard phase equivalence, because the set of allowed resource systems contains those allowed for standard phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Moreover, it is strictly coarser;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' for instance, a stack of 2D topologically ordered layers is non-trivial as a standard phase, but is in the trivial F-phase (with appropriate foliation structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Therefore each F-phase can contain multiple standard phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The reason we define F-phases is to be able to define foliated fracton phases, which are those F-phases that contain a representative system that is a fixed point of the foliated RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Because two (infinite) systems related by a foliated RG step are in the same F-phase, it is expected that the fixed point of a foliated fracton phase captures certain universal properties that are the same throughout the foliated fracton phase, and are referred to as its foliated fracton order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' It is important to emphasize that the foliated fracton order consists of properties that are the same even within different standard phases, so long as these standard phases are F-equivalent and belong to a foliated fracton phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Foliated fracton phases are a useful concept in the study of fracton physics because, in some fracton systems, they restore a connection between universal properties of a phase and a RG fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This connection fails when we study fracton models using standard phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' It should be noted that not every F-phase is a foliated fracton phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For instance, the F-phase containing Haah’s cubic code model is not a foliated fracton phase for any choice of foliation structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' It is not clear that such F-phases are interesting objects of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Appendix B: Review of string-net models In Appendix B 1, we review the basics of the string- net models that are relevant for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We follow the original construction as introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For more comprehensive introductions, we refer the readers to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 21, 22, 24, and 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In Appendix B 2, we discuss the string-net models on the minimal lattice on the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' String-net models The input data of a string-net model is a unitary fusion category46, which includes an index set {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=', N} and the associated data set (δijk, ds, F ijm kℓn ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A string-net model is defined on a trivalent lattice, where the local DOF live on the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Each edge has a Hilbert space of spanC{|0⟩ , |1⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=', |N⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Usually, an edge of the string-net is represented by a directed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For a directed edge, i∗ represents the edge in the state i pointing in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' That is i∗ i = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B1) In particular, 0∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The δ-symbol specifies the vertex rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' δijk takes values in {0, 1} and it is symmetric under permutation of the indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' δijk determines the allowed states on edges 20 at a trivalent vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A vertex is stable25 if i j k (B2) satisfies δijk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A vertex is not stable if δijk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The d- and F-symbols define the graphical rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The d-symbols evaluate loops to real numbers as s = ds = ds∗ = s∗ , (B3) where d0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' They satisfy the equation didj = � k δijk∗dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B4) The F-symbols define the transformations i j m ℓ k = � n F ijm kℓn i ℓ n j k , (B5) where the F-symbols are nonzero if all the vertices satisfy the vertex rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' They are normalized as F ijk j∗i∗0 = � dk didj δijk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B6) For the cases of interest in this paper, the F-symbols satisfy the tetrahedral symmetry F ijm kℓn = F ℓkm∗ jin = F jim ℓkn∗ = F imj k∗nℓ � dmdn djdℓ , (B7) as well as the pentagon equation � n F ijm kℓn F psℓ∗ inq F pqn jkr∗ = F psℓ∗ m∗kr∗F rsm∗ ijq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B8) From the pentagon equation, we can derive the orthogo- nality relation of the F-symbols that � n F ijm′ kℓn � F ijm kℓn �∗ = δmm′, (B9) where the complex conjugation on the F-symbol is given by � F ijm kℓn �∗ = F i∗j∗m∗ k∗ℓ∗n∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B10) The ground state wave-function of the string-net model |Ψ⟩ is given by |Ψ⟩ = � |X⟩∈HSN Qv Ψ(X) |X⟩ , (B11) where Ψ(X) = ⟨X|Ψ⟩, and |X⟩ denotes a string-net configuration in the stable vertex subspace HSN Qv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' A vector |X⟩ is a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Note that the set of all different |X⟩’s form an orthonormal basis for this subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The graphical rules define a set of relations between the amplitudes Ψ � i i′ k j � = �� dkdj di �∗ δii′Ψ � i � , Ψ � i j ℓ k m � = � n � F ijm kℓn �∗ Ψ � i j ℓ k n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B12) Moreover, the graphical rules can be used to define transformations for a generic string-net configuration ket- vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (7) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (19) are examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The commuting projector Hamiltonian, which has the above wave-function as the ground state, is given by HSN = − � v Qv − � p Bp, (B13) where Qv is the vertex projector enforcing the vertex rules δijk, and Bp = � s(ds/D)Bs p with D = � s(ds)2 being the total quantum dimension is the plaquette projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Each Bs p adds a counter-clockwise directed loop of s in the interior of a plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Its action can be evaluated by the F-symbols as illustrated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B4) implies that the Bs p’s satisfy Bi pBj p = � k δijk∗Bk p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B14) The ground state satisfies Qv |Ψ⟩ = |Ψ⟩ , Bp |Ψ⟩ = |Ψ⟩ (B15) for all v and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The minimal lattice A string-net model can be defined on the minimal trivalent lattice on the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The minimal lattice consists of three edges, two vertices, and one plaquette as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The ground states first have to satisfy the vertex constraints Qv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Recall that a vertex is called stable if the vertex constraint is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We denote a basis vector of the stable vertex subspace on the minimal lattice by |ijk⟩ = ����� i i j j k � ∈ HSN Qv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B16) 21 i j k y x FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 23: The minimal trivalent lattice on the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The lattice has three edges colored by red, blue, and green;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' two vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' and one plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' i, j, and k are state labels on the colored edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' i i j j k Cx Cy y x FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 24: An illustration of the two non-contractible loops Cx and Cy on the minimal lattice, taken by the logical operators W Ci α where α is the excitation label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The logical operators act within HSN Qv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 24 are the two different paths taken by the logical operators {W Ci α } where i ∈ {x, y} and α ∈ {excitations}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The action of the logical operators on a basis vector |ijk⟩ can be computed by the method introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 21, which we will not discuss in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We will review a string-operator construction for the double-Ising in Appendix C 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Next, the ground states need to satisfy the one plaquette term on the minimal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Consider the action of Bs p on a basis vector |abc⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Instead of directly fusing the s-loop into the edges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' we can first fuse different parts of the s-loop together and map it into a trivalent diagram Bs p ����� a a b b c � = ����� a a b b c s � = ����� a a b b c s � = � h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='g F s∗s0 s∗sh∗F s∗s0 s∗sg ����� a a b b c s g h � = � h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='g F s∗s0 s∗sh∗F s∗s0 s∗sg ����� a a b b c s g g h h � = � h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='t2 F s∗s0 s∗sh∗F s∗s0 s∗sg F s∗hs gst1 F sg∗s∗ h∗s∗t∗ 2 ����� a a b b c s g g h h t2 t1 � = � h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='t1 ds (ds)2 � dhdg dt1 F s∗hs gst1 F sg∗s∗ h∗s∗t∗ 1 ����� a a b b c g g h h t1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B17) where in the second equality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' we have brought the s-loop over the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We can also bring the loop below the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The choice does not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The action of the Bs p term can then be determined by fusing the trivalent diagram (orange) into the underlying lattice (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This will help us to show that for abelian string-net models, the plaquette term on the minimal lattice is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For abelian string-net models, the fusion of s 22 and s∗ is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' So, the above equation reduces to Bs p ����� a a b b c � abelian = ds (ds)2 ����� a a b b c 0 0 0 0 0 � abelian = ds ����� a a b b c � abelian , (B18) where to get to the last line, we have used (ds)2 = 1 for abelian models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Hence, we see that Bp acts as identity on HSN Qv in abelian models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In other words, on the minimal lattice, HSN Qv is the ground space of the abelian models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Appendix C: Other details of the doubled-Ising In Appendix C 1, we discuss the ground states of the doubled-Ising string-net on the minimal lattice (introduced in Appendix B 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In Appendix C 2, we show that when we put the doubled-Ising state on a lattice with smooth boundary, ψ ¯ψ and σ¯σ are condensed on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Doubled-Ising on minimal lattice The doubled-Ising string-net on the minimal lattice has a 10-dimensional stable vertex subspace HD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Qv , spanned by |000⟩, |220⟩, |202⟩, |022⟩, |110⟩, |112⟩, |101⟩, |011⟩, |211⟩, and |121⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' One of the dimensions is not part of the ground space, because the projector Bp has the eigenvalue of 0 on this state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' To find the ground space, we calculate the action of B1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Following from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B17) (the actions of B0 p and B2 p are trivial), we find B1 p ����� a a b b c � = 1 √ 2 ����� a a b b c 0 0 0 0 0 � + 1 √ 2 ����� a a b b c 2 0 0 2 2 � + 1 √ 2 ����� a a b b c 2 2 2 0 0 � + 1 √ 2 ����� a a b b c 0 2 2 2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (C1) That is, B1 p |abc⟩ = 1 √ 2 � I + W Cy ψ + W Cx ψ − W Cx ψ W Cy ψ � |abc⟩ = 1 √ 2 � I + W Cy ¯ ψ + W Cx ¯ ψ − W Cx ¯ ψ W Cy ¯ ψ � |abc⟩ , (C2) where we have identified the logical operators by the construction discussed in Appendix C 2, and the second equality follows from that we can bring the s-loop below the lattice in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We can further compute the action of B1 p by fusing the orange trivalent diagram into the underlying lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' At this point, it is clear that a product state |abc⟩ is generally not an eigenstate of B1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' An explicit calculation shows that |ψ ¯ψ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩ = − 1 2 |000⟩ + 1 2 |202⟩ + 1 2 |022⟩ + 1 2 |220⟩ has the eigenvalue − √ 2 under B1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Hence, it is an excited state, which car- ries a ψ ¯ψ fluxon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The other 9 dimensions have the eigen- value √ 2 under B1 p and are, thus, ground states on the minimal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' An orthonormal basis for the nine-dimensional ground space can be chosen to be the common eigenstates of logical operators W Cx ψ , W Cy ψ , W Cx ¯ ψ , and W Cy ¯ ψ , which all commute with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The nine common eigenstates are |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩1 = 1 2 |000⟩ + 1 2 |202⟩ + 1 2 |022⟩ − 1 2 |220⟩ , |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩2 = 1 √ 2 |011⟩ + i √ 2 |211⟩ , |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩3 = 1 √ 2 |101⟩ − i √ 2 |121⟩ , |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩4 = 1 √ 2 |011⟩ − i √ 2 |211⟩ , |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩5 = 1 2 |000⟩ − 1 2 |202⟩ + 1 2 |022⟩ + 1 2 |220⟩ , |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩6 = e− iπ 8 √ 2 |110⟩ + ie− iπ 8 √ 2 |112⟩ , |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩7 = 1 √ 2 |101⟩ + i √ 2 |121⟩ , |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩8 = e iπ 8 √ 2 |110⟩ − ie iπ 8 √ 2 |112⟩ , |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩9 = 1 2 |000⟩ + 1 2 |202⟩ − 1 2 |022⟩ + 1 2 |220⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The 10th dimension |ψ ¯ψ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' min⟩ is also a common eigenstate of W Cx ψ , W Cy ψ , W Cx ¯ ψ , and W Cy ¯ ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In this basis, the logical 23 operators takes the diagonal form W Cx ψ = � �� � � � 1 0 0 0 1 0 0 0 −1 � � � ⊗ I3×3 � �� ⊕ (−1), (C3) W Cy ψ = � �� � � � 1 0 0 0 −1 0 0 0 1 � � � ⊗ I3×3 � �� ⊕ (−1), (C4) W Cx ¯ ψ = � ��I3×3 ⊗ � � � 1 0 0 0 1 0 0 0 −1 � � � � �� ⊕ (−1), (C5) W Cy ¯ ψ = � ��I3×3 ⊗ � � � 1 0 0 0 −1 0 0 0 1 � � � � �� ⊕ (−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (C6) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Condensation on smooth boundary FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 25: An illustration of an open ended fluxon string-operator W path fluxon acting on a wave-function of the doubled-Ising string-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' W path fluxon creates the fluxon and its antiparticle on the plaquettes p1 and p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' It does not create any excitations along the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' W path fluxon is constructed by a loop of s-string that vertically penetrates the square-octagon lattice at the two plaquettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The string-operators of ψ ¯ψ and σ¯σ are given by a loop of 2-string and a loop of 1-string respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Using the string-operator construction discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 22 and 23, we can readily show that on the smooth boundary, the bosonic fluxons ψ ¯ψ and σ¯σ condense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Let us start with a review of the fluxon string- operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Consider a large square-octagon lattice placed on the xy-plane as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' An open ended fluxon string-operator is given by a loop of s-string which vertically penetrates the lattice through the center of the plaquettes p1 and p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The fluxon excitations are created at p1 and p2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The ψ ¯ψ string-operator is given by a loop of 2-string, and that of σ¯σ is given by a loop of 1-string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' An open ended string-operator of a chargeon is constructed by a line segement, which can be either above or below the lattice, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 26: The open ended chargeon string-operators of the doubled-Ising string-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' They are constructed by line segements of the strings, which can be either above or below the lattice depending on the type of the chargeon excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' On a ground state of the doubled-Ising, An open chargeon string-operator creates a vertex violation at each end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The ψ and σ string-operators correspond to a 2-string and a 1-string above the lattice respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The string-operators for ¯ψ and ¯σ correspond to those below the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' To compute the action of the string-operators, we need the R-symbols and the S-matrix26,27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The R-symbols define the braiding transformations b a c = Rba c b a c , (C7) where Rba c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The inverse transformations are defined by b a c = � Rba c �∗ b a c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (C8) Same as the F-symbols, Rba c ̸= 0 if δbac ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Elements of the S-matrix are given by Sab = 1 √ D � c dcRba c Rab c , (C9) where √ D = �� s(ds)2 = 2 is the total quantum dimension of the Ising unitary modular tensor category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The full S-matrix is S = 1 2 � � � 1 √ 2 1 √ 2 0 − √ 2 1 − √ 2 1 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (C10) Using the R- and F-symbols, we can always fuse the chargeon string-operators into the lattice at the cost of Z do0l-s p1 p2Z S-string X24 violating the vertex constraints at the ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' On the other hand, the fluxon string-operators can be fused into the lattice without introducing any vertex violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For example, consider the action of a non-trivial fluxon string-operator on the string-net wave-function ����� ℓ1 ℓ4 e ℓ2 ℓ3 s � = � j F ss0 eej ����� ℓ1 ℓ4 e e ℓ2 ℓ3 s j � = � j F ss0 eej � Res j �∗ � Rse j �∗ ����� ℓ1 ℓ4 e e ℓ2 ℓ3 s j � = � j dj de � Res j �∗ � Rse j �∗ ����� ℓ1 ℓ4 e ℓ2 ℓ3 � = √ D de Sse ����� ℓ1 ℓ4 e ℓ2 ℓ3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (C11) Note that the expression above differs from that of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 22 by a normalization factor, which is not important for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' For an example of a chargeon string-operator, let us consider a line segment below the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We compute ����� ℓ1 ℓ4 e ℓ2 ℓ3 s � = � j F ss0 eej ����� ℓ1 ℓ4 e e ℓ2 ℓ3 s s j � = � j F ss0 eej � Res j �∗ ����� ℓ1 ℓ4 e e ℓ2 ℓ3 s s j � = � j F ss0 eej � Res j �∗ ����� ℓ1 ℓ4 e e ℓ2 ℓ3 j � , (C12) where in the last step we have removed the s-strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' So, we see that an open chargeon string-operator always creates violations to the vertex constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Moreover, different fluxons correspond to different sets of eigenvalues of Ba p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' To determine these eigenvalues, we need the graphical rule a b = Sab S0b b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (C13) As an example, let us compute Ba pW path ψ ¯ ψ |ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='⟩ = Ba p ����� 2 � ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' = ����� 2 a � ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' = Sa2 S02 ����� 2 � ΨD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' , (C14) from which we see that the ψ ¯ψ fluxon has the eigenvalues of 1, − √ 2, and 1 for B0 p, B1 p, and B2 p respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' This result is exactly what we found for the ψ ¯ψ fluxon state in Appendix C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' It is easy to see that the σ¯σ fluxon has B0 p = 1, B1 p = 0, and B2 p = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We now show that, on the smooth boundary, the fluxons ψ ¯ψ and σ¯σ are condensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Without loss of generality, let us consider the doubled-Ising string-net with a single plaquette and everywhere else is set to |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Consider the action of an open ended fluxon string- operator passing through the lattice just outside the plaquette s Since the s-loop does not pass through the region enclosed by the plaquette, the string-operator does not create any fluxon excitation on the plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Via the F- and R-symbols, we can fuse the s-loop into the edges of the plaquette without changing any of the edges outside the plaquette or introducing any vertex violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Therefore, we see that the fluxons all condense on the smooth boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' On the other hand, because the chargeon string-operators necessarily introduce vertex violations, the chargeons remain as excitations on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Thus, we reach the conclusion that, on the smooth boundary, the condensed excitations are the fluxons ψ ¯ψ and σ¯σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Appendix D: Controlled gate details In this appendix, we present the details of the graphical definition of Gs p, its inverse, the commutation relations, 25 and the proof for the central equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Graphical definition We perform the graphical calculation that leads to the last line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We compute Gs p ����� ¯¯ℓ3 ¯ℓ3 ℓ4 ¯ℓ4 ¯ℓ1 ℓ1 ¯¯ℓ2 ¯ℓ2 ℓ∗ 1 ℓ2 ℓ3 ℓ4 a b c s′ � = δss′ ����� ¯¯ℓ3 ¯ℓ3 ℓ4 ¯ℓ4 ¯ℓ1 ℓ1 ¯¯ℓ2 ¯ℓ2 s ℓ∗ 1 ℓ2 ℓ3 ℓ4 a b c 0 � = δss′ � α F ℓ1ℓ∗ 10 ss∗α ����� ¯¯ℓ3 ¯ℓ3 ℓ4 ¯ℓ4 s α ℓ∗ 1 ¯ℓ1 ℓ1 ¯¯ℓ2 ¯ℓ2 ℓ2 ℓ3 ℓ4 a b c � =δss′ � α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='β F ℓ1ℓ∗ 10 ss∗α F ¯ℓ1aℓ1 sα∗β ����� ¯¯ℓ3 ¯ℓ3 ℓ4 ¯ℓ4 α ¯ℓ1 ℓ1 β a ¯¯ℓ2 ¯ℓ2 s ℓ2 ℓ3 ℓ4 b c � = δss′ � α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='γ F ℓ1ℓ∗ 10 ss∗α F ¯ℓ1aℓ1 sα∗β F ¯ℓ2ℓ2a∗ sβ∗γ ����� ¯¯ℓ3 ¯ℓ3 ℓ4 ¯ℓ4 α ¯ℓ1 ℓ1 γℓ2 ¯¯ℓ2 ¯ℓ2 s ℓ3 ℓ4 β b c � = · · · (repeat similar steps across the subsequent vertices around the plaquette) =δss′ � α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='τ F ℓ1ℓ∗ 10 ss∗α F ¯ℓ1aℓ1 sα∗β F ¯ℓ2ℓ2a∗ sβ∗γ F ¯¯ℓ2bℓ∗ 2 sγ∗δ F ¯ℓ3ℓ3b∗ sδ∗ε F ¯¯ℓ3cℓ∗ 3 sε∗η F ¯ℓ4ℓ4c∗ sη∗τ ����� ¯¯ℓ3 ¯ℓ3 ℓ4 ¯ℓ4 ¯ℓ1 ℓ1 ¯¯ℓ2 ¯ℓ2 s α γ ε τ β δ η � =δss′ � α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='τ F ℓ1ℓ∗ 10 ss∗α F ¯ℓ1aℓ1 sα∗β F ¯ℓ2ℓ2a∗ sβ∗γ F ¯¯ℓ2bℓ∗ 2 sγ∗δ F ¯ℓ3ℓ3b∗ sδ∗ε F ¯¯ℓ3cℓ∗ 3 sε∗η F ¯ℓ4ℓ4c∗ sη∗τ ����� ¯¯ℓ3 ¯ℓ3 ℓ4 ¯ℓ4 ¯ℓ1 ℓ1 ¯¯ℓ2 ¯ℓ2 α γ ε τ β δ η s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (D1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Isometric property We now show that Gs p is an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' That is, Gs p †Gs p equals identity on the input space, which we called VSN p,s in section VI A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Graphically, Gs p † removes a string of s from the edges of the plaquette when the controlled edge is in the state 26 |s⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Let us denote the matrix elements of Gs p by [Gs p](s,α,β,γ,δ,ε,η,τ) (s′,ℓ∗ 1,a,ℓ2,b,ℓ3,c,ℓ4)(ℓ1, ¯ℓ1, ¯ℓ2, ¯¯ℓ2, ¯ℓ3, ¯¯ℓ3, ¯ℓ4, ℓ4) = δss′F ℓ1ℓ∗ 10 ss∗α F ¯ℓ1aℓ1 sα∗β F ¯ℓ2ℓ2a∗ sβ∗γ F ¯¯ℓ2bℓ∗ 2 sγ∗δ F ¯ℓ3ℓ3b∗ sδ∗ε F ¯¯ℓ3cℓ∗ 3 sε∗η F ¯ℓ4ℓ4c∗ sη∗τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (D2) Then, Gs p † has an algebraic expression of Gs p † ����� ¯¯ℓ3 ¯ℓ3 ℓ4 ¯ℓ4 ¯ℓ1 ℓ1 ¯¯ℓ2 ¯ℓ2 α′ γ′ ε′ τ ′ β′ δ′ η′ s′ � = � t1,t2,t3, t4,t5 � [Gs p](s′,α′,β′,γ′,δ′,ε′,η′,τ ′) (s,ℓ∗ 1,t5,t4,t3,t2,t1,ℓ4) (ℓ1, ¯ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=', ℓ4) �∗ × ����� ¯¯ℓ3 ¯ℓ3 ℓ4 ¯ℓ4 ¯ℓ1 ℓ1 ¯¯ℓ2 ¯ℓ2 ℓ∗ 1 t4 t2 ℓ4 t5 t3 t1 s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (D3) Using the orthogonality relation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B9), we find Gs p †Gs p ����� ¯¯ℓ3 ¯ℓ3 ℓ4 ¯ℓ4 ¯ℓ1 ℓ1 ¯¯ℓ2 ¯ℓ2 ℓ∗ 1 ℓ2 ℓ3 ℓ4 a b c s′ � = δss′ � t1,t2,t3, t4,t5 δt1cδt2ℓ3δt3bδt4ℓ2δt5a ����� ¯¯ℓ3 ¯ℓ3 ℓ4 ¯ℓ4 ¯ℓ1 ℓ1 ¯¯ℓ2 ¯ℓ2 ℓ∗ 1 t4 t2 ℓ4 t5 t3 t1 s � , (D4) thereby establishing Gs p as an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Hence, Gp = � s Gs p is also an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Commutation relations The commutation relations of the Gs p operators immediately follow from the graphical definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Any two Gs p and Gs′ p′ commute, provided that they do not act on each other’s controlled edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' When p and p′ are the same plaquette, Gs p and Gs′ p commute trivially, because they act on orthogonal spaces with the control edge in |s⟩ and |s′⟩ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' If the plaquettes p and p′ are not next to each other, Gs p and Gs′ p′ obviously commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' When p and p′ are adjacent, the proof of commutation amounts to showing the order, in which the string s and s′ are fused into the bordering edges, does not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Consider Gs p and Gs′ p′ acting on two adjacent plaquettes p and p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We focus on the bordering edges of these two plaquettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We will show that the F-symbols associated with the two diagrams (the thickened red arrows indicate the direction of motion of the s- and s′-strings), s′ s 1st 2nd p′ p and s′ s 2nd 1st p′ p are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The left diagram corresponds to computing Gs pGs′ p′ on a reference ket-vector, and the right diagram corresponds to Gs′ p′Gs p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' In the case where s′ moves first, we find ����� s′ s ε′ d α b c a e � = � ρ F b∗ca s′ε′∗ρ ����� s′ s ε′ d α b c ρ e � = · · · = � q �� ρ F b∗ca s′ε′∗ρF e∗dc∗ s′ρ∗τ ′F τ ′ρ∗e∗ sα∗q∗ F ε′∗b∗ρ sqγ � ����� s′ s ε′ d α b q∗ γ τ ′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (D5) In the case where s moves first, we have ����� s′ s ε′ d α b c a e � = � ρ F dc∗e∗ sα∗ρ ����� s′ s ε′ d α b ρ c a � = · · · = � q �� ρ F dc∗e∗ sα∗ρ F ab∗c sρ∗γF γρ∗a s′ε′∗qF α∗dρ s′q∗τ ′ � ����� s′ s ε′ d α b q γ τ ′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (D6) For each fixed q, we want to show that the coefficients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' free sums over ρ, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (D5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (D6) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' To do this, we consider an alternative way of moving the strings s and s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' We will show that this alternative expression can be simplified, via the pentagon equation, 27 to produce either Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (D5) or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (D6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The alternative expression is obtained by moving both s and s′ to the central edge, ����� s′ s ε′ d α b c a e � = � η′,β F b∗ca s′ε′∗η′F dc∗e∗ sα∗β ����� s′ s ε′ d α b c η′ β � = � η′,β,q F b∗ca s′ε′∗η′F dc∗e∗ sα∗β F sβ∗c∗ s′η′∗q ����� s′ s ε′ d α b q η′ β � = · · · = � q Ccoef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (q) ����� s′ s ε′ d α b q γ τ ′ � , where, for each q, the coefficient of the alternative expression is Ccoef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (q) = � η′,β F b∗ca s′ε′∗η′F dc∗e∗ sα∗β F sβ∗c∗ s′η′∗q F α∗dβ s′q∗τ ′F ε′∗b∗η′ sqγ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (D7) Manipulating the F-symbols via the tetrahedral symme- try Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B7) and performing the above sum over either η′ or β via the pentagon equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B8), we obtain � � � � � � � � ρ F b∗ca s′ε′∗ρF e∗dc∗ s′ρ∗τ ′F τ ′ρ∗e∗ sα∗q∗ F ε′∗b∗ρ sqγ if sum over β, � ρ F dc∗e∗ sα∗ρ F ab∗c sρ∗γF γρ∗a s′ε′∗qF α∗dρ s′q∗τ ′ if sum over η′, (D8) which are exactly the coefficients in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (D5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (D6) for fixed q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Hence, the order, in which the strings s and s′ are fused into the bordering edges does not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' That is, [Gs p, Gs′ p′] = 0, as long as they do not act on each other’s controlled edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The proofs for the remaining commutation relations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (22) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (23) are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The central equation Let us prove the central equation on a triangular plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The proof on any polygon-shaped plaquette is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' With the graphical definitions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' we find Gγ p (|γ⟩ ⟨c|)ct Gc p † ����� ℓ2 ℓ3 ℓ1 c′ a b � =Gγ p (|γ⟩ ⟨c|)ct δcc′ � t1 F ℓ2a∗b cℓ3t∗ 1 ����� ℓ2 ℓ3 ℓ1 a t1 ℓ3 c � =Gγ p (|γ⟩ ⟨c|)ct δcc′ � t1 F ℓ2a∗b cℓ3t∗ 1 F ℓ1c∗a ct10 δt1ℓ∗ 1 ����� ℓ2 ℓ3 ℓ1 t1 ℓ3 c � =δcc′ � F ℓ2a∗b cℓ3ℓ1 F ℓ1c∗a cℓ∗ 10 � ����� ℓ2 ℓ3 ℓ1 ℓ∗ 1 ℓ3 γ � =δcc′ � F ℓ2a∗b cℓ3ℓ1 F ℓ1c∗a cℓ∗ 10 � � α F ℓ∗ 1ℓ10 γ∗γα∗ ����� ℓ2 ℓ3 ℓ1 α ℓ∗ 1 ℓ3 γ � =δcc′ � α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='β F ℓ2a∗b cℓ3ℓ1 F ℓ1c∗a cℓ∗ 10 F ℓ∗ 1ℓ10 γ∗γα∗F ℓ2ℓ3ℓ1 γα∗β ����� ℓ2 ℓ3 ℓ1 γ α β � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (D9) Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' it remains to show that the coefficient for every basis ket-vector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' fixing α and β) is the same as that of � k P γ ct � dk dγdc Bk p � P c ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' To do this, first, let us write the pentagon equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B8) in a different form via the tetrahedral symmetry Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B7) F ijp kq∗rF iq∗r ℓ∗sm = � n F rj∗k∗ nℓs∗ F ipj nsmF q∗kp∗ nm∗ℓ∗ dndr � dkdsdjdℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (D10) Focusing on the F-symbols of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (D9), we find F ℓ2a∗b cℓ3ℓ1 F ℓ1c∗a cℓ∗ 10 F ℓ∗ 1ℓ10 γ∗γα∗F ℓ2ℓ3ℓ1 γα∗β = � dα dℓ1dγ � da dℓ1dc F ℓ2a∗b cℓ3ℓ1 F ℓ2ℓ3ℓ1 γα∗β = � k dk dγdc F ℓ1ac∗ kγ∗α F ℓ2ba∗ kα∗β F ℓ3cb∗ kβ∗γ , (D11) The first equality follows from the normalization of F- symbols in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (B6), and we have used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' (D10) in 28 the second equality to get the last line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' The last line is exactly the coefficient of � k P γ ct � dk dγdc Bk p � P c ct on the same ket-vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAyT4oBgHgl3EQfTPfP/content/2301.00103v1.pdf'} +page_content=' Hence, we have proven the central equation on the triangular plaquette.' metadata={'source': 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b/JdE3T4oBgHgl3EQfvAtY/content/tmp_files/2301.04689v1.pdf.txt @@ -0,0 +1,3272 @@ +arXiv:2301.04689v1 [math.PR] 11 Jan 2023 +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +GUILLAUME BARRAQUAND, ORIANE BLONDEL, AND MARIELLE SIMON +Abstract. We consider the facilitated exclusion process, an interacting particle system on the +integer line where particles hop to one of their left or right neighbouring site only when the other +neighbouring site is occupied by a particle. A peculiarity of this system is that, starting from +the step initial condition, the density profile develops a downward jump discontinuity around +the position of the first particle, unlike other exclusion processes such as the asymmetric simple +exclusion process (ASEP). In the weakly asymmetric regime, we show that the field of particle +positions around the jump discontinuity converges to the solution of the multiplicative noise +stochastic heat equation (i.e. the exponential of a solution to the KPZ equation) on a half-line +subject to Dirichlet boundary condition, with initial condition given by the derivative of a Dirac +delta function. We prove this result by reformulating the problem in terms of ASEP on a half- +line with a boundary reservoir, for which we extend known proofs of convergence to deal with +Dirichlet boundary condition and the very singular type of initial condition that arises in our +case. +1. Introduction +1.1. Facilitated exclusion process. The facilitated exclusion process (FEP) was introduced +in the physics literature [RPSV00] as a representative of a universality class for absorbing phase +transitions. +It is an interacting particle system on a lattice in which particles can jump to +empty neighbors provided there is a particle in their neighborhood. So far our understanding +of this model is restricted to dimension 1, where it has been studied under different lights. Its +main feature is the absorbing transition mentioned above, at the critical particle density 1/2: +for particle densities below 1/2 (subcritical regime), the system fixates on a configuration with +isolated particles which cannot move, while for densities above 1/2 (supercritical regime) it +remains active forever and holes become eventually isolated. +The totally asymmetric version of this process (where particles only jump to the right) has +been studied in [BM09] (approach to the phase transition) and [CZ18,GLS19,GLS21] (identifica- +tion of the stationary states). Starting from a step initial condition, contrary to the well-studied +totally asymmetric simple exclusion process (TASEP), a downstep leads to a rarefaction fan +with a discontinuity [GKR10]. For the same initial condition, at large time t, particle posi- +tions fluctuate on the t1/3 scale with Tracy-Widom GUE statistics, while the fluctuations of the +rightmost particle, i.e. at the discontinuity, have Tracy-Widom GSE statistics [BBCS18a]. The +symmetric FEP has been studied as well, on the periodic lattice. It was found [BESS20,BES21] +that in the diffusive space-time scaling, and under the hydrodynamic limit, the macroscopic +density ρ evolves according to a Stefan problem written as ∂tρ = ∂uu +� 2ρ−1 +ρ +1ρ>1/2 +�, with the +space variable u belonging to the one-dimensional torus of size 1. In other words, starting the +microscopic dynamics from a density profile with both supercritical and subcritical regions, the +diffusive supercritical phase progressively invades the subcritical phase via moving interfaces, +until one or the other phase disappears. For the partially asymmetric version, where particles +jump to the right at rate p < 1 and to the left at rate q < p, invariant measures have been +characterized on the torus [GKR10] and on the line [AGLS22]. Recently, [ESZ22] showed that +the hydrodynamic limit (in the hyperbolic scaling) is given by the unique entropy solution of +∂tρ + (2p − 1)∂x +� (1−ρ)(2ρ−1) +ρ +1ρ>1/2 +� +, with x ∈ R. +The next natural question concerns the fluctuations of the FEP in the asymmetric case +(FASEP). As it has been noted for the totally asymmetric case in [BBCS18a], the problem +can be reformulated in terms of the asymmetric simple exclusion process (ASEP) on the half- +line with jump rates p > q, and with a specific boundary condition, where particles enter the +1 + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +2 +system at a rate p and can never exit (see Figure 1). For fixed p and q, we expect that, up to +scaling constants, the fluctuations should be similar as for the totally asymmetric FEP1. +In the present paper, we study the fluctuations of particle positions in the FASEP in a weakly +asymmetric asymptotic regime. In the bulk (that is, far from the jump discontinuity of the +density profile), we do not expect that the facilitation rule will have any effect on the scaling +limit of fluctuations. In particular, we expect that if the asymmetry is properly scaled with +time, the field of particle positions, appropriately rescaled, should converge to the solution of +the Kardar-Parisi-Zhang (KPZ) equation on the real line with narrow wedge initial condition. +This is consistent with the Tracy-Widom GUE asymptotics observed in the totally asymmetric +setting. However, for the step initial condition, if we consider the field of particle positions in +the FASEP around the first particle, the situation is more interesting and the facilitation rule +plays a role. The limit should be described by a stochastic PDE on a semi-infinite interval with +a specific boundary condition. The main goal of the present paper is to describe this stochastic +PDE. +1.2. KPZ equation and Hopf-Cole transform. In order to state our main result, let us first +recall how to solve the KPZ equation on the full one-dimensional line, which reads as +∂th = 1 +2∂uuh + 1 +2(∂uh)2 + ξ, +(1) +with ξ the standard space-time white noise on R+ × R. One usually considers the Hopf-Cole +transform of a putative solution h, namely Z(t, u) = eh(t,u). If we apply the chain rule in (1), +ignoring all issues of regularity, the function Z solves the Stochastic Heat Equation (SHE) with +multiplicative noise +∂tZ = 1 +2∂uuZ + Zξ. +(2) +The latter equation can be solved through standard SPDE techniques, and whenever it can be +shown that Z > 0 [Mue91], this procedure yields a Hopf-Cole solution to (1). In their seminal +paper [BG97], Bertini and Giacomin noticed that the discrete Hopf-Cole transform (introduced +by Gärtner [Ga87]) +Zt(x) := e−λht(x)+νt +(3) +of ASEP height function ht(x) (x ∈ Z), satisfies, for well-chosen parameters λ, ν, a martingale +problem that is a discrete analogue of the martingale problem satisfied by Z. In the weakly +asymmetric regime, say p = 1 +2eε, q = 1 +2e−ε with 0 < ε ≪ 1, as assumed in this paper, it can +then be showed that solutions of the discrete martingale problem converge to solutions of the +continuous one. Given uniqueness of the solution to the continuous martingale problem, this is +enough to conclude that ASEP height function, suitably rescaled, converges to a solution of the +KPZ equation. Further, [DT16] identifies a whole class of models to which this method may +apply, in the sense that a generalization of the discrete Hopf-Cole transform can be found and +convergence to the SHE can be proved. +Other approaches to solving (1) that do not require a detour through the SHE were also looked +for in the last decades, and can be useful in cases where there is no applicable discrete Hopf- +Cole transform (unlike the present paper). Let us mention regularity structures [Hai13,Hai14]; +energy solutions [GJ14,GP18], which have been applied e.g. in [BGS16,GJS17,GPS20,Yan18]; +and paracontrolled distributions [GIP15,GP17]. +1.3. KPZ equation on the positive half-line. A half-space analogue of the result from +[BG97] was proved in [CS18]. More precisely, under some condition on injection and ejection +rates at the origin of ASEP on the half-line, and assuming that the effective density at the origin +imposed by those rates scales as ρ ≈ 1 +2(1 + (A + 1 +2)ε), [CS18] shows that the height function +of ASEP on the half-line converges to the KPZ equation on R+ with Neumann type boundary +1Proving this would however require an analogous statement for ASEP on a half-line, and this has so far remained +out of reach. Indeed, some exact formulas characterizing the distribution of particles have been proved in [TW13b, +TW13a], and more recently in [BC22a], but these formulas are not amenable for asymptotic analysis. There exists +one exception where the fluctuations of half-line ASEP have been analyzed asymptotically, in [BBCW18], but +the boundary condition considered in that paper is different from the one which is relevant for the study of the +FASEP, and the methods used there crucially depend on this specific boundary condition. + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +3 +condition ∂xh(t, 0) = A. As in [BG97], this result is proved via the discrete Hopf-Cole transform +(3) of ASEP height function ht(x) (x ∈ N), which is shown to converge to the SHE on R+ with +Robin type boundary condition +� +∂tZ = 1 +2∂uuZ + Zξ, +∂uZ(t, 0) = AZ(t, 0). +(4) +Since Z(t, ·) is not differentiable, the boundary condition should rather be imposed on the half- +space heat kernel which is used to define the solution, we refer to [CS18] for details. The result +of [CS18] was restricted to A ⩾ 0 and near equilibrium initial conditions (see Definition 3.8 +below). It was then extended to all A ∈ R and the empty initial condition in [Par19]. Let +us note that on the full-line, the extension of the convergence result of [BG97] to step initial +condition was first discussed in [ACQ11], with considerably less details than in [Par19]. Some of +these results were further extended in [Yan22] to generalizations of ASEP, in the spirit of [DT16]. +An alternative way to make sense of the KPZ equation on a half-line via regularity structures +was also considered in [GH19]. +In the physics literature, the solution Z to the SHE is understood as the partition function for +a continuous Brownian directed polymer in a white noise potential ξ. The boundary parameter +A can then be understood as controlling an extra energy collected by polymer paths, given by +reflected Brownian motions, along the boundary [BBC16]. Alternatively, we may assume that +there is no extra potential on the boundary, but the Brownian paths in the polymer partition +function have elastic reflection on the boundary controlled by the parameter A. Discrete directed +polymers are another family of models, with exclusion processes, which converge to the KPZ +equation in full-space [AKQ14] or half-space [Wu20,Par22,BC22b]. +Remark 1.1. The KPZ equation (and the associated SHE via Hopf-Cole transform) has also +been considered on an interval with Neumann type boundary conditions. The discrete Hopf-Cole +transform of open ASEP height function satisfies Robin type boundary conditions [GLM17] and +converges to the KPZ equation on an interval [CS18,Par19,GPS20,Yan22]. +1.4. Main results. As previously mentioned, the problem of fluctuations of the FASEP reduces +to the fluctuations of ASEP on the half-line with, at the origin, injections of particles at the +right-jump rate p and no ejection of particles. +This translates into a microscopic boundary +condition for the discrete Hopf-Cole transform (3) given by +Zt(−1) = µZt(0), +with µ ≈ 1 − ε, +as stated more precisely in (11) below. In contrast, in the setting of [CS18,Par19], the boundary +parameter µ is scaled as µ ≈ 1 − ε2A. We will prove below that in our case, the appropriate +scaling of the solution is different from [CS18, Par19]: we will set Zε +t (u) = ε−2Zε−4t(ε−2u) for +any macroscopic point u, and prove that starting from the step initial condition for the FASEP +(i.e. empty initial condition for the half-line ASEP), Zε +t (u) weakly converges as a continuous +process (see Theorem 3.7 for a precise statement) to the solution of the SHE + + + + + + + +∂tZ = 1 +2∂uuZ + Zξ, +Z(t, 0) = 0, +Z(0, u) = −2δ′ +0(u). +(5) +We also prove a similar statement for near-equilibrium initial conditions (Theorem 3.9), under +the same scaling as in [CS18]. The SHE with Dirichlet boundary condition (Z(t, 0) = 0) was +already considered in [Par22] in the context of directed polymer models, but only for another +type of initial condition. Here the initial condition that we consider is very singular, this is the +derivative of a delta function at 0. We provide a more precise definition of this stochastic PDE +in Definition 3.3 and prove existence and uniqueness of the solution in Proposition 3.5 below. +Dealing with this very singular initial condition is one of the main novelties of this paper. +Eventually, our main result is therefore the following: the macroscopic fluctuations of the field +of the first particles’ positions in the weakly asymmetric FASEP are given (via the Hopf-Cole +transform) by the solution to the KPZ equation on the half-line R+, with initial condition being +the derivative of a Dirac distribution, and with Dirichlet boundary condition at the origin. + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +4 +1.5. Comparison with previous literature on half-space KPZ equation. In a sense, the +microscopic boundary condition Zt(−1) = µZt(0) with µ ≈ 1 − ε that we are considering is the +A → ∞ limit of the setting in [CS18, Par19] which considered µ ≈ 1 − Aε2, and it poses no +additional difficulty in terms of handling the boundary condition. However, as already said, our +initial condition is much more singular than the ones in [CS18,Par19,Par22]. Proving tightness +requires us to develop precise estimates about the Dirichlet heat kernel and its discrete analogue. +Further, the rigorous identification of the initial condition requires some control on the second +moment of Zε +t (u), and the estimate that we need turned out to be surprisingly difficult to prove. +Eventually, we use an explicit second moment integral formula for half-line ASEP from the recent +preprint [BC22a], coming from a Markov duality (such use of Markov duality also arises in the +proof of convergence of the stochastic six-vertex model height function to the KPZ equation +in [CGST20]). +The continuous directed polymer model corresponding to the solution of (5) was considered +in the physics paper [GLD12]. The polymer paths are conditioned not to hit the boundary, +and [GLD12] studied the distribution of the partition function of polymers starting and ending +at a location η, after letting η → 0 and appropriately rescaling the partition function by η2. +Restricting on test functions f : R+ → R such that f(0) = 0, the distribution 1 +ηδ0(· − η), where +δ0 is the delta Dirac distribution, converges to −δ′ +0. This explains why the initial condition +that we consider in the present paper is the physically natural one to consider for the SHE with +Dirichlet boundary condition, though the solution was never mathematically constructed before. +Finally, let us mention that the KPZ equation on a half-line with Dirichlet boundary condition +is also considered in [GH19], but there the Dirichlet boundary condition h(t, 0) = 0 is imposed +on the KPZ equation itself and not on the SHE, so that this corresponds to a completely different +stochastic PDE than the one we consider in the present paper. +1.6. Outline of the paper. In Section 2, we define the FASEP and ASEP and construct a +mapping that connects the two. Section 3 is devoted to the presentation of the Hopf-Cole trans- +form, in terms of which we state our main results on the fluctuations of the particle positions in a +weakly asymmetric regime. We then provide several preliminary results: first, the existence and +uniqueness of the macroscopic SHE with δ′ +0 initial condition and Dirichlet boundary condition +(Section 4), and second, some explicit estimates on the discrete heat kernel with diverging Robin +boundary condition (Section 5). Finally, the main convergence results cover two types of initial +conditions which require different scalings: the step (resp. derivative of delta) initial condition +(Theorem 3.7) and near-equilibrium initial conditions (Theorem 3.9), which both require new +arguments. We actually need the understanding of the latter as an intermediate step towards +Theorem 3.7, so we start by proving Theorem 3.9 in Section 6. Our main result, Theorem 3.7 +is finally proved in Section 7. +Acknowledgments. This project is partially supported by the ANR grant MICMOV (ANR- +19-CE40-0012) of the French National Research Agency (ANR), and by the European Union +with the program FEDER “Fonds européen de développement régional” with the Région Hauts- +de-France. It has also received funding from the European Research Council (ERC) under the +European Union’s Horizon 2020 research and innovative program (grant agreement n° 715734), +and from Labex CEMPI (ANR-11-LABX-0007-01). +This article is also partially based upon +work supported by the National Science Foundation under Grant No. DMS-1928930 while G.B. +participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, +California, during the Fall 2021 semester. +2. Microscopic models and mapping +In the following, N denotes the set of non-negative integers, N∗ the set of positive integers. +The facilitated asymmetric exclusion process in dimension one (FASEP) is a Markov process +on the state space Ω := {0, 1}Z which is denoted by {ηt(x) ; x ∈ Z}t⩾0. +Each component +ηt(x) ∈ {0, 1} is the occupation variable of the configuration of particles at site x ∈ Z. + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +5 +Let p, q ∈ (0, 1) be two asymmetry parameters. The time evolution of the particle configura- +tions is ruled by the Markov generator LF which acts on functions f : Ω → R as follows: +LF f(η) = +� +x∈Z +pη(x − 1)η(x)(1 − η(x + 1)) +�f(ηx,x+1) − f(η) +� ++ +� +x∈Z +qη(x + 1)η(x)(1 − η(x − 1)) +�f(ηx,x−1) − f(η) +� +(6) +where ηx,y is the configuration obtained from η after the exchange of the occupation variables +η(x) ↔ η(y), namely: ηx,y(z) := η(x)1z=y +η(y)1z=x +η(z)1z /∈{x,y}. In other words, as it can be +read on the generator, particles are displayed on the lattice Z and jump to their neighbouring +sites at rates which encode the following rules: +• a jump to the right from site x to site x + 1 occurs with rate p if and only if site x is +occupied by a particle, site x + 1 is empty and site x − 1 is occupied ; +• a jump to the left from site x to site x − 1 occurs with rate q if and only if site x is +occupied by a particle, site x − 1 is empty and site x + 1 is occupied. +We say that site x ∈ Z is occupied by an active particle if it can jump either to x − 1 or x + 1 +with positive rate, in other words, it is such that +η(x − 1)η(x)(1 − η(x + 1)) + (1 − η(x − 1))η(x)η(x + 1) = 1. +Let us now map this process onto another exclusion process. We need to introduce some notation. +Let L < R be two integers, and define: +ΩL,R := +� +η ∈ Ω ; + + + + + + + +η(x) = 0 +if x ⩾ R, +η(x) = 1 +if x ⩽ L, +η(x) + η(x + 1) ⩾ 1 +if x < R. +� +and +Ω := +� +L 0, ηt ∈ Ω. +For any η ∈ Ω, we can label particles from right to left, by the following recursive procedure: +assume that η ∈ ΩL,R for some L < R, and define2 +X1(η) := R − 1, +Xi+1(η) := max +�j < Xi(η) ; η(j) = 1 +�. +We are now ready to construct the mapping: let S : Ω → {0, 1}N∗ be the application such that, +for any η ∈ Ω, +σ := S(η) +satisfies +σ(i) = 1 − η(Xi − 1). +In other words, site i is occupied by a particle in σ (i.e. σ(i) = 1) if and only if the i-th particle +in η has an empty site to its left. +In particular, the step initial condition η0 given by +η0(x) = 1x⩽x0 for some x0 ∈ Z, +belongs to Ω, and it corresponds to an empty configuration σ0 = S(η0), namely +σ0(x) = 0 +for any x ∈ N∗. +(7) +These initial conditions will be important in what follows. The dynamics of the mapped process +is described in the following lemma. Note that with the step initial condition, for all t ⩾ 0, +Xi(ηt) − Xi+1(ηt) ∈ {1, 2}, and S(ηt) encodes all the information about these spacings. +2Note that R is a function of η since it is the position of the right-most particle, therefore X1 is well defined. + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +6 +11 +10 +9 +8 +7 +6 +5 +4 +3 +2 +1 +L +R +p +p +p +q +q +FASEP on Z +half-line ASEP +reservoir +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +p +p +p +q +q +Figure 1. The top figure represents a configuration η in Ω and the possible +transitions with their respective rates. The bottom figure represents the mapped +configuration S(η) on {0, 1}N∗ and the possible transitions, in the same color as +the corresponding transitions in η. +Lemma 2.2. Assume that the generator of the Markov process {ηt}t⩾0 is LF given in (6), +then {S(ηt)}t⩾0 is an asymmetric simple exclusion process on the infinite half-line N∗ with a +boundary reservoir which injects particles at rate p. More precisely its generator is given as +follows: for any σ ∈ Σ := {0, 1}N∗, for any f : Σ → R, +Lf(σ) = +∞ +� +x=1 +� +pσ(x)(1 − σ(x + 1)) + qσ(x + 1)(1 − σ(x)) +�� +f(σx,x+1) − f(σ) +� ++ p(1 − σ(1)) +� +f(σ1) − f(σ) +� +, +where σ1 is obtained after the creation of one particle at site 1, namely σ1(x) = 1x=1+σ(x)1x̸=1. +Proof. The proof is straightforward by looking at every possible transition, see Figure 1. +□ +3. Main results and strategy of the proof +From now on we consider the half-line ASEP denoted by {σt}t⩾0, which is a Markov process +on Σ = {0, 1}N∗ generated by L given in (8). +3.1. Hopf-Cole transform. Let us define the height function associated with the particle +system, in the following (standard) way: for any x ∈ N∗, +ht(x) := ht(0) + +x +� +k=1 +(2σt(k) − 1), +with ht(0) = 2 +∞ +� +k=1 +σ0(k) − 2 +∞ +� +k=1 +σt(k). +This is well-defined when �∞ +k=1 σ0(k) < ∞. In that case, note that −ht(0) is equal to twice the +number of particles that have entered into the system between times 0 and t (no particle can +exit the system by hypothesis). For any initial condition σ0 ∈ Σ and any t > 0, this number is +bounded by a Poisson random variable with parameter pt, and in particular, it is almost surely +finite. In the case where the initial number of particles in the system is infinite, we can still +define ht(0) as minus twice the number of particles that have entered the system. Thus, starting +from any initial condition σ0 ∈ Σ, the height function satisfies: for any t > 0, any x ∈ N, +ht(x) < ∞ a.s. For ν, λ ∈ R which will be chosen later we then define the Hopf-Cole transform +Zt(x) := e−λht(x)+νt, +x ∈ N. +(8) +Alternatively, Zt can be defined as a function of the positions of the particles in the FASEP. +It is straightforward to check that, under the coupling described in Section 2, for the FASEP +started from the step initial condition, the position of the j-th particle in the FASEP is related +to the height function with empty initial condition through +Xj(ηt) = +∞ +� +k=j +σt(k) − j. + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +7 +Consequently, the Hopf-Cole transform can be recast as +Zt(x) = e2λXx+1(ηt)+(3x+2)λ+νt. +(9) +We will now mainly work with the Hopf-Cole transform Z and we state our results in terms of +this quantity. In the following, for any function f : Z → R we define its left and right gradients +by +∇+f(x) := f(x + 1) − f(x), +∇−f(x) := f(x − 1) − f(x) +and its discrete Laplacian by ∆f(x) := f(x + 1) + f(x − 1) − 2f(x). One knows that Z satisfies +dZt(x) = (νZt(x) + LZt(x))dt + dMt(x), +x ∈ N, +where {Mt(x)}t⩾0 are martingales whose quadratic variations will be computed below. +As +in [CS18,Par19] (see also [GLM17]), we look for conditions on ν, λ so that νZt(x) + LZt(x) can +be rewritten as D∆Zt(x) for some diffusion coefficient D. After straightforward computations +(given in Appendix A) we choose +λ = 1 +2 log q +p, +ν = q + p − 2√pq, +(10) +which imply: for any x ∈ N∗, +dZt(x) = D∆Zt(x)dt + dMt(x), +with D = √pq. +(11) +It remains to define Zt(−1) in order that (11) remains valid at the boundary point x = 0, which +can be done if we let +Zt(−1) := µZt(0), +with µ = +�q +p = eλ. +(12) +We therefore obtain the following result. +Lemma 3.1. We assume the choice of parameters (10) and define µ = +� +q/p. Let ∆µ be the +discrete Laplacian on N with the following boundary condition: +∆µf(x) := +� +f(x + 1) + f(x − 1) − 2f(x) +if x > 0 +f(1) + µf(0) − 2f(0) +if x = 0. +(13) +Then, for any x ̸= y ∈ N, the following three quantities are martingales: +Mt(x) := Zt(x) − Z0(x) − D +� t +0 +∆µZs(x)dx, +Mt(x)2 − +� t +0 +d +ds[M(x)]s ds, +Mt(x)Mt(y), +and moreover, +d +dt[M(x)]t = + + + +Zt(x)2 � +ηt(x)(1 − ηt(x + 1))(p−q)2 +p ++ ηt(x + 1)(1 − ηt(x))(p−q)2 +q +� +if x > 0, +Zt(0)2(1 − ηt(1))(p−q)2 +p +if x = 0. +Proof. This is straightforward, using the following identity: +Zt(x + 1) +Zt(x) += +�q +p +�1 − ηt(x + 1) +� + +�p +q ηt(x + 1). +□ +3.2. Weak asymmetry. From now on, we consider the half-line ASEP in the weak asymmetry +regime where +p = 1 +2eε +and +q = 1 +2e−ε, +for ε > 0. +(14) +Rewriting everything in terms of ε, the Hopf-Cole transform reads as +Zt(x) = eεht(x)+νt, +where ν = 1 +2eε(e−ε − 1)2 = 1 +2ε2 + o(ε2). +The boundary parameter µ appearing in (12) and the diffusion coefficient D are equal to +µ = e−ε, +D = 1 +2. +In this weak asymmetry regime, the quadratic variation of the above martingale satisfies: + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +8 +Lemma 3.2. As ε → 0 we have +d +dt[M(x)]t = ε2Zt(x)2 + ∇+Zt(x)∇−Zt(x) + o(ε2)Zt(x)2, +for any x > 0 +(15) +d +dt[M(0)]t = ε2Zt(0)2 − εZt(0)∇+Zt(0) + o(ε2)Zt(0)2. +(16) +Note that the term εZt(0)∇+Zt(0) at the boundary is new in our case, it did not appear in +the case of [CS18]. +Proof. It is straightforward using +∇+Zt(x) = εZt(x) +�2ηt(x + 1) − 1 + o(1) +� +and Lemma 3.1 which implies that +d +dt[M(x)]t = ε2� 1 +2Zt(x)2 + 1 +2Zt(x + 1)Zt(x − 1) +� + ∇+Zt(x)∇−Zt(x) + o(ε2)Zt(x)2. +□ +3.3. Main theorems. Before stating our main results, let us start by defining the notion of +solution for the stochastic heat equation which is at the core of the convergence results of this +paper. +Definition 3.3. Let ξ be the standard space-time white noise on R+ × R+, on some probability +space (Ω, F, P). We say that Zt(u) solves the stochastic heat equation (SHE) +∂tZ = 1 +2∂uuZ + Zξ +(17) +on the time interval [0, T], with Dirichlet boundary condition, and initial condition Zini, if for +any t ∈ (0, T], +Zt(u) = +� +R+ +dvP Dir +t +(u, v)Zini(v) + +� t +0 +ds +� +R+ +dvP Dir +t +(u, v)Zs(v)ξ(s, v), +(18) +where the Dirichlet half-space heat kernel P Dir +t +is defined by +P Dir +t +(u, v) = +1 +√ +2πt +� +e−(u−v)2/(2t) − e−(u+v)2/(2t)� +. +(19) +3.3.1. Empty initial condition. We define, for any u ∈ ε2N, the scaled process +Zε +t (u) := ε−2Zε−4t(ε−2u) +(20) +and we extend Zε +t (·) to the continuous half-line R+ by linear interpolation. +Remark 3.4. Let us already emphasize here that the scaling in (20) is not the one which will +appear later when the initial condition is supposed to be near-equilibrium (see Section 3.3.2, +(24) and Definition 3.8), nor the one in [Par19], where the macroscopic initial condition is the +delta Dirac function δ0 and the prefactor there is ε−1 instead of ε−2. +Let us consider the space of test functions +H = +�φ ∈ C∞ +c (R) : φ(0) = 0 +�. +(21) +The initial condition (7) implies that Z0(x) = µx, for any x ∈ N, and therefore, for any φ ∈ H, +�Zε +0, φ +� −−−→ +ε→0 2φ′(0), +(22) +where (Z, φ) := +� +R Zφ denotes the usual scalar product in L2(R). In other words, the initial +condition of the continuous limit of the rescaled process Zε +t is −2δ′ +0 where δ′ +0 is the derivative of +the delta Dirac distribution. +Our first result, which will be proved in Section 4, is the existence and uniqueness of solutions +to (17) for the δ′ +0 initial condition and Dirichlet boundary condition: + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +9 +Proposition 3.5. Let ξ be the standard space-time white noise on R+×R+, on some probability +space (Ω, F, P). There exists a C(R+)–valued process (Zt)t>0, which is adapted to the filtration +Ft = σ ({ξ(s, ·)}s⩽t), and solves the SHE in the sense of Definition 3.3 with initial condition +Zini = −2δ′ +0. This solution is unique in the class of adapted continuous processes satisfying +sup +X∈R+ +s∈(0,t) +� +s2E +� +Zs(X)2�� +< ∞. +(23) +Remark 3.6. The existence and uniqueness of solutions to (17) are already proved in [Par22, +Theorem 4.1, Prop. 4.2] when Zini is near equilibrium (see Definition 3.8 below). This is not +the case of the δ′ +0 initial condition that we consider here, and therefore we need to provide a +new proof. Existence and uniqueness is also proved in [Par19, Proposition 4.3] for Robin type +boundary condition and δ0 initial condition. As we will see below in Section 4, the proof of +Proposition 3.5 involves different estimates than [Par19,Par22]. +The main result of this paper is the following convergence: +Theorem 3.7. Fix T > 0. Assume the initial particle configuration is empty as in (7). Then the +rescaled process {Zε +s}s∈(0,T] converges as ε → 0 to the solution of the stochastic heat equation (17) +on the time interval [0, T] with Dirichlet boundary condition, and initial condition Zini = −2δ′ +0 +(as defined in Definition 3.3), in the sense of weak convergence of probability measures on the +path space D((0, T], C(R+)) endowed with the Skorokhod topology. +3.3.2. Near-equilibrium initial condition. We also study the simpler case of near-equilibrium +initial condition. In this section we consider the scaled process +Z ε +t (u) := ε2Zε +t (u) = Zε−4t(ε−2u), +(24) +which differs from Zε +t by a factor ε2, and we allow Z0(x) to be different from µx. +Definition 3.8. We say that a sequence of random functions F ε ∈ C(R+) is near-equilibrium +if it satisfies the following: there exists a > 0 such that, for any n ∈ N, any α ∈ (0, 1 +2), there +exists some constant C = C(α, n) > 0 such that, for any u, u′ ∈ R+, any ε > 0, +∥F ε(u)∥n ⩽ Ceau +(25) +∥F ε(u) − F ε(u′)∥n ⩽ C|u − u′|αea(u+v), +(26) +where ∥G∥n := E[|G|n]1/n denotes the Ln–norm with respect to the probability measure. +An intermediate – although important – result is the following: +Theorem 3.9. Fix T > 0. Assume that the initial condition Z ε +0 ∈ C(R+) is near-equilibrium +(in the sense of Definition 3.8), and that Z ε +0 weakly converges in C(R+) to some initial condition +Zini as ε → 0. +Then the rescaled process (Z ε +s )s∈[0,T] converges as ε → 0 to the solution of the stochastic +heat equation (17) with initial condition Zini in the time interval [0, T] (as defined in Definition +3.3), in the sense of weak convergence of probability measures on the path space D([0, T], C(R+)) +endowed with the Skorokhod topology. +3.3.3. An example of near-equilibrium initial condition. Theorem 3.9 may be useful indepen- +dently from its application to the proof of Theorem 3.7. Let us consider the half-line ASEP +generated by L in the weakly asymmetric regime (14), but with initial condition given by prod- +uct Bernoulli, that is, we assume that the variables σ0(k), k ∈ N∗, are independent Bernoulli +variables with parameter ̺. Let us scale ̺ = (1 − ε(B + 1 +2))/2, so that εh0(ε−2u) converges to a +Brownian motion with drift −(B + 1/2). Then, it can be shown that Z ε +0 is near-equilibrium3, +and thus Theorem 3.9 can be applied, to find that (Z ε +s ) converges to the SHE with Dirichlet +boundary condition, and initial condition given by the exponential of a Brownian motion with +drift −(B + 1/2). +3The term near-equilibrium actually comes from the fact that such initial condition is stationary for ASEP on Z. + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +10 +Denoting by Z (t, u) this solution, [Par22] showed that we have the identity in distribution +lim +u→0 +Z (t, u) +u += � +Z (t, 0) +(27) +where � +Z is the solution of the SHE on R+ with Robin boundary parameter B and delta Dirac +function as initial condition. In the special case B = −1/2, that is ̺ = 1/2, the law of +� +Z (t, 0) +is explicitly known and related to eigenvalue GOE (Gaussian Orthogonal Ensemble) statistics +[BBCW18,Par19]. For other values of B, the law of � +Z (t, 0) was computed very recently [IMS22] +(see also [KLD20]). +Remark 3.10. Discrete analogues of the identity (27), allowing to exchange the roles of the +boundary and initial condition parameters, also exist for directed polymers [BBC20, Prop. 8.1], +last passage percolation [BBCS18b, Lemma 6.1] or more general models defined through Pfaffian +Schur measures [BR01, Corollary 7.6]. This suggests that the height function at the origin for +ASEP on N∗ may have the same distribution in the following two situations: +(1) The reservoir has injection rate p, ejection rate 0, and the initial condition is i.i.d. +Bernoulli with parameter ̺; +(2) The reservoir has injection parameter α = p̺, ejection parameter γ = q(1 − ̺), and the +initial configuration is empty. +There exists such an identity in distribution when p = 1, q = 0, that is in the case of TASEP4. +In the weakly asymmetric regime ε → 0, such an identity would yield (27) (which is proved +in [Par22] through another route). We leave this as an open question in the case of ASEP. +3.4. Strategy of the proof. As stated in Definition 3.3, a mild solution Zt(x) to the stochastic +heat equation (17) with Dirichlet boundary condition satisfies (18). Solutions can equivalently +be characterized by the following martingale problem. The equivalence of the two notions of +solutions is proved in [Par19, Prop. 4.4] in the case of Neumann type boundary condition. The +argument applies mutatis mutandis in the Dirichlet case. +Definition 3.11. A solution Zt(u) to the martingale problem for the stochastic heat equation +(17) with Dirichlet boundary condition, initial condition Zini and time interval [0, T], is a random +variable Z with values in C([0, T], C(R+)), such that for all φ ∈ H and 0 < t ⩽ T, the following +quantities are martingales: +Nt(φ) := (Zt, φ) − (Zini, φ) − 1 +2 +� t +0 +(Zs, φ′′)ds, +(28) +Qt(φ) := Nt(φ)2 − +� t +0 +(Z 2 +s , φ2)ds. +(29) +We divide the proof of Theorem 3.7 (Section 7) and Theorem 3.9 (Section 6) into several +steps, following the strategy of [Par19]: +(1) For a process Z ε +t starting from a near-equilibrium initial condition Z ε +0 , which satisfies +that Z ε +0 ⇒ Zini as ε → 0, we prove its convergence towards the solution to the heat +equation with Dirichlet boundary condition starting from Zini (a.k.a. Theorem 3.9). This +is split in two steps: proof of tightness (Section 6.1), and identification of the limit point +(Section 6.2). The latter step uses the martingale problem above: we show that the +discrete martingale problem gives the continuous one in the limit, and the control of the +error terms is a consequence of the tightness estimates. +(2) We come back to the initial condition Z0(x) = µx of Theorem 3.7. We prove that at +time δ > 0, Zε +δ defined in (20) is near-equilibrium in the sense of Definition 3.8. This is +the purpose of Section 7.1. Similarly to the first point, this property gives us tightness +in D([δ, T], C(R+)) for any δ ∈ (0, T), and that any limit point is solution to (17) with +Dirichlet boundary condition. +4More precisely, one needs to consider the TASEP on N∗ with an initial condition such that the site 1 is occupied, +and all other sites are occupied according to Bernoulli i.i.d. random variables. Then, the height function for +this model has the same distribution [PS01] as the height function associated to the last passage percolation +model with boundary considered in [BBCS18b]. The symmetry between the boundary and the initial condition +parameters follows from [BBCS18b, Lemma 6.1] (choosing the parameter α there as α = ̺). + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +11 +(3) The missing point is to push δ to 0. We use consistency, and identify the initial condition. +The structure of the argument is similar to [Par19] but we use a different method in order +to obtain second moment bounds (Lemma 7.5). We rely on the analysis of exact integral +formulas for the moments of Zt(x) obtained using a Markov duality in [BC22a]. +4. Existence and uniqueness of the solution +This section is devoted to the proof of existence and uniqueness of the solution to the SHE +(Definition 3.3) with Dirichlet boundary condition and initial condition Zini = −2δ′ +0. Although +we follow a standard argument from [Wal86] of Picard iteration (see also [Par19, Section 4]), +the initial condition that we consider is much more singular than initial condition considered in +previous works, and this requires us to use refined estimates. +It will be convenient to introduce the notation +dP Dir +t +(u, 0) := −P Dir +t +(u, ·) ∗ 2δ′ +0(·) = 2∂vP Dir +t +(u, v) +�� +v=0 = +2 +� +2 +πue− u2 +2t +t3/2 +. +(30) +We start with an estimate involving the quantity dP Dir +t +(u, 0), which we will reuse in Sections +7.2 and 7.3. +Lemma 4.1. Define, for any s ∈ (0, t) and u ∈ R+, +Gt(s, u) = +� +R+ +dv +� +P Dir +t−s(u, v) +�2 � +dP Dir +s +(v, 0) +�2 . +(31) +There exists a constant C such that for any s ∈ (0, t) and for any u ∈ R+, +Gt(s, u) ⩽ C +√ +t +� +s(t − s)dP Dir +t +(u, 0)2. +(32) +Proof. Using the explicit expressions for P Dir +t−s(u, v) from (19) and dP Dir +s +(v, 0) in (30), we can +perform the integral (31) and obtain +Gt(s, u) = +2e +−u2 +t +� +t(t−s) +s +� +1 − e +−su2 +t(t−s) +� ++ 2u2 +� +π3/2t5/2� +s(t − s)u2 +. +Dividing by dP Dir +t +(u, 0)2, we get +Gt(s, u) +dP Dir +t +(u, 0)2 = +√ +t +� +t(t−s) +s +� +1 − e +−su2 +t(t−s) +� ++ 2u2 +� +4√π +� +s(t − s)u2 +. +Using the bound 1 − e−x ⩽ x, and simplifying the resulting expression, we obtain that +Gt(s, u) +dP Dir +t +(u, 0)2 ⩽ +3 +4√π +√ +t +� +s(t − s). +□ +Let us now turn to the proof of Proposition 3.5. +Proof of Proposition 3.5. Fix a terminal time T > 0 and consider the Banach space B of adapted +processes (Zt) satisfying +∥Z∥2 +B := +sup +X∈R+ +s∈(0,T) +� +s2E +� +Zs(X)2�� +< ∞. +We define a sequence of processes defined for t ⩽ T, u ∈ R+, by +U0(t, u) := dP Dir +t +(u, 0). +Un+1(t, u) := +� t +0 +� +R+ +P Dir +t−s(u, v)Un(s, v)ξ(s, v)dvds. + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +12 +This implies that, if Sn = �n +k=0 Un, then +Sn+1(t, u) = dP Dir +t +(u, 0) + +� t +0 +� +R+ +P Dir +t−s(u, v)Sn(s, v)ξ(s, v)dvds. +In order to show the existence of the solution to (108), it suffices to show that the series Sn con- +verges in the space B, and for that we will show that � ∥Un∥B converges. Regarding uniqueness, +it follows from the same argument as in [Par19, Proposition 4.2]. +In order to estimate ∥Un∥B, we introduce +fn(s) = sup +v∈R+ +� +E +�Un(s, v)2� +dP Dir +s +(v, 0)2 +� +. +By Itô isometry, we have, +E +� +Un+1(t, u)2� += +� t +0 +ds +� +R+ +dv +� +P Dir +t−s(u, v) +�2 E +� +Un(s, v)2� +. +Thus, we may write, recalling the definition of Gt(s, u) in (31), +E +� +Un+1(t, u)2� +⩽ +� t +0 +ds +� +R+ +dv +� +P Dir +t−s(u, v) +�2 dP Dir +s +(v, 0)2fn(s) +⩽ +� t +0 +dsGt(s, u)fn(s). +Using Lemma 4.1, we obtain that +E +� +Un+1(t, u)2� +⩽ C +� +dP Dir +t +(u, 0) +�2 2 +√ +t +� t +0 +ds +fn(s) +� +s(t − s). +(33) +Dividing both sides of (33) by (dP Dir +t +(u, 0))2, we obtain that for t ∈ (0, T), +fn+1(t) ⩽ C +√ +t +� t +0 +ds +fn(s) +� +s(t − s). +Iterating this inequality, we get +fn+2(t) ⩽ C +√ +t +� t +0 +dr√r +� r +0 +ds +fn(s) +� +r(t − r) +� +s(r − s) += C +√ +t +� t +0 +dsfn(s) +√s +� t +s +dr +√t − r√r − s += C +√ +t +� t +0 +dsfn(s) +√s , +by exchanging the integration order, from which we deduce by induction that fn(t) ⩽ Cntn/2/(⌊n/2⌋)!. +Hence, we have obtained that +∥Un∥2 +B ⩽ C +sup +u∈R+ +s∈(0,T) +� +E +�Un(s, u)2� +dP Dir +s +(u, 0)2 +� += fn(T) ⩽ CnT n/2 +(⌊n/2⌋)!, +where in the first inequality we have simply used that dP Dir +s +(u, 0) ⩽ Cs−1 (which is easy to +check using the explicit expression (30)). This shows that �+∞ +n=0 ∥Un∥B < ∞ so that the series +�+∞ +n=0 Un exists in the space B, and it concludes the proof. +□ +5. Heat kernel with diverging Robin boundary conditions +In the proofs of Theorems 3.7 and 3.9, we will need sharp estimates on the heat kernel pR +t +with Robin boundary conditions. We therefore collect in this section all preliminary results on +pR +t , which will be invoked later on. We start with a definition. +Definition 5.1. Let µ < 1. +The discrete heat kernel with Robin boundary condition (and +parameter µ), is defined as the solution to: for any x, y ∈ N +∂tpR +t (x, y) = 1 +2∆ypR +t (x, y), +pR +0 (x, y) = 1x=y, +pR +t (−1, y) = µpR +t (0, y), + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +13 +where ∆y denotes the discrete Laplacian acting on functions of the variable y. +Since we assume µ < 1, pR +t (x, y) corresponds to the transition probability for a continuous time +random walk on N which behaves as the symmetric simple random walk on positive integers, +while at 0, after an exponentially distributed waiting time with mean 1, it jumps to 1 with +probability 1 +2, it stays at 0 with probability µ +2, and it is killed with probability 1−µ +2 . This gives +us a representation of pR in terms of the transition probabilities (pn) of the underlying discrete +random walk (which moves similarly but at integer times): +pR +t (x, y) = +∞ +� +n=0 +e−t tn +n!pn(x, y). +(34) +Moreover, we have the following representation for pR +t , in terms of the kernel pt for the continuous +time symmetric simple random walk on Z, see [CS18, Section 4.1]: +pR +t (x, y) = pt(x − y) + µpt(x + y + 1) + (1 − µ−2) ++∞ +� +z=2 +µzpt(x + y + z). +(35) +In particular, pR +t (x, y) is increasing in µ. +Definition 5.2. Let {pε +t(x, y)}x,y∈N be the discrete heat kernel with Robin boundary condition +and parameter µ = e−ε. +5.1. First properties. +Lemma 5.3 (Heat kernel bounds). Fix T > 0. We have the following estimates. +(i) For any a > 0, there exists C = C(a, T) ∈ R+ such that, for all t ⩽ ε−4T, x ∈ N, +∞ +� +y=0 +pε +t(x, y)eaε2y ⩽ Ceaε2x. +(36) +(ii) There exists C = C(T) ∈ R+ such that, for all t ⩽ ε−4T, x, y, z ∈ N, for all v ∈ [0, 1], +for |x − y| ⩽ ⌈ +√ +t⌉ +|pε +t(x, z) − pε +t(y, z)| ⩽ C +� +1 ∧ +1 +√ +t1+v +� +|x − y|v. +(37) +Moreover, for all b > 0, there exists C = C(T, b) ∈ R+ such that for all t ⩽ ε−4T, +x, z ∈ N, for all v ∈ [0, 1], +|∇±pε +t(x, z)| ⩽ C +� +1 ∧ +1 +√ +t1+v +� +e−b|z−x|(1∧t−1/2), +(38) +and consequently, for all a ⩾ 0, there exists C = C(T, a) ∈ R+ such that +∞ +� +y=0 +��∇±pε +t(x, y) +�� eaε2yea|x−y|(1∧t−1/2) ⩽ Ceaε2xt−1/2. +(39) +(iii) For all t ⩾ s ⩾ 0, x, y ∈ N, +pε +s(x, y) ⩽ et−spε +t(x, y). +(40) +Proof. A number of these bounds follow from those established for instance in [Par19]. Therein, +the author considers the discrete heat kernel with Robin boundary condition with parameter +µA = 1 − Aε2 (note that ε for us corresponds to √ε therein). In particular, the monotonicity in +µ of pR allows us to use directly the bounds established in [Par19]. More precisely: +• (36) follows from Corollary 3.3 in [Par19], where we take a1 = 0 and a2 = a; +• (37) and (38) follow from similar bounds that hold for the standard heat kernel pt on +the whole line Z, and from (35), as in the proof of Proposition 3.2 in [Par19]. In fact, +the monotonicity in µ of (35) implies a monotonicity in the upper bounds used in the +proof of [Par19], so that we can use the estimates therein as an upper bound; +• (40) is an immediate consequence of (34). +□ + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +14 +Lemma 5.4 (Key cancellation and consequences). We have the following estimates: +(i) For any x, x′ ∈ N, +∞ +� +y=0 +� ∞ +0 +∇+pε +t(x, y)∇+pε +t(x′, y)dt = 1x=x′. +(41) +(ii) For any a > 0, there exist ε0 = ε0(a, T) > 0, c = c(a, T) ∈ (0, 1) such that for all ε < ε0, +x ∈ N∗, +∞ +� +y=1 +� ε−4T +0 +���∇+pε +r(x, y)∇−pε +r(x, y) +��� eaε2|x−y|dr ⩽ c. +(42) +(iii) For any a > 0, there exist ε0 = ε0(a, T) > 0, C = C(a, T) ∈ R+ such that for all +ε < ε0, x ∈ N∗, t ⩽ T, +∞ +� +y=1 +� ε−4t +0 +���∇+pε +r(x, y)∇−pε +r(x, y) +��� eaε2|x−y| +1 +√ +ε−4t − r +dr ⩽ Cε2. +(43) +Proof. The last two bounds (42) and (43) follow from the first one (41), together with (38) and +(39), by the same arguments as in [CS18, Proof of Corollary 5.4]. +Let us show (41), which can be proved in a more elementary way than is done in [CS18] +(where it is Proposition 5.1). Indeed, note that �∞ +y=0 pε +t(x, y)pε +t(x′, y) = pε +2t(x, x′) by symmetry +of pε +t and the Markov property of the random walk described after Definition 5.1. Moreover, +� ∞ +0 pε +2t(x, x′)dt = 1 +2G(x, x′) is the Green’s function associated with that random walk, i.e. the +expected number of times the random walk started from x goes through x′ before being killed. +Elementary computations therefore yield +∞ +� +y=0 +� ∞ +0 +∇+pε +t(x, y)∇+pε +t(x′, y)dt = G(x+1, x′ +1)+G(x, x′)−G(x+1, x′)−G(x, x′ +1). (44) +By the interpretation of G in terms of number of visits, it is clear that G(x + 1, x′ + 1) = +G(x, x′ + 1) if x > x′ and G(x + 1, x′) = G(x, x′) if x ⩾ x′. Moreover, by first-step analysis, +G(x + 1, x + 1) = 1 + 1 +2 [G(x, x + 1) + G(x + 2, x + 1)] = 1 + 1 +2 [G(x, x + 1) + G(x + 1, x + 1)], +and therefore G(x + 1, x + 1) = 2 + G(x, x + 1). Then (41) follows. +□ +5.2. Moment estimates. In this section, we establish moment estimates on the heat kernel +pR +t and its convolution with the empty initial condition Z0(x) = µx. +Proposition 5.5. There exists a constant C > 0 such that for any ε ∈ (0, 1), t > 0, α ∈ [0, 1 +2) +and x, y ∈ N, we have that for any µ ∈ (0, 1), +pR +t (x, y) ⩽ C +√ +t, +(45) +and for µ = e−ε, we have that +ε−2E [Zt(x)] = ε−2 � +y≥0 +pε +t(x, y)µy ⩽ Cmin +�(ε4t)−1, ε−2� +(46) +and +��(pε +t ∗ ε−2Z0)(x) − (pε +t ∗ ε−2Z0)(y) +�� = +����ε−2 � +z≥0 +pε +t(x, z)µz − ε−2 � +z≥0 +pε +t(y, z)µz +���� +⩽ C +� +ε2|x − y| +�α (ε4t)−1−α/2. +(47) +Remark 5.6. Note that the constant C in (45) is universal, and does not depend on t nor on +the terminal time T. This estimate is called long-time estimate in [Par19], see Proposition 3.6 +therein. Here, we provide a different proof and obtain an even better estimate. + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +15 +Proof. Let us first establish explicit formulas for the quantities to bound in (45), (46) and (47). +For t ⩾ 0 and x ∈ Z, we have the integral representation of the heat kernel +pt(x) = +1 +2iπ +� +e +1 +2(ξ+ξ−1−2)tξx dξ +ξ , +(48) +where the contour is a positively oriented circle around 0. From (48) and (35), we deduce +pR +t (x, y) = +1 +2iπ +� +e +1 +2(ξ+ξ−1−2)tξx +� +ξ−y + ξy+1 µ − ξ +1 − µξ +� dξ +ξ , +(49) +where the contour is a positively oriented circle around 0 with radius smaller than µ−1. Hence, +we have that +� +y≥0 +pR +t (x, y)µy = E [Zt(x)] = +1 +2iπ +� +e +1 +2(ξ+ξ−1−2)tξx (1 − µ2)(1 − ξ2) +(1 − µ/ξ)(1 − µξ)2 +dξ +ξ +(50) +where the contour is a positively oriented circle around 0 with radius comprised between µ and +µ−1, and +� +z≥0 +pR +t (x, z)µz − +� +z≥0 +pR +t (y, z)µz = +1 +2iπ +� +e +1 +2(ξ+ξ−1−2)t(ξx − ξy) (1 − µ2)(1 − ξ2) +(1 − µ/ξ)(1 − µξ)2 +dξ +ξ . +(51) +Now we estimate the integrals above. Since µ ∈ (0, 1), we may assume that the contour (in +each formula above) is a circle of radius 1. Now, since ξ has modulus 1, we have that |ξx−y| = +|ξx+y+1| = 1 and for any µ ∈ (0, 1), |µ − ξ| = |µ − ξ| = |µξ − 1| so that +��� µ−ξ +1−µξ +��� = 1. Using the +change of variables ξ = eiθ in (49), we get +pR +t (x, y) ⩽ 2 +2π +� π +−π +e +t +2 ℜ[eiθ+e−iθ−2]dθ. +Then, we use the estimate cos(x) − 1 ⩽ −x2 +5 , valid for x ∈ (−π, π), so that +ℜ[eiθ + e−iθ − 2] = 2 cos(θ) − 2 ⩽ −2θ2 +5 +. +(52) +Hence, setting t = ε−4T, we obtain +pR +t (x, y) ⩽ ε2 +π +� π +π +e +−θ2t +5 dθ. +Finally, using the change of variables θ = ˜θt−1/2, +pR +t (x, y) ⩽ +1 +π +√ +t +� +R +e +−˜θ2 +5 d˜θ +which proves (45) with C = 1 +π +� +R e +−z2 +5 dz. Now we turn to the proof of (46) and assume henceforth +that µ = e−ε. Since pR +t (x, ·) defines a measure on N with mass at most 1, and 0 < µ < 1, we +have the trivial bound E [Zt(x)] ⩽ 1, so that ε−2E [Zt(x)] ⩽ ε−2. It remains to show that +ε−2E [Zt(x)] = ε−2 � +y≥0 +pε +t(x, y)µy ⩽ C(ε4t)−1. +Using the change of variables ξ = eiθε2, and setting x = ε−2X, (50) becomes +ε−2 � +y≥0 +pε +t(x, y)µy = 1 +2π +� ε−2π +−ε−2π +e +1 +2 +� +eiε2θ+e−iε2θ−2 +� +teiθX +(1 − e−2ε)(1 − e2iε2θ) +(1 − e−ε−iε2θ)(1 − e−ε+iε2θ)2 dθ. +(53) +Using x − x2 +2 ⩽ 1 − e−x ⩽ x for x > 0, and 0 ⩽ 1 − cos(x) ⩽ x2 +2 for x ∈ (−π, π), we have +����� +(1 − e−2ε)(1 − e2iε2θ) +(1 − e−ε−iε2θ)(1 − e−ε+iε2θ)2 +����� = (1 − e−2ε) +� +2(1 − cos(2ε2θ)) +(1 + e−2ε − 2e−ε cos(ε2θ))3/2 ⩽ +4ε3|θ| +(ε − ε2/2)3 = +4|θ| +(1 − ε/2)3 . +Thus, using additionally the estimate (52) and setting t = ε−4T, we obtain the bound +ε−2 � +y≥0 +pε +t(x, y)µy ⩽ 1 +2π +� ε−2π +−ε−2π +e +−θ2T +5 +4|θ| +(1 − ε/2)3 dθ. + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +16 +The change of variables θ = ˜θT −1/2 now yields (for ε ∈ (0, 1)) +ε−2 � +y≥0 +pε +t(x, y)µy ⩽ 1 +T +16 +π +� +R +|˜θ|e +−˜θ2 +5 d˜θ, +which proves (46) with C = 16 +π +� +R |z|e−z2/5dz. Now we turn to the proof of (47). Using (51) and +the same steps as above (with y = ε−2Y ), +������ +ε−2 � +z≥0 +pε +t(x, z)µz − ε−2 � +z≥0 +pε +t(y, z)µz +������ +⩽ 1 +T +16 +π +� +R +|˜θ|e +−˜θ2 +5 +���eiX ˜θT −1/2 − eiY ˜θT −1/2��� d˜θ +(54) +Hence, using the change of variables u = 2θ and choosing a constant C such that 16 +π |˜θ|e +−˜θ2 +5 +⩽ +Ce +−u2 +2 , ������ +ε−2 � +z≥0 +pε +t(x, z)µz − ε−2 � +z≥0 +pε +t(y, z)µz +������ +⩽ C +T +� +R +e +−u2 +2 +���e2iXuT −1/2 − e2iY uT −1/2��� du, +⩽ C +T +� +R +e +−u2 +2 +���1 − e2i(X−Y )uT −1/2��� du, +⩽ 2C +T max{|X − Y |T −1/2, 1} +� +R +ue +−u2 +2 du. +Thus, (47) holds for any α ∈ (0, 1]. +□ +6. Near-equilibrium initial condition: proof of Theorem 3.9 +In this section we assume that the initial condition is near-equilibrium, and we prove Theorem +3.9. We are thus interested in the rescaled process Z ε +t defined in (24) for any u ∈ ε2N and +then extended to R+ by linear interpolation. Together with the uniqueness of the solution to +the martingale problem (28)–(29), the next two propositions prove the desired result stated in +Theorem 3.9. +Proposition 6.1 (Tightness). Under the assumptions of Theorem 3.9, the sequence of pro- +cesses (Z ε +s )s∈[0,T] is tight in the space D([0, T], C(R+)). Moreover, any limit point belongs to +C([0, T], C(R+)). +Proposition 6.2 (Identification of limit points). Under the assumptions of Theorem 3.9, any +limit point (Zs)s∈[0,T] of (Z ε +s )s∈[0,T] in the space D([0, T], C(R+)) satisfies the continuous mar- +tingale problem (28)–(29). +The rest of the section is devoted to the proof of Proposition 6.1 and Proposition 6.2. +6.1. Tightness: +proof of Proposition 6.1. The main ingredient is the following lemma, +which is a consequence of estimates established in [CS18, Par19], adapted to the case µ = e−ε +instead of 1 − ε2A. +Lemma 6.3. Fix T > 0 and assume that the initial condition Z ε +0 ∈ C(R+) is near-equilibrium +(Definition 3.8). Then for all n ∈ N and α ∈ [0, 1 +2), there exists C = C(α, n, T) ∈ R+ such that +for all ε > 0, u, u′ ∈ R+, s, s′ ∈ [0, T], +∥Z ε +s (u)∥n ⩽ Ceau, +(55) +∥Z ε +s (u) − Z ε +s (u′)∥2n ⩽ C|u − u′|αea(u+v), +(56) +∥Z ε +s (u) − Z ε +s′(u′)∥2n ⩽ C +� +ε2α ∨ |t − t′|α/2� +e2au. +(57) +These estimates, together with Arzela-Ascoli’s Theorem, imply Proposition 6.1 (see [Bil68, +Chapter 3]). In order to prove Lemma 6.3, we use the bounds on the heat kernel pR which have +been proved in Lemma 5.3 and Lemma 5.4. Sine the proof closely follows that in [Par19], we +do not detail it, but just point out which ingredients are needed where. We repeatedly use the +following lemma (proved in [DT16, Lemma 3.1], [CS18, Lemma 4.18], as stated in [Par19, Lemma +5.3]). + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +17 +Lemma 6.4. For any n ∈ N∗, there exists C = C(n) < ∞ such that, for any F bounded on +R+ × N and any t > 1, +���� +� t +0 +∞ +� +y=0 +F(s, y)dMs(y) +���� +2 +n +⩽ Cε2 +� t +0 +∞ +� +y=0 +F(s, y)2��Zs(y) +��2 +nds, +(58) +where F(s, y) = sup|s′−s|<1 |F(s′, y)|. Moreover, when F(s, y) = pε +s(x, y) for some x ∈ N, +pεs(x, y)2 ⩽ Cs−1/2pε +s+1(x, y). +(59) +The first bound (55) in Lemma 6.3 is obtained by an iteration argument, and is a consequence +of (25), (36) and Lemma 6.4 (same proof as in [Par19, Proof of Proposition 5.4]). +For the second bound (56), write the consequence of Lemma 3.1 +Zt(x) − Zt(y) = +∞ +� +z=0 +[pε +t(x, z) − pε +t(y, z)] Z0(z) + +� t +0 +∞ +� +z=0 +�pε +t−s(x, z) − pε +t−s(y, z) +� dMs(z). +(60) +To control the first sum, we can proceed as in [Par19] and extend Z0 into a function over Z by +imposing that z �→ Z0(z − 1) − µZ0(z) is odd. Note that Z0 now depends implicitly on ε. Then +it is easy to check that Z0 still satisfies (26) (possibly changing C). The rest of the proof of (56) +is exactly similar to [Par19] and uses Lemma 6.4, (36), (37), (40), and (55). +The last bound (57) is obtained as in [Par19]. It is even simpler for us because the term +labeled J2 therein vanishes. The estimates rely on Lemma 6.4, on (55) and on (36). +6.2. Identification of limit points: proof of Proposition 6.2. +6.2.1. Discrete martingale problem. Let us denote by ∆ε := ∆µ the discrete Laplacian with +boundary condition defined in (13) with µ = e−ε. We also introduce, for any φ, ψ : R+ → R +which are square summable, the following notation: +(ψ, φ)ε := ε2 +∞ +� +x=0 +φ(ε2x)ψ(x). +(61) +From Lemma 3.1, for any φ : R+ → R with compact support, +N ε +t (φ) := (Zε−4t, φ)ε − (Z0, φ)ε − 1 +2 +� ε−4t +0 +(∆εZs, φ)ε ds +(62) +is a martingale. Let us compute +ε−2(∆εZs, φ)ε = +∞ +� +x=0 +Zs(x) +� +φ(ε2(x + 1)) + φ(ε2(x − 1)) − 2φ(ε2x) +� ++ Zs(−1)φ(0) − Zs(0)φ(−ε2) += +∞ +� +x=0 +Zs(x)ε4φ′′(ε2x) ++ +∞ +� +x=0 +Zs(x) +� +∆φ(ε2·)(x) − ε4φ′′(ε2x) +� ++ +�q +pZs(0)φ(0) − Zs(0)φ(−ε2), += ε4 +∞ +� +x=0 +Z ε +ε4s(ε2x)φ′′(ε2x) ++ +∞ +� +x=0 +Z ε +ε4s(ε2x) +� +∆φ(ε2·)(x) − ε4φ′′(ε2x) +� ++ Z ε +ε4s(0) +��q +pφ(0) − φ(−ε2) +� +. +Therefore N ε +t (φ) can be rewritten as +N ε +t (φ) = ε2 +∞ +� +x=0 +φ(ε2x)Z ε +t (ε2x) − ε2 +∞ +� +x=0 +φ(ε2x)Z ε +0 (ε2x) − 1 +2 +� t +0 +ε2 +∞ +� +x=0 +Z ε +s (ε2x)φ′′(ε2x)ds +− 1 +2ε−2 +��q +pφ(0) − φ(−ε2) +� � t +0 +Z ε +s (0)ds + E1(ε), +(63) + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +18 +where the error term is +E1(ε) := −1 +2ε−2 +∞ +� +x=0 +� +∆εφ(ε2·)(x) − ε4φ′′(ε2x) +� � t +0 +Z ε +s (ε2x)ds. +(64) +Let φ : R → R be smooth and compactly supported, such that φ(0) = 0. Fix ψ : R → R+ +smooth, compactly supported, such that ψ(0) = 1 and ψ′(0) = 0, and define +φε = φ + εφ′(0)ψ, +(65) +so that φε(0) = εφ′ +ε(0). From the previous computation (63), the following is a martingale: +N ε +t (φε) = ε2 +∞ +� +x=0 +φ(ε2x)Z ε +t (ε2x) − ε2 +∞ +� +x=0 +φ(ε2x)Z ε +0 (ε2x) +− ε2 +2 +� t +0 +∞ +� +x=0 +Z ε +s (ε2x)φ′′(ε2x)ds + R1 + R2 + R3, +where +R1 := −1 +2ε−2 +∞ +� +x=0 +� +∆εφε(ε2·)(x) − ε4φ′′ +ε(ε2x) +� � t +0 +Z ε +s (ε2x)ds +(66) +R2 := −1 +2ε−2 � +e−εφε(0) − φε(−ε2) +� � t +0 +Z ε +s (0)ds, +(67) +R3 := ε3φ′(0) +∞ +� +x=0 +ψ(ε2x) +� +Z ε +t (ε2x) − Z ε +0 (ε2x) +� +. +(68) +Moreover, the quadratic variation of N ε +t (φε) is given (see Lemma 3.1 and Lemma 3.2) by: +[N ε(φε)]t = ε4 +∞ +� +x=1 +φ2 +ε(ε2x) +� ε−4t +0 +� +ε2Zs(x)2 + ∇+Zs(x)∇−Zs(x) + o(ε2)Zs(x)2� +ds +(69) ++ ε4φ2 +ε(0) +� ε−4t +0 +� +ε2Zs(0)2 + εZs(0)∇+Zs(0) + o(ε2)Zs(0)2� +ds. +(70) +In order to conclude, we prove in the next section that R1, R2, R3 vanish in probability as ε → 0, +which then establishes that Nt(φ) (defined in (28)) is a martingale for any Z limit point of Z ε. +Then, we verify (29). +6.2.2. Proof of Proposition 6.2. In order to prove that R1, R2, R3 vanish in probability, we +estimate their ∥ · ∥n-norms. Let us start with R2, where we will see why we chose φε as in (65). +(R2) For any n ∈ N, by Lemma 6.3, +∥R2∥n ⩽ Cε−2t +� +e−εεφ′(0) − φ(−ε2) − εφ′(0)ψ(−ε2) +� +(71) +⩽ Cε−2t +� +(1 − ε)εφ′(0) + ε2φ′(0) − εφ′(0) + o(ε2) +� += o(1). +(72) +Therefore R2 vanishes in any Ln, n ∈ N. +(R3) Moreover, the same holds for R3, again thanks to Lemma 6.3, and because the series +ε2 �∞ +x=0 ψ(ε2x)e2aε2x converges to +� +R+ ψ(u)e2audu (the finiteness of this quantity is the +reason we do not take ψ ≡ 1). +(R1) Let us now consider R1. By Lemma 6.3, it is enough to show that, for all x ∈ N∗, +���∆εφε(ε2·)(x) − ε4φ′′ +ε(ε2x) +��� = o(ε4), +and +���∆εφε(ε2·)(0) − ε4φ′′ +ε(0) +��� = o(ε2), +which can be checked by elementary computations and thanks to the compactness of the +support of φ. + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +19 +It remains to show that any limit point Z of Z ε satisfies (29). Let us rewrite +[N ε(φε)]t = +� t +0 +ε2 +∞ +� +x=0 +φ2(ε2x)Z ε +s (ε2x)2ds + R′ +1 + R′ +2 + R′ +3 + R′ +4, +(73) +where +R′ +1 = +� t +0 +ε2 +∞ +� +x=0 +[φ2 +ε(ε2x) − φ2(ε2x)]Z ε +s (ε2x)2ds, +(74) +R′ +2 = ε4 +� ε−4t +0 +∞ +� +x=1 +φ2 +ε(ε2x)∇+Zs(x)∇−Zs(x)ds, +(75) +R′ +3 = −ε5 +� ε−4t +0 +φ2 +ε(0)Zs(0)∇+Zs(0)ds, +(76) +R′ +4 = o(1) +� t +0 +ε2 +∞ +� +x=0 +φ2(ε2x)Z ε +s (ε2x)2ds. +(77) +We have to show that R′ +1, R′ +2, R′ +3 vanish (e.g. in L2) as ε → 0. +(R′ +1) For the first term, the result follows from the bound ∥φ2 +ε − φ2∥∞ = O(ε), the fact that +φ, φε have compact support, and from (55). +(R′ +3 and R′ +4) Both terms R′ +3 and R′ +4 are also controlled with (55) and (56). +(R′ +2) Similarly to [CS18] and [Par19, Proof of Theorem 5.7], we split R′ +2 in two parts: write +R′ +2 = r1 + r2, where +r1 = ε4 +� ε−3 +0 +∞ +� +x=1 +φ2 +ε(ε2x)∇+Zs(x)∇−Zs(x)ds +(78) +r2 = ε4 +� ε−4t +ε−3 +∞ +� +x=1 +φ2 +ε(ε2x)∇+Zs(x)∇−Zs(x)ds. +(79) +By (56), ∥∇±Zs(x)∥2 ⩽ Cε2αe2aε2x for any α < 1 +2 and some C, a > 0. Therefore, we can +bound +∥r1∥2 ⩽ Cε4ε−3ε4α +∞ +� +x=1 +φ2 +ε(ε2x)e4aε2x ⩽ C′ε4α−1, +(80) +which goes to 0 if we choose for instance α = 1/3. On the other hand, +E[r2 +2] = 2ε8 +� ε−4t +ε−3 +� s +ε−3 +∞ +� +x,x′=1 +φ2 +ε(ε2x)φ2 +ε(ε2x′)E +�∇+Zs(x)∇−Zs(x)∇+Zr(x′)∇−Zr(x′) +�drds (81) += 2ε8 +� ε−4t +ε−3 +� s +ε−3 +∞ +� +x,x′=1 +φ2 +ε(ε2x)φ2 +ε(ε2x′)E +�∇+Zr(x′)∇−Zr(x′)U(x, r, s) +�drds, +(82) +where +U(x, r, s) := E +� +∇+Zs(x)∇−Zs(x)|Fr +� +, +Fr := σ (Zs(x); x ∈ N, s ⩽ r) . +(83) +Therefore, +E[r2 +2] ⩽ Cε8ε4α +� ε−4t +ε−3 +� s +ε−3 +∞ +� +x,x′=1 +φ2 +ε(ε2x)φ2 +ε(ε2x′)e4aε2x′E +�|U(x, r, s)| +�drds +(84) +⩽ Cε6+4α +� ε−4t +ε−3 +� s +ε−3 +∞ +� +x=1 +φ2 +ε(ε2x)E +�|U(x, r, s)| +�drds. +(85) +We will prove in Lemma 6.5 below an estimate on E +�|U(x, r, s)| +� which shows that +E[r2 +2] ⩽ Cε4+6α +� ε−4t +ε−3 +� s +ε−3 +1 +√s − rdrds ⩽ Cε1+6α. +(86) + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +20 +Therefore, R′ +i goes to 0 in L2 for i = 1, ..., 4, and both properties (28) and (29) are satisfied, +which conclude the proof of Proposition 6.2. We now prove the last needed estimate: +Lemma 6.5. For any α < 1 +2, there exist a, C = C(α, t) > 0 such that, if ε−3 ⩽ r ⩽ s ⩽ ε−4t, +x ∈ N∗, +E +�|U(x, r, s)| +� ⩽ Cε2α eaε2x +√s − r. +(87) +Proof of Lemma 6.5. As in [BG97,CS18,Par19], we write +U(x, r, s) =∇−Is(x)∇+Is(x) + ∇−Is(x)∇+N s +r (x) + ∇−N s +r (x)∇+Is(x) + ∇−N s +r (x)∇+N s +r (x) ++ E + + +� s +r +∞ +� +y=0 +Ks−τ(x, y)d[M(y)]τ +������ +Fr + + , +(88) +where +Is(x) = +∞ +� +y=0 +pε +s(x, y)Z0(y) +N s +r (x) = +� r +0 +∞ +� +y=0 +pε +s−τ(x, y)dMτ (y), +Kt(x, y) = ∇+pε +t(x, y)∇−pε +t(x, y). +By (39), (25) and Lemma 3.1, we have (see e.g. [CS18, Proof of Lemma 5.7]) +max +� +E +��∇±Is(x) +�2� +, E +��∇±N s +r (x) +�2�� +⩽ Cε2α eaε2x +√s − r. +(89) +Moreover, the expectation term in (88) can be split into +(ε2 + o(ε2)) +∞ +� +y=0 +� s +r +Ks−τ(x, y)E[(Zτ (y))2|Fr]dτ − ε +� s +r +Ks−τ(x, 0)E[Zτ (0)∇+Zτ(0)|Fr]dτ +− +∞ +� +y=1 +� s +r +Ks−τ(x, y)E[∇−Zτ(y)∇+Zτ(y)|Fr]dτ. +If we can bound the first two terms by Cε2αeaε2x/√s − r, using (42), (43), the same iterative +procedure as described in [CS18] yields the desired result. +Let us consider the second term (which is new with respect to the case treated in [CS18]). +By (55), (56) and (38), we have for any α < 1 +2, b ⩾ 0, ν ⩽ 1 +����ε +� s +r +Ks−τ(x, 0)E +�Zτ(0)∇+Zτ(0)|Fr +�dτ +���� ⩽ Cε1+4αe−2b|x| +� s +r +� +1 ∧ (s − τ)1+ν� +dτ += Cε1+4αe−2b|x|(s − r)−ν. +It thus remains to check that +ε2 +∞ +� +y=0 +� s +u +Ks−τ(x, y)E[(Zτ(y))2|Fu]dτ ⩽ Cε2α eaε2x +√s − u, +(90) +which is exactly Lemma 5.8 in [CS18]. In turn, this inequality is a consequence of bounds (38), +(39) and Lemma 6.3 (see [CS18]). +□ +7. Delta prime initial condition: proof of Theorem 3.7 +In this section we come back to our initial problem and assume that the initial condition is +empty so that Z0(x) = µx, where we recall that µ = +� +q/p. Therefore, the initial condition is +not near-equilibrium, and we now use the different scaling +Zε +t (u) := ε−2Zε−4t(ε−2u). +(91) +Let us now prove Theorem 3.7. + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +21 +Recall that we already know from Section 4 existence and uniqueness of the solution to the +SHE (Definition 3.11) with Dirichlet boundary condition and initial condition Zini = −2δ′ +0. +Also, we have obtained in Section 5 explicit estimates involving the discrete Dirichlet heat +kernel, relying on exact computations. +The estimates from Section 5 enable us to show in Section 7.1 that at time δ > 0, Zε +δ(·) is +near-equilibrium in the sense of Definition 3.8. This allows to follow the same strategy as in +Section 6 to have tightness in the space D([δ, T], C(R+)) for any 0 < δ < T, and show that +the limit points satisfy the SHE (Proposition 7.2). +Following [Par19], we will then show in +Section 7.2 that there exists a limit point in the space D((0, T], C(R+)), which is solution to the +SHE with Dirichlet boundary condition. We also prove in Section 7.2 second moment bounds +satisfied by this limit point, using exact formulas from [BC22a]. We finally determine the initial +condition in Section 7.3, using the moment bounds from Section 7.2. +7.1. Near-equilibrium property. Recall definition (91). We now prove the following esti- +mates which tell us that at any positive time t, Zε +t is near-equilibrium: +Proposition 7.1. Fix T ⩾ 0. Let α ∈ [0, 1 +2) and n ∈ N. There exists C = C(α, T, n) ∈ R+ such +that, for all ε ∈ (0, 1), x, x′ ∈ N, and t, t′ ∈ [0, ε−4T] with t′ < t: +∥ε−2Zt(x)∥n ⩽ C(ε4t)−1 +(92) +∥ε−2(Zt(x) − Zt(x′))∥n ⩽ C(ε2|x − x′|)α(ε4t)−1−α/2 +(93) +∥ε−2(Zt(x) − Zt′(x))∥n ⩽ C(ε4t′)−1−α/2 ε2α(1 ∨ |t − t′|α/2). +(94) +Proof. The proof uses the same argument as in [Par19, Proposition 6.2]. +First of all, let us note that, as in [Par19, Lemma 6.1]: for any τ ⩾ 0, there exists Cτ such +that, for any ε ∈ (0, 1), +sup +t⩽τ +∥Zt(x)∥n ⩽ CτZ0(x). +(95) +This is due to the fact that the initial configuration of particles is empty and the jump rate +satisfies p ⩽ 1 +2 + ε, therefore the position of the largest occupied site at time t is stochastically +dominated by a Poisson random variable N(3t/2) with mean 3t/2. +Then, we write the martingale decomposition +Zt(x) = +� +z⩾0 +pε +t(x, z)Z0(z) + +� t +0 +� +z⩾0 +pε +t−s(x, z)dMs(z). +(96) +We bound the second term in the right-hand-side for t ⩾ 1, using both estimates (58) and (59) +from Lemma 6.4, and get +���� +� t +0 +� +z⩾0 +pε +t−s(x, z)dMs(z) +���� +2 +n +⩽ Cε2 +� t +0 +(t − s)−1/2 � +z⩾0 +pε +t−s+1(x, z)∥Zs(z)∥2 +n ds. +(97) +For the first term, from Proposition 5.5, (46) recall that +ε−2 � +z⩾0 +pε +t(x, z)Z0(z) ⩽ C min +�(ε4t)−1, ε−2�, +(98) +where C > 0 is a universal constant, and therefore +����ε−2 � +z⩾0 +pε +t(x, z)Z0(z) +���� +2 +⩽ C min +�(ε4t)−1, ε−2�(pε +t ∗ ε−2Z0)(x). +(99) +Summarizing, using the convexity inequality (a + b)2 ⩽ 2a2 + 2b2 we have proved, for any +t ∈ [1, ε−4T], +��ε−2Zt(x) +��2 +n ⩽ Cmin +�(ε4t)−1, ε−2�(pε +t ∗ ε−2Z0)(x) ++ Cε2 +� t +0 +(t − s)−1/2 � +z⩾0 +pε +t−s+1(x, z) +��ε−2Zs(z) +��2 +n ds. +(100) + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +22 +Now, for t ⩽ 1, we use (95) together with (40), and we obtain +∥Zt(x)∥n ⩽ CZ0(x) ⩽ Cetpε +t(x, x)Z0(x) ⩽ C(pε +t ∗ Z0)(x) +which implies from (99): for any t ⩽ 1, +��ε−2Zt(x) +��2 +n ⩽ Cmin +�(ε4t)−1, ε−2�(pε +t ∗ ε−2Z0)(x). +(101) +Since (101) is stronger than (99), we have established that (100) holds for any t ∈ [0, ε−4T]. We +are now able to iterate this inequality using the semigroup property of pε +t. We get +∥ε−2Zt(x)∥2 +n ⩽ Cmin +�(ε4t)−1, ε−2�(pε +t ∗ ε−2Z0)(x) + ++∞ +� +k=0 +Ck+2ε2(k+1)Ik(t) (pε +t+k+1 ∗ ε−2Z0)(x) +where +Ik(t) = +� +∆k(t) +min +�(ε4t0)−1, ε−2� +k +� +j=1 +(tj − tj−1)−1/2(t − tk)−1/2dt0 · · · dtk +∆k(t) = {(t0, . . . , tk) ∈ Rk+1 : 0 < t0 < t1 < · · · < tk < t} +and with the convention � +∅ = 1. We estimate the Ik(t) with a change of variables, and we +obtain +Ik(t) ⩽ 2k+1t(k+1)/2 +(k/2)! +. +Now, recalling the bound (98), we can conclude that +∥ε−2Zt(x)∥2 +n ⩽ C min +�ε−2, (ε4t)−1�(pε +t ∗ ε−2Z0)(x) + C′ � +k⩾0 +Ck (ε4t)k/2 +(k/2)! (pε +t+k+1 ∗ ε−2Z0)(x) +⩽ C(ε4t)−1�(pε +t ∗ ε−2Z0)(x) + 1 +�. +where we used that ε4t ⩽ T and therefore C depends on the terminal time T. We then obtain +(92) from (98). +We now turn to (93). In a similar fashion we have +��ε−2(Zt(x) − Zt(x′)) +�� +n ⩽ +��(pε +t ∗ ε−2Z0)(x) − (pε +t ∗ ε−2Z0)(x′) +�� ++ +����ε−2 +� t +0 +� +z⩾0 +�pε +t−s(x, z) − pε +t−s(x′, z) +�dMs(z) +���� +n +. +(102) +We call the terms on the right-hand-side J1 and J2, respectively. We get from (47) that +J1 ⩽ C(ε2|x − y|)α(ε4t)−1−α/2 +Now, for J2, we have +J2 +2 ⩽ Cε2|x − y|α +� t +0 +(t − s)−(1+α)/2 � +z⩾0 +�pε +t−s+1(x, z) + pε +t−s+1(y, z) +���ε−2Zs(z) +��2 +nds +⩽ Cε−2|x − y|α +� t +0 +(t − s)−(1+α)/2s−1 � +z⩾0 +�pε +t−s+1(x, z) + pε +t−s+1(y, z) +��(pε +s ∗ ε−2Z0)(x) + 1 +�ds += Cε−2t−1/2t−α/2|x − y|α�(pε +t+1 ∗ ε−2Z0)(x) + (pε +t+1 ∗ ε−2Z0)(y) + 1 + 1 +� +⩽ C(ε4t)−1/2(ε4t)−1−α/2(ε2|x − y|)α. +So +J2 ⩽ C(ε2|x − y|)α/2(ε4t)−3/4−α/4 +and we conclude that (93) holds since (ε4t)−3/4−α/4 ⩽ T 1/4+α/4(ε4t)−1−α/2. +Finally we prove (94), noting similarly that +��ε−2(Zt(x)−Zt′(x)) +�� +n ⩽ +��(pε +t−t′ ∗ε−2Zt′)(x)−ε−2Zt′(x) +�� +n + +����ε−2 +� t +t′ +� +z⩾0 +pε +t−u(x, z)dMt′(z)du +���� +n +. +(103) + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +23 +We denote the terms on the right-hand side above by I1 and I2 respectively. We have, using +(92) and (93), +I1 = +����ε−2 � +z⩾0 +pε +t−t′(x, z)Zt′(z) − Zt′(x) +���� +n +⩽ +����ε−2 � +z⩾0 +pε +t−t′(x, z) +�Zt′(z) − Zt′(x) +����� +n ++ +���� +� +z⩾0 +pε +t−t′(x, z) − 1 +���� +��ε−2Zt′(x) +�� +n +⩽ +� +z⩾0 +pε +t−t′(x, z)(ε4t′)−1−α/2�ε2|x − z| +�α + Cε|t − t′|1/2(ε4t′)−1 +⩽ C(ε4t′)−1−α/2 ε2α(1 ∨ |t − t′|α/2) + Cε|t − t′|1/2(ε4t′)−1 +⩽ C(ε4t′)−1−α/2 ε2α(1 ∨ |t − t′|α/2) + C(ε4t′)−1−α/2ε2α|t − t′|α/2. +Let us explain the steps used in the above series of inequalities. +• In the second line we have simply used the triangle inequality. +• In the third line, we have used Proposition 7.1 and the bound +���� +� +z⩾0 +1 − pε +t−t′(x, z) +���� ⩽ Cε2|t − t′|1/2. +(104) +Recall that pε +t−t′(x, z) correspond to transition probabilities of continuous-time discrete- +space random walks (see the discussion below Definition 5.1). Hence, the LHS of (104) is +the probability, for a random walk started at x at time t′ (let t′ = 0 for simplicity), to have +been killed by time t. This probability is bounded by the killing rate 1−µ +2 +< Cε times the +expectation of the time spent at 0 by an auxiliary random walk having transition kernel +pR with µ = 1 (that is with no killing). For the latter random walk, the expectation +of the time spent at 0 is the same as for the discrete space-discrete time random walk, +with kernel denoted pn(x, z) in (34). It is well-known that the local time at 0 for such +random walks is of order +√ +t (one may for instance consider that when at 0, the random +walk stays there for an independent time, geometrically distributed, while outside from +0 it behaves as the absolute value of the simple random walk). Hence, the bound (104) +holds. +• In the fourth line, we have used the bound +� +z⩾0 +pε +t−t′(x, z)|x − z|α ⩽ C(1 ∨ |t − t′|α/2). +(105) +This bound is proved in [Par19, Eq. (30) and Corollary 3.3] for the Robin heat kernel +pR under a different scaling of µ, and in particular, it holds for µ = 1. Now recall that +for any t > 0, x, z ∈ N, the Robin heat kernel pR is increasing in µ (see (35)), so that +(105) holds when µ = e−ε. +• In the last line we used ε4t, ε4t′ ⩽ T, and the fact that +ε2|t − t′|1/2 × (ε4t′)−1 = +� +(ε4(t − t′))1/2−α/2ε2α(t − t′)α/2� +× +� +(ε4t)α/2(ε4t)−1−α/2� +. +(106) +This proves the desired bound for I1. Now we consider I2. We have, using (92) +I2 +2 ⩽ Cε2 +� t +t′ (t − s)−1/2 � +z⩾0 +pε +t−s+1(x, z) +��ε−2Zs(z) +��2 +n ds +⩽ Cε2(ε4t)−2 +� t +t′ (t − s)−1/2 � +z⩾0 +pε +t−s+1(x, z)ds +⩽ Cε2(ε4t)−2 +� t +t′ (t − s)−1/2ds = C(ε4t)−2ε2(t − t′)1/2, +where we have used that +� +z⩾0 +pε +s(x, z) ⩽ 1, +(107) + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +24 +since the pε +s(x, z) correspond to transition probabilities of a random walk with killing. Therefore +I2 ⩽ C(ε4t)−1 × ε(t − t′)1/4 ⩽ C(ε4t′)−1−α/2ε2α|t − t′|α/2, +by the same argument as in (106). This proves (94). +□ +7.2. Construction and properties of the limit point. We are now able to obtain tightness +of {Zε +t }, and identify the law of any of its limit point, as follows: +Proposition 7.2 (Tightness). For any 0 < δ ⩽ τ, the laws of {Zε} are tight on the Skorokhod +space D([δ, τ], C(R+)). Moreover, any limit point P is an element of C([δ, τ], C(R+)). +For any θ ∈ [δ, τ], define Lθ : C([δ, τ], C(R+)) → C(R+) as the evaluation map at time θ. +Then, the process {Lθ+δ ; θ ∈ [0, τ − δ]} has the same distribution under P as the solution of +the stochastic heat equation (17) as defined in Definition 3.11, with initial condition Zini whose +distribution is the same as the one of Lθ under P. +Proof. The argument is exactly the same as in [Par19, Proof of Corollary 6.3]. The tightness +property is based on Proposition 7.1, together with Arzela-Ascoli’s Theorem. +Continuity of +limit points follow from Proposition 7.1 and Kolmogorov’s continuity criterion. +Finally, the +identification of limit points follows from the same arguments as in Section 6.2, replacing Z ε +by Zε in (63) (Proposition 7.1 replacing Lemma 6.3 in the control of the error terms). +□ +The next step consists in defining a limit point in C((0, +∞), C(R+)) (Lemma 7.3 below) and +identifying the initial condition. This means showing that the Duhamel form of the SHE (18) +is satisfied for all t > 0, with Zini = −2δ′ +0, that is, +Zt(u) = dP Dir +t +(u, 0) + +� t +0 +ds +� +R+ +dvP Dir +t +(u, v)Zs(v)ξ(s, v). +(108) +Lemma 7.3. Let Qε denote the law of Zε on C((0, +∞), C(R+)). Then, there exists a measure +Q on C((0, +∞), C(R+)) which is a limit point of the sequence {Qε}ε on D((0, +∞), C(R+)). +Proof. This can be proved exactly as in [Par19, Lemma 6.5] thanks to Kolmogorov’s extension +Theorem. +□ +The identification of the initial condition follows a general argument, given in [Par19], based +on estimates on the first two moments given in the next two lemmas, which however are specific +to the δ′ +0 initial condition and use arguments different from [Par19]. We start with some exact +computation for the first moment. +Lemma 7.4. Let {Zt} be distributed according to the measure Q defined in Lemma 7.3. For +any t > 0, and u ∈ R+, we have +lim +ε→0 E [Zε +t (u)] = E [Zt(u)] = dP Dir +t +(u, 0) = 1 +2π +� +∞ +−∞ +e +−t +2 θ2+iθu(−4iθ)dθ, +(109) +where we recall that dP Dir +t +(u, 0) is defined in (30). +Proof. Using the same steps as in the proof of Proposition 5.5 – see (53) in particular – +E [Zε +t (u)] = ε−2 � +v≥0 +pε +tε−4(ε−2u, ε−2v)µy += 1 +2π +� ε−2π +−ε−2π +e +1 +2 +� +eiε2θ+e−iε2θ−2 +� +ε−4teiθu +(1 − e−2ε)(1 − e2iε2θ) +(1 − e−ε−iε2θ)(1 − e−ε+iε2θ)2 dθ. +(110) +Using the estimate (52), we see that the exponential term in the integrand of (110) is dominated +by e−2θ2t/5, so that we may apply dominated convergence to take the ǫ → 0 limit. It is easy to +see that the integrand converges pointwise to +e− θ2t +2 +iθu(−4iθ), +so that +lim +ε→0 E [Zε +t (u)] = +� +∞ +−∞ +e− θ2t +2 +iθu(−4iθ)dθ. + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +25 +This Gaussian type of integral can be computed explicitly as 2 +� +2 +π +u +t3/2 e− u2 +2t , which, by the +definition of the Dirichlet heat kernel from (19), equals 2∂vP Dir +t +(u, 0). Furthermore, we have +E +�(Zε +t (u))2� ⩽ (C/T)2 by (92), so that the sequence is uniformly integrable. Hence we may +exchange the limit with the expectation, and obtain (109). +□ +We can conclude from Lemma 7.4 that if Zε +t (u) does converge as ε → 0 to some Zt(u) solving +the stochastic heat equation (17) with some deterministic initial condition Zini, then for all +t > 0, +� +R+ +dvP Dir +t +(u, v)Zini(v) = dP Dir +t +(u, 0). +(111) +This suggests that Zini = −2δ′. However, in order to rigorously identify the initial condition, +we need to establish that (108) holds, and for that we will also need a second moment estimate. +Lemma 7.5. Fix T > 0 and consider {Zt}t∈(0,T] distributed according to the measure Q defined +in Lemma 7.3 (restricted to the time interval (0, T]). There exists a constant C = C(T) such +that, for any u ∈ R+ and 0 < t ⩽ T, +��Zt(u) +��2 +2 ⩽ C +� +dP Dir +t +(u, 0) +�2 +(112) +��Zt(u) − dP Dir +t +(u, 0) +��2 +2 ⩽ C +√ +t +� +dP Dir +t +(u, 0) +�2 , +(113) +where ∥F∥2 := +�� |F|2dQ. +Remark 7.6. The bounds (112) and (113) are different from those in [Par19, Lemma 6.6] which +is the analogue of Lemma 7.5 in the case of the Neumann boundary condition with narrow wedge +initial condition. The proof of [Par19, Lemma 6.6] uses the martingale decomposition (96) and +equation (100) above. These ingredients allow to mimic at the microscopic level the construction +of the solution to the SHE (similar to our Section 4). In our case, we did not manage to obtain +the discrete analog of (32) which would have been needed to complete the proof, because this +required much sharper heat kernel estimates. We eventually opted to use the exact computation +of lim E +�Zε +t (u)2� as ε → 0 which was done in the separate article [BC22a]. +Proof. From [BC22a, Prop. 4.8], we have that for u1 ⩽ u2, +E[Zt(u1)Zt(u2)] = 42 +� +iR+1+η +dz1 +2iπ +� +iR +dz2 +2iπ +z1 − z2 +z1 − z2 − 1 +z1 + z2 +z1 + z2 − 1e +tz2 +1 +2 −u1z1+ +tz2 +2 +2 −u2z2z1z2, +(114) +where η > 0 is any positive real number, and the expectation is taken with respect to the +measure Q. More precisely, [BC22a, Prop. 4.8] in the case n = 2 states that under the measure +Qε from Lemma 7.3, the limit as ε → 0 of E[Zε +t (u1)Zε +t (u2)] is given by the right-hand side of +(114). However, the case n > 0 of the same equation [BC22a, Eq. (4.20)] shows that higher +moments are uniformly bounded as ε → 0, which implies that the sequence of random variables +Zε +t (u1)Zε +t (u2) is uniformly integrable as ε → 0. Hence, the weak convergence to Zt(u1)Zt(u2) +implied by Proposition 7.2 holds as well in L1, so that (114) holds. +The evaluation of this double Gaussian integral is not trivial. To simplify its estimation, it is +convenient to use the shorthand notation +D(t, u) = 4 +� +iR +dz +2iπe +tz2 +2 −uzz = 4 +� +iR+1+η +dz +2iπ e +tz2 +2 −uzz = dP Dir +t +(u, 0). +(115) +The second equality follows from Cauchy’s theorem5 and the lack of residues between the vertical +lines iR and iR + 1 + η. +Using that for Real[z1 ± z2 − 1] > 0, +1 +z1 ± z2 − 1 = +� ∞ +0 +dye−y(z1±z2−1), +5More precisely, we may apply the Cauchy theorem on the rectangle formed by the points −iR, iR, iR + 1 + η, +−iR + 1 + η. Since the function z �→ e +tz2 +2 −uzz has no residues inside the rectangle, the integral over the rectangle +equal zero. Moreover, due to the exponential decay of the integrand as the imaginary part increases, we see that +the difference between the integration along the segment [−iR, iR] or the segment [−iR + 1 + η, iR + 1 + η] goes +to zero as R goes to infnity. This proves that the two integral formulas in (115) are equal. + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +26 +we have that +E[Zt(u1)Zt(u2)] = +� ∞ +0 +dλ +� ∞ +0 +dµeλ+µ∂λ∂µD(t, u1 + λ + µ)D(t, u2 − λ + µ), +so that using the exact explicit expression for dP Dir +t +(u, 0) in (30), +∥Zt(u)∥2 +2 +dP Dir +t +(u, 0)2 = 1 +4 +� ∞ +0 +dλ +� ∞ +0 +dµeλ+µ∂λ∂µ +� +(u + µ)2 − λ2 +u2 +e− λ2+µ2+2µu +t +� +. +Computing the derivatives in λ and µ, the integral can be evaluated using Mathematica, which +yields the explicit formula +∥Zt(u)∥2 +2 +dP Dir +t +(u, 0)2 = +e−u +�√πet/4√ +t +� +eu � +erf +� √ +t +2 +� ++ 1 +� �t(u − 1) + 2u2� + te +u2 +t +� +erf +� +t−2u +2 +√ +t +� ++ 1 +�� ++ 2euu(t + 2u) +� +4u2 +. +(116) +We see that the maximum of this function is attained as u → 0, so that +0 ⩽ +∥Zt(u)∥2 +2 +dP Dir +t +(u, 0)2 ⩽ 1 +8 +� +√πet/4√ +t(t + 6) +� +erf +�√ +t +2 +� ++ 1 +� ++ 2(t + 4) +� +. +(117) +It can be checked that this expression behaves as 1 + 3√π +4 +√ +t + o( +√ +t) as t → 0, so that on an +interval [0, T], (117) is bounded by 1+C +√ +t where the constant C depend on T. This immediately +implies (112), and, using Lemma 7.4, it also implies (113). +□ +7.3. Conclusion. Provided with Proposition 7.2 and Lemma 7.3, one can follow the argument +of [Par19, Lemma 6.7 and Theorem 6.8], and one obtains that {Zε +s}s∈(0,T] converges as ε → 0 +to a solution of the stochastic heat equation (17) on the time interval [0, T] with Dirichlet +boundary condition, in the sense of weak convergence of probability measures on the path space +D((0, T], C(R+)) endowed with the Skorokhod topology. +It remains to formally check that the initial condition is Zini = −2δ′ +0 as we have claimed. We +need to check that the limit of {Zε +s}s∈(0,T], denoted {Zs}s∈(0,T], satisfies (108). Since Zs is a +solution of (2) on the space D((0, T], C(R+)), we already know that for any 0 < s < t < T, +Zt(u) = P Dir +t−s(u, ·) ∗ Zs(·) + +� t +s +dτ +� +R+ +dvP Dir +t−τ (u, v)Zτ(v)ξ(τ, v). +Hence, following [Par19, Section 6], we may write +�����Zt(u) − dP Dir +t +(u, 0) − +� t +0 +dτ +� +R+ +dvP Dir +t−τ(u, v)Zτ (v)ξ(τ, v) +����� +2 +⩽ +���P Dir +t−s(u, ·) ∗ Zs(·) − dP Dir +t +(u, 0) +��� +2 + +����� +� s +0 +dτ +� +R+ +dvP Dir +t−τ(u, v)Zτ (v)ξ(τ, v) +����� +2 +. +(118) +This inequality holds for any 0 < s < t, so that in order to show that the LHS of (118) equal +zero, it suffices to show that the RHS vanishes as s → 0. Using the semi-group property, +���P Dir +t−s(u, ·) ∗ Zs(·) − dP Dir +t +(u, 0) +��� +2 ⩽ +����� +� +R+ +dvP Dir +t−s(u, v) +� +Zs(v) − dP Dir +s +(v, 0) +������ +2 +⩽ +� +R+ +dvP Dir +t−s(u, v) +���Zs(v) − dP Dir +s +(v, 0) +��� +2 +⩽ +� +R+ +dvP Dir +t−s(u, v)s1/4dP Dir +s +(v, 0) +⩽ s1/4dP Dir +t +(v, 0) + +WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS +27 +where in the third inequality we have used Lemma 7.5. Hence we have obtained that +lim +s→0 +���P Dir +t−s(u, ·) ∗ Zs(·) − dP Dir +t +(u, 0) +��� +2 = 0. +Now we turn to the second term to bound. By Ito isometry, +����� +� s +0 +dτ +� +R+ +dvP Dir +t−τ(u, v)Zτ (v)ξ(τ, v) +����� +2 +2 += +� s +0 +dτ +� +R+ +dvP Dir +t−τ (u, v)2E +� +Zτ(v)2� +⩽ C +� s +0 +dτ +� +R+ +dvP Dir +t−τ (u, v)2dPτ(v, 0)2 += C +� s +0 +dτGt(τ) +⩽ C +� +t +t − s +� s +0 +dτ 1 +√τ dPt(u, 0)2 +⩽ C +� +st +t − s dPt(u, 0)2, +where in the first inequality we have used Lemma 7.5, and in the second inequality we have used +Lemma 4.1 (recall that the function Gt(τ) is defined in (31)). We conclude that +lim +s→0 +����� +� s +0 +dτ +� +R+ +dvP Dir +t−τ(u, v)Zτ (v)ξ(τ, v) +����� +2 += 0, +so that (108) is satisfied for all 0 < t ⩽ T. This concludes the proof of Theorem 3.7. +Appendix A. Microscopic Cole-Hopf transform +Here we explain our choice of parameters (10) which permits to obtain the discrete stochastic +heat equation as in (11). We observe that Zt(x) is affected only by exchanging values η(x), +η(x + 1). Therefore, one can check that +LZt(x) = +� +pσt(x)(1 − σt(x + 1))(e2λ − 1) + qσt(x + 1)(1 − σt(x))(e−2λ − 1) +� +Zt(x) +for any x > 0 and moreover +LZt(0) = p(1 − σt(1))(e2λ − 1)Zt(0). +Besides, the discrete Laplacian acts as: +∆Zt(x) = +�e−λ(2σt(x+1)−1) + eλ(2σt(x)−1) − 2 +�Z(x) +for any x > 0. By identification we obtain the following conditions +ν += +D(eλ + e−λ − 2) +(119) +ν += +D(2eλ − 2) − p(e2λ − 1) +(120) +ν += +D(2e−λ − 2) − q(e−2λ − 1). +(121) +which, after resolution, give +λ = 1 +2 log q +p, +D = √pq, +ν = q + p − 2√pq. +(122) +Finally, we need to define Z(−1) such that, at the boundary point x = 0, we get (νZ(0) + +LZ(0)) = D∆Z(0). 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Current fluctuations for the totally asymmetric simple exclusion process. +In and Out of Equilibrium (Mambucaba 2000) Progress in Probability, 51:185–204, 2001. +[RPSV00] +M. Rossi, R. Pastor-Satorras, and A. Vespignani. Universality class of absorbing phase transitions +with a conserved field. Phys. Rev. Lett., 85:1803–1806, Aug 2000. +[TW13a] +C. A. Tracy and H. Widom. The asymmetric simple exclusion process with an open boundary. J. +Math. Phys., 54(10):103301, 2013. +[TW13b] +C. A. Tracy and H. Widom. The Bose gas and asymmetric simple exclusion process on the half-line. +J. Stat. Phys., 150(1):1–12, 2013. +[Wal86] +J. B. Walsh. An introduction to stochastic partial differential equations. In École d’Été de Probabilités +de Saint Flour XIV-1984, pages 265–439. Springer, 1986. +[Wu20] +X. Wu. Intermediate disorder regime for half-space directed polymers. J. Stat. Phys., 181:2372–2403, +2020. +[Yan18] +K. Yang. Stochastic Burgers Equation via energy solutions from non-stationary particle systems. +arXiv:1810.02836, 2018. +[Yan22] +K. Yang. Kardar-Parisi-Zhang equation from non-simple variations on open-ASEP. Probab. Theor. +Rel. Fields, 183, 06 2022. +G. Barraquand, Laboratoire de physique de l’école normale supérieure, ENS, Université PSL, +CNRS, Sorbonne Université, Université Paris-Cité, Paris, France +Email address: guillaume.barraquand@ens.fr +O. Blondel, Univ Lyon, CNRS, Université Claude Bernard Lyon 1, UMR5208, Institut Camille +Jordan, F-69622 Villeurbanne, France +Email address: blondel@math.univ-lyon1.fr +M. Simon, Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille +Jordan, F-69622 Villeurbanne, France +Email address: msimon@math.univ-lyon1.fr + diff --git a/JdE3T4oBgHgl3EQfvAtY/content/tmp_files/load_file.txt b/JdE3T4oBgHgl3EQfvAtY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aed7571a18fea5656137f4441ef8a050a565377e --- /dev/null +++ b/JdE3T4oBgHgl3EQfvAtY/content/tmp_files/load_file.txt @@ -0,0 +1,1239 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf,len=1238 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='04689v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='PR] 11 Jan 2023 WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS GUILLAUME BARRAQUAND, ORIANE BLONDEL, AND MARIELLE SIMON Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' We consider the facilitated exclusion process, an interacting particle system on the integer line where particles hop to one of their left or right neighbouring site only when the other neighbouring site is occupied by a particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' A peculiarity of this system is that, starting from the step initial condition, the density profile develops a downward jump discontinuity around the position of the first particle, unlike other exclusion processes such as the asymmetric simple exclusion process (ASEP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' In the weakly asymmetric regime, we show that the field of particle positions around the jump discontinuity converges to the solution of the multiplicative noise stochastic heat equation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' the exponential of a solution to the KPZ equation) on a half-line subject to Dirichlet boundary condition, with initial condition given by the derivative of a Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' We prove this result by reformulating the problem in terms of ASEP on a half- line with a boundary reservoir, for which we extend known proofs of convergence to deal with Dirichlet boundary condition and the very singular type of initial condition that arises in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Facilitated exclusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' The facilitated exclusion process (FEP) was introduced in the physics literature [RPSV00] as a representative of a universality class for absorbing phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' It is an interacting particle system on a lattice in which particles can jump to empty neighbors provided there is a particle in their neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' So far our understanding of this model is restricted to dimension 1, where it has been studied under different lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Its main feature is the absorbing transition mentioned above, at the critical particle density 1/2: for particle densities below 1/2 (subcritical regime), the system fixates on a configuration with isolated particles which cannot move, while for densities above 1/2 (supercritical regime) it remains active forever and holes become eventually isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' The totally asymmetric version of this process (where particles only jump to the right) has been studied in [BM09] (approach to the phase transition) and [CZ18,GLS19,GLS21] (identifica- tion of the stationary states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Starting from a step initial condition, contrary to the well-studied totally asymmetric simple exclusion process (TASEP), a downstep leads to a rarefaction fan with a discontinuity [GKR10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' For the same initial condition, at large time t, particle posi- tions fluctuate on the t1/3 scale with Tracy-Widom GUE statistics, while the fluctuations of the rightmost particle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' at the discontinuity, have Tracy-Widom GSE statistics [BBCS18a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' The symmetric FEP has been studied as well, on the periodic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' It was found [BESS20,BES21] that in the diffusive space-time scaling, and under the hydrodynamic limit, the macroscopic density ρ evolves according to a Stefan problem written as ∂tρ = ∂uu � 2ρ−1 ρ 1ρ>1/2 �, with the space variable u belonging to the one-dimensional torus of size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' In other words, starting the microscopic dynamics from a density profile with both supercritical and subcritical regions, the diffusive supercritical phase progressively invades the subcritical phase via moving interfaces, until one or the other phase disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' For the partially asymmetric version, where particles jump to the right at rate p < 1 and to the left at rate q < p, invariant measures have been characterized on the torus [GKR10] and on the line [AGLS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Recently, [ESZ22] showed that the hydrodynamic limit (in the hyperbolic scaling) is given by the unique entropy solution of ∂tρ + (2p − 1)∂x � (1−ρ)(2ρ−1) ρ 1ρ>1/2 � , with x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' The next natural question concerns the fluctuations of the FEP in the asymmetric case (FASEP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' As it has been noted for the totally asymmetric case in [BBCS18a], the problem can be reformulated in terms of the asymmetric simple exclusion process (ASEP) on the half- line with jump rates p > q, and with a specific boundary condition, where particles enter the 1 WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS 2 system at a rate p and can never exit (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' For fixed p and q, we expect that, up to scaling constants, the fluctuations should be similar as for the totally asymmetric FEP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' In the present paper, we study the fluctuations of particle positions in the FASEP in a weakly asymmetric asymptotic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' In the bulk (that is, far from the jump discontinuity of the density profile), we do not expect that the facilitation rule will have any effect on the scaling limit of fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' In particular, we expect that if the asymmetry is properly scaled with time, the field of particle positions, appropriately rescaled, should converge to the solution of the Kardar-Parisi-Zhang (KPZ) equation on the real line with narrow wedge initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' This is consistent with the Tracy-Widom GUE asymptotics observed in the totally asymmetric setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' However, for the step initial condition, if we consider the field of particle positions in the FASEP around the first particle, the situation is more interesting and the facilitation rule plays a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' The limit should be described by a stochastic PDE on a semi-infinite interval with a specific boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' The main goal of the present paper is to describe this stochastic PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' KPZ equation and Hopf-Cole transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' In order to state our main result, let us first recall how to solve the KPZ equation on the full one-dimensional line, which reads as ∂th = 1 2∂uuh + 1 2(∂uh)2 + ξ, (1) with ξ the standard space-time white noise on R+ × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' One usually considers the Hopf-Cole transform of a putative solution h, namely Z(t, u) = eh(t,u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' If we apply the chain rule in (1), ignoring all issues of regularity, the function Z solves the Stochastic Heat Equation (SHE) with multiplicative noise ∂tZ = 1 2∂uuZ + Zξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' (2) The latter equation can be solved through standard SPDE techniques, and whenever it can be shown that Z > 0 [Mue91], this procedure yields a Hopf-Cole solution to (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' In their seminal paper [BG97], Bertini and Giacomin noticed that the discrete Hopf-Cole transform (introduced by Gärtner [Ga87]) Zt(x) := e−λht(x)+νt (3) of ASEP height function ht(x) (x ∈ Z), satisfies, for well-chosen parameters λ, ν, a martingale problem that is a discrete analogue of the martingale problem satisfied by Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' In the weakly asymmetric regime, say p = 1 2eε, q = 1 2e−ε with 0 < ε ≪ 1, as assumed in this paper, it can then be showed that solutions of the discrete martingale problem converge to solutions of the continuous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Given uniqueness of the solution to the continuous martingale problem, this is enough to conclude that ASEP height function, suitably rescaled, converges to a solution of the KPZ equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Further, [DT16] identifies a whole class of models to which this method may apply, in the sense that a generalization of the discrete Hopf-Cole transform can be found and convergence to the SHE can be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Other approaches to solving (1) that do not require a detour through the SHE were also looked for in the last decades, and can be useful in cases where there is no applicable discrete Hopf- Cole transform (unlike the present paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Let us mention regularity structures [Hai13,Hai14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' energy solutions [GJ14,GP18], which have been applied e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' in [BGS16,GJS17,GPS20,Yan18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' and paracontrolled distributions [GIP15,GP17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' KPZ equation on the positive half-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' A half-space analogue of the result from [BG97] was proved in [CS18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' More precisely, under some condition on injection and ejection rates at the origin of ASEP on the half-line, and assuming that the effective density at the origin imposed by those rates scales as ρ ≈ 1 2(1 + (A + 1 2)ε), [CS18] shows that the height function of ASEP on the half-line converges to the KPZ equation on R+ with Neumann type boundary 1Proving this would however require an analogous statement for ASEP on a half-line, and this has so far remained out of reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Indeed, some exact formulas characterizing the distribution of particles have been proved in [TW13b, TW13a], and more recently in [BC22a], but these formulas are not amenable for asymptotic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' There exists one exception where the fluctuations of half-line ASEP have been analyzed asymptotically, in [BBCW18], but the boundary condition considered in that paper is different from the one which is relevant for the study of the FASEP, and the methods used there crucially depend on this specific boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS 3 condition ∂xh(t, 0) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' As in [BG97], this result is proved via the discrete Hopf-Cole transform (3) of ASEP height function ht(x) (x ∈ N), which is shown to converge to the SHE on R+ with Robin type boundary condition � ∂tZ = 1 2∂uuZ + Zξ, ∂uZ(t, 0) = AZ(t, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' (4) Since Z(t, ·) is not differentiable, the boundary condition should rather be imposed on the half- space heat kernel which is used to define the solution, we refer to [CS18] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' The result of [CS18] was restricted to A ⩾ 0 and near equilibrium initial conditions (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='8 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' It was then extended to all A ∈ R and the empty initial condition in [Par19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Let us note that on the full-line, the extension of the convergence result of [BG97] to step initial condition was first discussed in [ACQ11], with considerably less details than in [Par19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Some of these results were further extended in [Yan22] to generalizations of ASEP, in the spirit of [DT16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' An alternative way to make sense of the KPZ equation on a half-line via regularity structures was also considered in [GH19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' In the physics literature, the solution Z to the SHE is understood as the partition function for a continuous Brownian directed polymer in a white noise potential ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' The boundary parameter A can then be understood as controlling an extra energy collected by polymer paths, given by reflected Brownian motions, along the boundary [BBC16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Alternatively, we may assume that there is no extra potential on the boundary, but the Brownian paths in the polymer partition function have elastic reflection on the boundary controlled by the parameter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Discrete directed polymers are another family of models, with exclusion processes, which converge to the KPZ equation in full-space [AKQ14] or half-space [Wu20,Par22,BC22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' The KPZ equation (and the associated SHE via Hopf-Cole transform) has also been considered on an interval with Neumann type boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' The discrete Hopf-Cole transform of open ASEP height function satisfies Robin type boundary conditions [GLM17] and converges to the KPZ equation on an interval [CS18,Par19,GPS20,Yan22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' As previously mentioned, the problem of fluctuations of the FASEP reduces to the fluctuations of ASEP on the half-line with, at the origin, injections of particles at the right-jump rate p and no ejection of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' This translates into a microscopic boundary condition for the discrete Hopf-Cole transform (3) given by Zt(−1) = µZt(0), with µ ≈ 1 − ε, as stated more precisely in (11) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' In contrast, in the setting of [CS18,Par19], the boundary parameter µ is scaled as µ ≈ 1 − ε2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' We will prove below that in our case, the appropriate scaling of the solution is different from [CS18, Par19]: we will set Zε t (u) = ε−2Zε−4t(ε−2u) for any macroscopic point u, and prove that starting from the step initial condition for the FASEP (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' empty initial condition for the half-line ASEP), Zε t (u) weakly converges as a continuous process (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='7 for a precise statement) to the solution of the SHE \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ∂tZ = 1 2∂uuZ + Zξ, Z(t, 0) = 0, Z(0, u) = −2δ′ 0(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' (5) We also prove a similar statement for near-equilibrium initial conditions (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='9), under the same scaling as in [CS18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' The SHE with Dirichlet boundary condition (Z(t, 0) = 0) was already considered in [Par22] in the context of directed polymer models, but only for another type of initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Here the initial condition that we consider is very singular, this is the derivative of a delta function at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' We provide a more precise definition of this stochastic PDE in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='3 and prove existence and uniqueness of the solution in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Dealing with this very singular initial condition is one of the main novelties of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Eventually, our main result is therefore the following: the macroscopic fluctuations of the field of the first particles’ positions in the weakly asymmetric FASEP are given (via the Hopf-Cole transform) by the solution to the KPZ equation on the half-line R+, with initial condition being the derivative of a Dirac distribution, and with Dirichlet boundary condition at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Comparison with previous literature on half-space KPZ equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' In a sense, the microscopic boundary condition Zt(−1) = µZt(0) with µ ≈ 1 − ε that we are considering is the A → ∞ limit of the setting in [CS18, Par19] which considered µ ≈ 1 − Aε2, and it poses no additional difficulty in terms of handling the boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' However, as already said, our initial condition is much more singular than the ones in [CS18,Par19,Par22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Proving tightness requires us to develop precise estimates about the Dirichlet heat kernel and its discrete analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Further, the rigorous identification of the initial condition requires some control on the second moment of Zε t (u), and the estimate that we need turned out to be surprisingly difficult to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Eventually, we use an explicit second moment integral formula for half-line ASEP from the recent preprint [BC22a], coming from a Markov duality (such use of Markov duality also arises in the proof of convergence of the stochastic six-vertex model height function to the KPZ equation in [CGST20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' The continuous directed polymer model corresponding to the solution of (5) was considered in the physics paper [GLD12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' The polymer paths are conditioned not to hit the boundary, and [GLD12] studied the distribution of the partition function of polymers starting and ending at a location η, after letting η → 0 and appropriately rescaling the partition function by η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Restricting on test functions f : R+ → R such that f(0) = 0, the distribution 1 ηδ0(· − η), where δ0 is the delta Dirac distribution, converges to −δ′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' This explains why the initial condition that we consider in the present paper is the physically natural one to consider for the SHE with Dirichlet boundary condition, though the solution was never mathematically constructed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Finally, let us mention that the KPZ equation on a half-line with Dirichlet boundary condition is also considered in [GH19], but there the Dirichlet boundary condition h(t, 0) = 0 is imposed on the KPZ equation itself and not on the SHE, so that this corresponds to a completely different stochastic PDE than the one we consider in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Outline of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' In Section 2, we define the FASEP and ASEP and construct a mapping that connects the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Section 3 is devoted to the presentation of the Hopf-Cole trans- form, in terms of which we state our main results on the fluctuations of the particle positions in a weakly asymmetric regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' We then provide several preliminary results: first, the existence and uniqueness of the macroscopic SHE with δ′ 0 initial condition and Dirichlet boundary condition (Section 4), and second, some explicit estimates on the discrete heat kernel with diverging Robin boundary condition (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Finally, the main convergence results cover two types of initial conditions which require different scalings: the step (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' derivative of delta) initial condition (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='7) and near-equilibrium initial conditions (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='9), which both require new arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' We actually need the understanding of the latter as an intermediate step towards Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='7, so we start by proving Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='9 in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Our main result, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='7 is finally proved in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' This project is partially supported by the ANR grant MICMOV (ANR- 19-CE40-0012) of the French National Research Agency (ANR), and by the European Union with the program FEDER “Fonds européen de développement régional” with the Région Hauts- de-France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' It has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovative program (grant agreement n° 715734), and from Labex CEMPI (ANR-11-LABX-0007-01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' This article is also partially based upon work supported by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' DMS-1928930 while G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2021 semester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Microscopic models and mapping In the following, N denotes the set of non-negative integers, N∗ the set of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' The facilitated asymmetric exclusion process in dimension one (FASEP) is a Markov process on the state space Ω := {0, 1}Z which is denoted by {ηt(x) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' x ∈ Z}t⩾0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Each component ηt(x) ∈ {0, 1} is the occupation variable of the configuration of particles at site x ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' WEAKLY ASYMMETRIC FACILITATED EXCLUSION PROCESS 5 Let p, q ∈ (0, 1) be two asymmetry parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' The time evolution of the particle configura- tions is ruled by the Markov generator LF which acts on functions f : Ω → R as follows: LF f(η) = � x∈Z pη(x − 1)η(x)(1 − η(x + 1)) �f(ηx,x+1) − f(η) � + � x∈Z qη(x + 1)η(x)(1 − η(x − 1)) �f(ηx,x−1) − f(η) � (6) where ηx,y is the configuration obtained from η after the exchange of the occupation variables η(x) ↔ η(y), namely: ηx,y(z) := η(x)1z=y +η(y)1z=x +η(z)1z /∈{x,y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' In other words, as it can be read on the generator, particles are displayed on the lattice Z and jump to their neighbouring sites at rates which encode the following rules: a jump to the right from site x to site x + 1 occurs with rate p if and only if site x is occupied by a particle, site x + 1 is empty and site x − 1 is occupied ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' a jump to the left from site x to site x − 1 occurs with rate q if and only if site x is occupied by a particle, site x − 1 is empty and site x + 1 is occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' We say that site x ∈ Z is occupied by an active particle if it can jump either to x − 1 or x + 1 with positive rate, in other words, it is such that η(x − 1)η(x)(1 − η(x + 1)) + (1 − η(x − 1))η(x)η(x + 1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Let us now map this process onto another exclusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' We need to introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' Let L < R be two integers, and define: ΩL,R := � η ∈ Ω ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 η(x) = 0 if x ⩾ R, η(x) = 1 if x ⩽ L, η(x) + η(x + 1) ⩾ 1 if x < R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE3T4oBgHgl3EQfvAtY/content/2301.04689v1.pdf'} +page_content=' � and Ω := � L 3σ +significantly positive for NQ,tot ≥ 4. The slope is roughly 0.1 +at this point, but four problems over the course of a full term +is a very minimal amount of practice, and grade earned at +this level is indeed lower than the T-conditioned mean by 0.2 +grade points. Gains in the top panel become evident above +ten questions, and the slope continues to grow, reaching 0.3 at +NQ,tot ≃ 300. This study volume represents roughly 25 ques- +tions per week, or five questions per school day. The positive +slope in grade gain continues up to the limit of our study vol- +ume data. In our sample, 34 students, or 0.5% of the popula- +tion, attempted over 1000 questions within a semester, a vol- +ume that slightly oversamples the number of questions avail- +able and that corresponds to a rate of roughly 15 questions per +school day. +Table VI quantifies for each study volume indicator (NQ,tot, +Nsess, and NQ,mean) the overall grade gain, max(∆GPE), de- +fined as the maximum difference in KLLR mean values across +TABLE VI. Maximum study gains, max(∆GPE), between high and +low PR study volume (see text) for each study measure along with the +fractional reduction in grade variance after accounting for both T and +study volume. The latter is calculated as ∆Var0→i ≡ (σi2−σ02)/σ02, +where σ0 is the scatter in GPE originally and σi is the scatter in grade +point after removing trends in µGPE(T) (i = 1) and both that and +⟨∆GPE⟩(X) for study measure X (i = 2). +Indicator +max(∆GPE) +∆Var0→1 +∆Var0→2 +NQ,tot ++0.78±0.12 +−0.16±0.03 +−0.19±0.03 +Nsess ++0.62±0.19 +−0.16±0.03 +−0.18±0.03 +NQ,mean ++0.71±0.11 +−0.16±0.03 +−0.19±0.03 +the study volume domain. Uncertainties in the difference are +propagated from the respective KLLR bootstrap errors on the +minimum and maximum values. The table also quantifies the +fractional reduction in variance ∆Var from the initial scatter in +GPE σ0 to the scatter after accounting for µGPE(T) (∆Var0→1) +and after additionally accounting for ⟨∆GPE⟩(X) in study mea- +sure X (∆Var0→2). The reduction in variance from the middle +to end stage is then ∆Var1→2. +For each study indicator, students with high study volume +tend to do significantly (> 3σ) better than those with low +study volume. Across the three study indicators, we see simi- +lar gains (within 1σ of each other) of 0.74±0.14 grade points. +This result is consistent with a two (minor) letter grade in- +crement, for example, moving from a B- to a B+ or from a +B+ to an A. These results are also robust with varying ker- +nel size: doubling or halving kernel width yields statistically +identical results. Furthermore, the results are robust with vary- +ing student math ability. Dividing the student sample into ter- +ciles by T, we find similar overall grade gains: 0.60 ± 0.21, +0.73±0.19, and 0.75±0.12 for the lowest, intermediate, and +highest T terciles, respectively, averaged across study indica- +tors. All students benefit similarly from increased study vol- +ume. +Across each study indicator, we see similar reductions of +variance, with ∼ 20% of the scatter in student grades ex- +plained by the joint µGPE(T) and ⟨∆GPE⟩(X) trends. +The +trend of mean grades running with T accounts for the ma- +jority of this reduction, causing on its own a 16% ± 3% re- +duction. The range of µGPE(T) is roughly 1.2 grade points, so +the running of ⟨∆GPE⟩(X) with maximal running of roughly +half the range has less capacity to reduce variance. The dis- +tribution of NQ,tot also clusters strongly towards lower values, +even further limiting its ability to reduce variance. However, +on sampling evenly in logNQ,tot, we find a significant variance +reduction on additionally including study volume in the model +of ∆Var1→2 = (9 ± 3)%. This suggests that our initial inabil- +ity to measure significant reduction of variance on accounting +for study trends is tied to the clustering of NQ,tot at relatively +modest values. Below, we incorporate both math test score +and study volume into a linear model of student grade. +We now turn our attention to several demographic subpopu- +lations, investigating relationships to mean trends in the space +of math test score, study volume, and final grade. + +Practice Makes Better +7 +B. +Demographic differences +2.6 +2.8 +3.0 +3.2 +GPE +Female +Male +HS +A,B +M,D +both +single +URM +NUR +ITL +1st +2nd +3rd +4th +sex +parental education +parents at home +N/URM +income bins +0.86 +0.88 +0.90 +0.92 +0.94 +0.96 +T +50 +60 +70 +80 +90 +100 +110 +NQ, tot +Female +Male +HS +A,B +M,D +both +single +URM +NUR +ITL +1st +2nd +3rd +4th +FIG. 4. +Points show mean GPE, NQ,tot, and T for the various de- +mographic subpopulations indicated in the legend, with error bars +showing standard deviation of the mean. Quadratic fits of GPE and +NQ,tot as functions of T for the full student population, shown in blue +with ±1, 2 and 3σ uncertainties, demonstrate starkly different trends. +Grades strongly correlate with T while study volume does not. +Differing educational experiences exist across lines of so- +cioeconomic status, sex, race, ethnicity, home environment, +and other factors.10–12,20,21 As detailed in §II C, we consider +several of these demographic indicators (see Tables III & IV) +to the extent and precision available to us; most indicators +are self-reported. We investigate the degree to which demo- +graphic groups are stratified in the space of T, GPE, and NQ,tot, +then quantify how each sub-group’s mean grade deviates from +expectations. +Figure 4 illustrates the relationship between mean measures +for demographic groups, shown as points, and the overall +trend for all students, shown as the line (with shaded regions +indicating ±3σ uncertainty on the fit). For this analysis, we +measure mean NQ,tot values for all students for whom we have +T scores, including those for whom NQ,tot = 0. Note that the +mean value, ⟨NQ,tot⟩ ≃ 72, differs from the median log value of +40 shown in Fig. 3 because the distribution of NQ,tot is close to +log-normal and has substantial width. This level of effort cor- +responds to attempting roughly five questions per week during +the term. +Stark differences exist between GPE versus NQ,tot in their +trends with T. +Mean grade earned (top panel) correlates +strongly with math test score (0.38 ± 0.01), rising by half a +grade point over just a 0.1 increase in T. In stark contrast, the +volume of study by students (lower panel) is effectively flat +(insignificantly correlated: 0.02±0.01); low T students study +very nearly as much as their high T counterparts. We return to +this issue when considering GPAO as an alternate to T below. +Mean values of math test score T for demographic groups +cover the 0.1 range shown, with underrepresented minority +(URM) students and students whose parents have the lowest +education level (≤HS) possessing the lowest scores and in- +ternational students the highest. These groups lie somewhat +off the overall trend in grade, with ITL students lying above +and URM and ≤HS below the population mean. Note that, +because of the positive curvature in mean grade with T, mean +values of sub-populations will tend to lie somewhat above the +trend line. +We find no global trend in study volume with T; students +across the spectrum of math test scores put in similar level of +practice effort. Differences between demographic groups are +apparent in Figure 4, with URM and ≤HS populations lying +below the trend while high income and international students +rest above it. +Do Math Score and Study Volume Explain Demographic +Grade Shifts? +The trends in the bottom panel of Figure 4 suggest an expla- +nation for demographic trends for grade deviation from mean +trends; vis. that the demographic shifts in study volume often +have similar sign to that of ∆GPE, the deviation of GPE from +µGPE(T). However, these shifts sometimes misalign; for ex- +ample, the highest income group tends to study more than av- +erage yet achieves only average grades. To address the degree +to which study explains demographic deviations from mean +trends, we employ a model that is a linear combination of the +expected mean grade conditioned on T (equation (2)), com- +bined with the grade shift as a function of study volume using +the full student population (the KLLR fit of Figure 3, largely +equivalent to the parameterized fit of equation (5)). +Figure 5 shows the mean deviation from these expectations +for the demographic sub-populations. The zero line here rep- +resents the expected value based on our model of the full stu- +dent population, and the shaded regions show median ±3σ +uncertainties in the theory mean (essentially the combined un- +certainties of the quadratic and KLLR fits for a median stu- +dent). Most demographic sub-populations have mean devia- +tions consistent with the model’s expectations, but significant +outliers are seen toward lower earned grade. Students whose +parents had a high school degree or less lie significantly be- +low expectations, at −0.27 ± 0.04 grade points, as do URM +students, at −0.14 ± 0.04 grade points. In addition, students +from households with only a single parent present as well as +students from high schools in the lowest income regions dis- +play a smaller, less significant deficit of −0.10 ± 0.04 grade +points. All other sub-populations were < 2σ deviant from ex- +pectations. + +Practice Makes Better +8 +Female +Male +HS +A,B +M,D +single +both +URM +NUR +ITL +1st +2nd +3rd +4th +0.3 +0.2 +0.1 +0.0 +0.1 +deviation +sex +parental education +parents at home +N/URM +income bins +FIG. 5. Mean GPE deviations of demographic groups from the over- +all trend based on combining math score T and study volume NQ,tot +for the full student population. The shaded region around the null +line shows ±3σ uncertainties of the predictions based on combin- +ing T and NQ,tot. The error bar for each demographic group is the +±1σ error in that group’s mean GPE deviation from the mean model +prediction. +C. +Comparison to GPAO +In this section, we investigate results when using GPAO as +a baseline for comparison, rather than T. First, we explain +some limitations of GPAO (with more issues outlined in Ap- +pendix B), Second, for PHYSICS 140 and 240 we show the +µGPE(GPAO) plot analogous to Figure 2. Third, we show for +the two courses combined a gains of study table, analogous to +Table VI. +1. +GPAO correlation with study +A core issue with using GPAO as a baseline for com- +parison (as is done, for example, with the “grade anomaly” +GPE − GPAO of W20) is its correlation with study habits. +Averaged over log or non-logged versions of all three study +volume indicators used in this study (a total of six possible +metrics), correlations between study and GPAO were statisti- +cally significant, 0.132±0.013, whereas correlations between +study and T were insignificant, only 0.013±0.013. For each +of the six possible metrics, study correlated with GPAO > 5σ +significantly. For T, while NQ,tot was consistent with null cor- +relation (0.014 ± 0.015), Nsess sightly favored a negative cor- +relation (−0.027 ± 0.014), and NQ,mean favored a slight posi- +tive correlation (0.054 ± 0.012). Regardless of study metric, +GPAO consistently shows stronger and more significant cor- +relations to study than T does. Because we seek to measure +how study volume influences final grade, this coupling makes +GPAO relatively untenable compared to T for this purpose. +Figure 6 shows the interplay between PR study volume, +GPAO, and composite test score T for PHYSICS 140 and 240 +combined. KLLR fits to study volume as a function of GPAO +show clear rising behavior for each individual T bin, while +2.75 +3.00 +3.25 +3.50 +3.75 +4.00 +GPAO +0 +50 +100 +150 +NQ, tot +T|[0.7, 0.8) +T|[0.8, 0.9] +T|(0.9, 1.0] +FIG. 6. +Study volume as a function of GPAO for students in bins +of T shown in the legend. Mean values are KLLR-derived using a +kernel width 0.40 and 1σ uncertainties are shown as shaded regions. +Population means for each T group are shown as points with ±1σ +error bars. Significant study trends with GPAO exist within each +math score range. +the means (displayed as points with error bars) show no sig- +nificant trend in study behavior with T. The striation of trend +lines shows that among students with the same GPAO, those +with lower T scores tended to study more on PR. In contrast +to the lack of correlation between T and study, at fixed GPAO +there is a significant trend of decreasing PR study volume for +students with higher math scores, T. +Though students with high T scores tend to have higher +grades and though students who studied more tended to have +higher grades, there was no significant correlation between +T and study volume. This seeming inconsistency is resolved +in the anticorrelation between study volume and T at fixed +GPAO. If we take PR study volume as a proxy for typical stu- +dent study habits, then this could perhaps be interpreted to +reveal that students with higher T scores tended to not need +to study as much to receive the same high grades as students +with lower T scores. +Noting that this correlation between study volume and +GPAO makes the latter a less reliable baseline for measuring +grade gains due to study, we move forward in this direction +with the expectation of finding a lower magnitude of grade +gain due to study volume at fixed GPAO. +2. +Fitting GPE as a function of GPAO +We begin by fitting student grades earned in each physics +class as a function of their end-of-term grade point average +in all other courses (GPAO). As in §II E, we use a quadratic +fitting for ease of comparison. +Figure 7 shows the quadratic fit of mean GPE as a func- +tion of GPAO for both physics courses individually. On split- +ting between students with and without T scores available, +we find identical fit parameterization besides normalization. +Students with T scores available tended to have higher GPAO +by 0.044 ± 0.017 grade points (2.6σ significant). All other + +Practice Makes Better +9 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +GPAO +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +GPE +PHYSICS 140 +PHYSICS 240 +FIG. 7. +As Figure 2, but fitting mean GPE for the two physics +courses as a function of GPAO rather than T. The GPAO distribu- +tion was similar between courses, with means around 3.4, standard +deviation of ±0.5, and GPAO ≳ 2.45 for 95% of students. +fit parameters were ≲ 2σ different, showing good agreement +between population dynamics of those with and without T +scores recorded. +As with µGPE(T), we see > 5σ signifi- +cantly positive curvature. +The bulk of the population has +GPE < GPAO (diagonal line), meaning that these physics +courses tend to be the lower grades on their transcripts, es- +pecially so for students with low GPAO already. For exam- +ple, a student with otherwise straight As (GPAO = 4.0) would +likely get an A- while a student with otherwise straight Bs +(GPAO = 3.0) would likely get a C+. +3. +Deviations from GPAO mean due to study volume +After calculating a deviation from expected grade at a given +GPAO, ∆ = GPE − µGPE(GPAO), we investigate whether a +trend exists in this deviation as a function of PR study vol- +ume, using KLLR fitting, analogous to that done in Figure 3. +We present gains of study in Table VII, analogous to Table VI. +Those with highest study volume tended to do better than ex- +pected by 0.27 ± 0.07 grade points (averaging over the three +study indicators) as compared to those with low study volume. +TABLE VII. As Table VI, but measuring gains and variance reduc- +tion with GPAO as a baseline for grade expectations (rather than T). +Indicator +max(∆GPE) +∆Var0→1 +∆Var0→2 +NQ,tot ++0.30±0.08 +−0.46±0.01 +−0.47±0.01 +Nsess ++0.20±0.06 +−0.46±0.01 +−0.47±0.01 +NQ,mean ++0.33±0.06 +−0.46±0.01 +−0.47±0.01 +This is less than half the grade gain measured in Table VI, +where T was used as a baseline for comparison. The stark dif- +ference in grade gains measured suggests that the significant +correlation between GPAO and study volume (see §III C 1) +has subsumed a large portion of the grade gains measured. +That is, because GPAO reflects in part an individual’s study +volume, subtracting it out in the initial baseline grade esti- +mate µGPE(GPAO) removes a significant fraction of the mea- +surable grade gains due to study. Despite GPAO predicting +grades better than T (reducing variance by 47% rather than +only 19%), GPAO has less utility in quantifying the grade ben- +efits of study, washing out over half of the measurable grade +benefit. +IV. +DISCUSSION +Our findings help demonstrate to educators and learners +the benefits of practice study in introductory physics courses. +While this finding is neither original nor surprising, the overall +magnitude in grade gain for students classified by pre-college +math ability is large. +We acknowledge that there is a good match between the +nature of the PR practice, which constitutes low-stakes for- +mative assessment, and the summative midterm and final ex- +aminations. After all, PR consists of nearly 1000 problems +used on past examinations in each course. +Given the me- +dian response time per problem of roughly 90 seconds3, we +do not think that memorization per se plays an important role. +Rather, other factors, such as the similarity of problem con- +struction and structure, the mix of quantitative and qualita- +tive questions, and particular “tricky” problem styles involv- +ing multiple physics concepts, is likely to be similar between +old and new exams. +Timed, multiple choice examinations have been criticized +as an obstacle to higher-level thinking22 and as a poten- +tial causal factor in gender inequality.23 Mixed forms of +assessment24 may improve inequity and promote higher-level +thinking. +Our findings with respect to demographic characteristics +motivate more study of how better to support physics learn- +ing for first-generation (our ≤HS category) and underrepre- +sented minority students. +Students from single-parent and +low income households also earn lower grades than expected +(Fig. 4). These categories overlap, inviting future work to +understand how students with multiple intersecting identities +perform.25 +Our comparison to GPAO study trends reveals far more +grade benefit measured when conditioning on T than on +GPAO. Because GPAO significantly correlates with study vol- +ume (while T does not), subtracting out a GPAO baseline from +grades subtracts out the effects of study behaviors to some ex- +tent. While GPAO may be the most accurate predictor of GPE, +it is a flawed baseline for the purposes of our analysis (see +also Appendix B) as it washes away the majority the the study +benefit trend measured in §III A. In essence, we find utility in +using T as a baseline for student comparison. +The utility of T could in part be due to its correlation +with general mental ability g.26–29 As g is largely intrinsic +(polygenic, yet ≳ 50% heritable and ≳ 90% consistent with +age30–33) to each student, it is largely orthogonal to a stu- +dent’s external, personal choice of study habits. In contrast, +GPAO represents a mix of both internal ability and external +choice, and thus subsumes a portion of the study signal we +attempt to measure in this study. Regardless of interpretation, + +Practice Makes Better +10 +this difference in grade gains (measured with GPAO vs. T +baselines) warrants further investigation and discussion. As +more colleges move to “test-optional” admissions in the fu- +ture, we may in the process be limiting the potential of future +investigations to quantify student learning gains. +Another potential reason for the strong utility of T could lie +in the similarity of assessment styles. Both standardized tests +and the assessment methods used in many large introductory +STEM courses take the form of high-stakes multiple-choice +questions in a timed setting. +V. +CONCLUDING SUMMARY +Using a large sample of practice study from the Prob- +lem Roulette service, we find that student final grades scale +quadratically with the logarithm of the term-aggregated num- +ber of questions encountered, with an overall gain of nearly +0.8 grade points. By comparison, the largest demographic de- +viation we find is ≲ 0.3 grade points, so with roughly a ques- +tion per day on average, this deficit can be overcome purely +through study. +We summarize our findings in the following advice to stu- +dents and teachers. Our advice to students is: +Do a problem for every time you brush your teeth. +Doing at least one or two problems a day each +school day of the term will likely give you a quar- +ter letter grade raise. Quadratic gains with log +study mean that doubling study volume from one +to two questions per week benefits you, but dou- +bling from two to four questions per week benefits +your grade even more! +Our advice to teachers is: +Differences between demographic groups, once +accounting for T and study, were ≲ 0.3 grade +points, with the largest outlier by far being first- +generation students. Study can benefit students +by up to three-quarters of a letter grade. Encour- +aging students to do at least one problem a day +puts them above median study habits, on average +putting them above expectations at a given T. +This study only scratches the surface of PR data avail- +able. Besides only working with physics courses, our model +was relatively simple, treating T and GPE separately and us- +ing other demographic, academic, and study-related variables +only tangentially. A more nuanced model such as multi-level +modeling or simultaneous fitting of many variables at once +(such as with the machine learning tool SHAP34) could help +disentangle which factors are more or less causal (though +it could not determine absolute causality). Future analyses +should also treat nationality, ethnicity, and race more care- +fully, rather than using the broad categories of nationality +and binary URM status. We also would like to investigate +the effects of study session length—early results suggest that +working for more than about 50 minutes tends to yield mini- +mal gains. Finally, we wish to compare findings between all +courses, observing which trends persist cross-subject. +We should never lose sight that the purpose of this work +is to help individual students. Teachers and students alike +can benefit from understanding how T and study affect their +grades. Teaching is a handshake, requiring earnest participa- +tion of both parties for best results. (We can’t fall into the trap +of subscribing to either a student deficit model or a teacher +deficit model alone; both parties need to improve their habits +and grow as individuals.) As students improve their scholastic +habits and teachers improve course structure and learn how to +best reach struggling individuals, we can grow together and +improve the education system. +Appendix A: Problem Roulette structure +After logging in and selecting a course, students can choose +from various modes of study: Individual, Group, or Practice +Exam (student-generated or faculty-generated as formal prac- +tice exams). Problem Roulette also collects questions from +topics that students struggle with most, which students can +study from. +In the Individual or Group study modes, students select +which topics to study (e.g., “Vector Algebra”, “Relative Mo- +tion”), whether or not to use a timer for the session, and how +many questions to pull (defaults: 10, 25, all). They are then +presented with an exam-like set of questions one at a time. Af- +ter the session, they are presented with a review of the ques- +tions attempted with correct answers indicated, along with an +overall accuracy score and the session duration. +Instructors can view course analytics for a given term. At +the instructor dashboard, they see student activity (e.g., which +days saw more PR use), student accuracy versus questions an- +swered for each topic, the accuracy of answers by topic, and +the accuracy of individual questions answered. This infor- +mation can reveal which topics or types of questions students +tend to struggle with most or least. +Appendix B: Additional issues with GPAO +Though GPAO correlates more with course grade earned +(GPE) than any other indicator we investigated, it has a sev- +eral issues. The most crucial issue of its correlation with study +is detailed in §III C; here we delineate several additional is- +sues with using GPAO as a baseline for GPE comparison (vs. +T as a baseline, as used in this paper), that is, using grade +anomaly GPE − GPAO instead of the metric GPE − µGPE(T) +used herein. +1. +Relation to personality traits +Because GPAO depends on study, significant differences +exist between different demographic populations. +As dis- +cussed in Richardson, Abraham, and Bond 35, male students +tend to study less than female students, and they tend to have +a lower GPAO than female students by roughly 0.07 to 0.14 + +Practice Makes Better +11 +points, despite having significantly higher T scores on aver- +age. This means that in a class where each student got iden- +tical grades, the grade anomaly for males and females could +still differ for causes extrinsic to the course in question. +2. +Differing course selection +GPAO depends on course selection of individual students, +but course selection differs drastically by demographics, and +not all course loads are equal. If all students took similar +courses regardless of demographic group, then GPAO would +be a fair measure of “anomaly”—how differently the course +was graded compared to expectations from other courses. +However, different groups tend to take different courses, lead- +ing to systemic shifts between different demographic sub- +groups. If a student tends to take easier courses, they could +have a 4.0 GPAO, then this would allow the student more free +time to study on PR. However, a similar signal comes from +a student who, despite taking incredibly challenging courses, +works very hard and still has a 4.0 GPAO, yet then has very +little free time to study on PR. While this exemplifies why +GPAO has power in predicting GPE, it also shows that inter- +pretation of GPAO is clouded. +3. +Deletion of campus-wide trends +Grade anomaly GPE − GPAO wipes out campus-wide bi- +ases. +Expressing this numerically: if every course in the +school consistently awarded lower grades by a bias b to a +given group g versus the rest of the population p, then the +grade anomaly of that group would be +(grade anomaly)g = GPEg −GPAOg +(B1) += (GPEp −b)−(GPAOp −b) += GPEp −GPAOp = (grade anomaly)p, +so the group then has identical grade anomaly to the rest of +the population. This then erases the effects of campus-wide +biases. +ACKNOWLEDGMENTS +WKB was partially funded by an Academic Innovation +Fund award from the UM Center for Academic Innovation +(CAI). 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Kremen, “Genetic and +environmental influences on general cognitive ability: Is g a valid latent +construct?” Intelligence 43, 65–76 (2014). +34S. Lundberg and S.-I. Lee, “A unified approach to interpreting model pre- +dictions,” (2017), arXiv:1705.07874 [cs, stat]. +35M. Richardson, C. Abraham, and R. Bond, “Psychological correlates of +university students’ academic performance: A systematic review and meta- +analysis,” Psychological Bulletin 138, 353–387 (2012). + diff --git a/RdE1T4oBgHgl3EQfHgN2/content/tmp_files/load_file.txt b/RdE1T4oBgHgl3EQfHgN2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..37bd0dc87afc6055b4b4604e3e4574ac171b0371 --- /dev/null +++ b/RdE1T4oBgHgl3EQfHgN2/content/tmp_files/load_file.txt @@ -0,0 +1,888 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf,len=887 +page_content='Practice Makes Better Practice Makes Better: Quantifying Grade Benefits of Study W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Black,1 Rebecca L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Matz,2 Mark Mills,2 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Evrard1, 3 1)Departments of Physics and Astronomy & Astrophysics, University of Michigan, Ann Arbor 2)Center for Academic Innovation, University of Michigan, Ann Arbor 3)Program in Computing for the Arts and Sciences, University of Michigan, Ann Arbor (*Electronic mail: wkblack@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='edu) (Dated: 10 January 2023) Problem Roulette (PR), an online study service at the University of Michigan, offers point-free practice to students preparing for examinations in introductory STEM courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Using four years of PR data involving millions of problem attempts by thousands of students, we quantify benefits of increased practice study volume in introductory physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' After conditioning mean final grade on standardized (ACT/SAT) math test score, we analyze deviations based on student study volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We find a strong effect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' mean course grade rises quadratically with the logarithm of the total number of PR questions encountered over the term (NQ,tot), with an overall gain of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='12 grade points between 1 < NQ,tot < 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The gains are persistent across the range of math test score represented in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' A linear model for final grade using test score and study volume largely accounts for demographic stratification, including by sex, parental education level, number of parents at home, nationality / underrepresented minority status, and regional income level, with two significant exceptions: students whose parents did not earn a college degree, who earn −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='04 grade points (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='2σ) below expectations and underrepresented minority students at −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='14±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='04 points (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='7σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Residual scatter in final grade remains comparable to the maximum gain from studying more, implying that the model is far from deterministic at the level of an individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Our findings can help motivate students to study more and help teachers to identify which types of students may especially need such encouragement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' INTRODUCTION Physics is a notoriously difficult subject in the eyes of many undergraduates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='1 Introductory physics courses are typically among the most-failed courses on college campuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Ac- cording to student evaluations of teaching at our university2 the workload of the first-semester physics course is perceived as considerably heavier than the workload of introductory courses in general chemistry or statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In a study of past examination problems for these three subjects3, we found that introductory physics questions are both more complex (take a longer time to solve on average) and harder (have a lower av- erage correct response rate) than questions in chemistry and statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Asked by students how to succeed in physics, in- structors often recommend practicing more problems, among other strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We seek here to measure how such practice study affects final grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Benefits of practice study are well-documented, though some forms of study benefit students more than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In 2013, Dunlosky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 4 performed a meta-analysis on hun- dreds of studies on learning, measuring the utility of various study methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Of the ten main study methods considered, they found that practice testing (“self-testing or taking prac- tice tests over to-be-learned material”) and distributed prac- tice (“implementing a schedule of practice that spreads out study activities over time”) were of the highest utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Both methods benefited learners of different ages and abilities and were shown to boost students’ performance across many dif- ferent kinds of tasks and contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In contrast, study methods such as summarizing, highlighting, and rereading were less effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' A more recent meta-analysis5 of several hundred studies finds significant gains from testing (a term they use to include quizzing), measuring boosts in student attainment that benefit all kinds of students in similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Such find- ings inspired the creation of a study tool at the University of Michigan known as Problem Roulette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='6 Problem Roulette (PR) is an optional, points-free study service at the University of Michigan (UM) which provides students open access to a large library of locally-authored, topically-organized problems in multiple subject areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Most of the content consists of multiple-choice questions used in past examinations in introductory science, technology, engi- neering, and mathematics (STEM) courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' “Roulette” refers to both its random selection of questions (an example is shown in Figure 1) as well as its reflection of high-risk assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Students can optionally activate settings which simulate timed tests, mirroring the difficulty and stress of actual exams;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' see Appendix A for more details on study modes and instruc- tor options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' PR’s equality of accessibility to UM students sidesteps paywalls of online repositories and bypasses access limits to large banks of past exams held by exclusive student groups such as fraternities, sororities, and honor societies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Our group previously examined study behaviors and grade benefits of practice based on PR usage data from 2013 to 2017 in three subject areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Weaverdyck, Anbajagane, and Evrard 3 (hereafter, W20) found that female students worked ∼25% more problems on average than male students in all three sub- jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Female students additionally received a slightly higher grade benefit than male students from higher volumes of PR practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Using a simple binary classification determined by the median study volume in each class, moderate benefits of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='15 grade points were found for high versus low volumes of practice study in chemistry and statistics, with a weaker benefit in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' To measure these study benefits, W20 used a particular measure called “grade anomaly”, comparing each student’s grade point earned (GPE) in a particular course relative to their end-of-term cumulative GPA computed from all other arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='02927v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='ed-ph] 7 Jan 2023 Practice Makes Better 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Example Problem Roulette question as viewed on a mobile device for the course PHYSICS 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' classes (termed GPAO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='7,8 “Grade anomaly” is then simply GPE−GPAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' While measures such as GPAO are commonly used to control for student performance, “grade anomaly” masks a large portion of grade benefit from practice study due to its correlation with study habits (see §III C), so subtracting off GPAO from GPE subsumes a significant portion of other- wise measureable study gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' It is therefore not optimal for our current study, which seeks to quantify gains of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In this paper, after modeling GPE as a function of stan- dardized math exam score (concordant ACT and SAT math subscores), we quantify the extent to which deviations from the mean trend correspond to increased PR study volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We then consider five demographic characteristics: sex, parental education level, single parent status, nationality and un- derrepresented minority (URM) status, and high school zip code median income (a proxy for estimating family income).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Grade differences within each of these subgroups have been noted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='9–14 we investigate to what extent these differences are explained or exacerbated by accounting for test scores and study volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We describe our data and methods in §II before present- ing our key findings in §III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' That section begins with grade gains from practice conditioned on ACT/SAT math score (§III A) followed by an exploration of demographic differ- ences (§III B) and a comparison to grade gains conditioned on GPAO (§III C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We offer some reflection in §IV followed by some succinct recommendations for teachers and students in §V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' METHODS In this section, we introduce the data and scope of our study, including the math test score measure (T) used to model mean GPE as a base condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We also describe de- mographic groupings and the flexible population modeling method, kernel localized linear regression (KLLR), that we use to model study gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The study was determined to be ex- empt from ongoing review by our Institutional Review Board (HUM00158291).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Courses investigated Problem Roulette currently supports sixteen courses with over ten thousand unique questions available in aggregate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Our primary focus here is on the introductory, calculus-based physics sequence for scientists and engineers: PHYSICS 140 (General Physics I: Mechanics) and its continuation, PHYSICS 240 (General Physics II: Electricity and Mag- netism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' These courses are primarily (∼ 80%) taken by engi- neering students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' UM also offers a similar sequence designed for life sciences students as well as an honors sequence for physics majors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The former sequence was rebuilt during the study period and so is not yet supported by PR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The latter sequence has a much smaller enrollment which limits its sta- tistical power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' While our focus is on study in physics, in this section only we offer some context by including basic PR usage and mean grade behavior in three other PR-supported STEM courses: General Chemistry: Macroscopic Investigations and Reaction Principles (CHEM 130), Elementary Programming Concepts (EECS 183), and Introduction to Statistics and Data Analysis (STATS 250).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Usage statistics for selected PR-supported courses over a seven-semester period (Winter 2018 to Winter 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We focus on PHYSICS 140 and 240 here but include three other courses for context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Statistics reflect the ∼ 93% of students with ACT or SAT scores available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Course NQ,coursea Nstudentsb fusec ∑NQ,totd ∑Nsesse PHYSICS 140 854 3856 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='5% 348208 27471 PHYSICS 240 931 2772 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='8% 131868 10312 CHEM 130 685 5786 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0% 1036986 59358 EECS 183 953 3803 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='3% 246239 13371 STATS 250 526 9435 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='9% 614700 38242 a Number of unique questions available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' b Number of students enrolled during the study period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' c Fraction of students encountering at least one question on PR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' d Total number of questions completed per term by all students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' e Total number of sessions completed per term by all students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=" An Atwood's Machine consists of two blocks suspended over a massless, frictionless pulley with a massless string." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The blocks have masses m and 3m, as shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' What is the magnitude of the tension in the string?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 3m A mg B 4mg/3 c 2mg D 3mg /2 E 3mg SubmitPractice Makes Better 3 Table I shows PR usage statistics for the seven-semester study period beginning Winter 2018 and ending Winter 2021 (inclusive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Spring and Summer terms during this period are also included, but these have much smaller enrollments than the traditional Fall/Winter terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The student count includes only those with either an ACT or SAT scores recorded, repre- senting ∼ 93% of the full student enrollment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Because it is an optional service that is not employed for summative course assessments, not all students use PR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The usage fractions are generally high, with values ranging from 60% in PHYSICS 240 to 90% in CHEM 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Note that the student numbers are unique only within each class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' many stu- dents enroll in several of these courses over the course of their careers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The intensity of study is highest in CHEM 130, with an average of 200 questions attempted per term by PR-using students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' By this metric, study volume in PHYSICS courses is considerably lower, with value of 110 and 78 questions per term by PR-engaged students in 140 and 240, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' These values are somewhat misleading, as the distributions of student study volume measures are more log-normal than normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Median volumes are smaller by more than a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Study volume indicators TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Study volume indicators for PHYSICS classes described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Values are the 50% (median), 75%, and 95% quantiles of each quantity for a sample of students with non-zero PR study (see Table I for usage fractions) and available ACT/SAT math scores (93% of total enrollment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Indicator PHYSICS 140 PHYSICS 240 50% 75% 95% 50% 75% 95% NQ,tot 50 103 389 25 36 296 Nsess 5 9 28 3 4 21 NQ,mean 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='4 26 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='3 10 24 We use several indicators of study volume in this paper, each calculated on a per term basis and summarized in Ta- ble II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The primary measure we use is the total number of unique questions encountered per session, summed over all sessions on Problem Roulette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' This measure, NQ,tot, is given for each student individually over the full academic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' For the combined sample of PR users in both physics courses, the log-mean value is exp⟨lnNQ,tot⟩ ≃ 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='5, with roughly a fac- tor of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='3 as standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' This value includes skipped questions, which represents roughly 20% of the total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We also examine two additional measures of practice: i) the number of PR sessions, Nsess, where a new session is triggered when a student logs on and responds to at least one question, either for the first time or after a period of inactivity since responding to the last question, and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' ii) the mean number of questions encountered per session, NQ,mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Over the study period, different approaches to midterm and final examinations were taken by instructors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In some terms three evening exams, spaced by roughly one month, were held, each with typically 20 questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In other terms, bi- weekly 10-question “quizzes” were held.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In some semesters these were held during class meetings while in others (par- ticularly during the COVID-19 shutdown period) they were held in the evening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Final examinations were cumulative and typically consisted of 25 questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' All questions were multiple-choice with typically five possible responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' All ex- aminations were proctored and timed, and across the study period student scores on these assessments contributed half or slightly more to their final letter grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Overall weighted stu- dent scores map to letter grades in a manner defined at the be- ginning of the semester;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' there is no “curve” in the traditional sense of using ranked ordering to define grade boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' With that context, the Nsess values in Table II reflect the fact that PR-engaged students typically use the service once per examination, but with quite a large spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Combining both courses, we find exp⟨lnNsess⟩ ≃ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='4, with nearly a factor of three spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Five percent of students had 25 or more ses- sions over the term, a frequency of roughly two sessions per week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Sessions typically average about 30 minutes in dura- tion, during which students attempt roughly eight problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Again, the dispersion is large (factor ∼ 3), and five percent of students average 25 problems per session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Our analysis using PR-based statistics is an incomplete proxy for total study effort by students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Students do use other methods of study, including doing homework problems from their online textbook service, using tutors, or by going to on- campus sites such as the Physics Help Room and Science Learning Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' While inherently only a partial picture of student study effort, the volume of PR activity is substantial enough to offer useful insight, as we show below (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Demographic groups We examine several demographic distinctions, delineated in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Table IV gives the fractions for each group among the ACT/SAT student populations in the two physics courses of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The University of Michigan’s definition of underrepre- sented racial/ethnic minority (URM) is tied to nationality and requires some exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' As defined in the Learning Ana- lytics Data Architecture (LARC)15 Data Dictionary, students are considered international (ITL) if they are neither U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' cit- izens nor U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' permanent residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The international stu- dent population is roughly two-thirds Asian, with the plural- ity of students from China (46%) and another 5% from the Republic of Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The rest of international students come to UM from over 100 countries, each individually accounting for < 4% of the remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Of the remaining domestic student population, students are considered underrepresented if they self-identify as Black or African American;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Hispanic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Native American;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' or Native Hawaiian or Other Pacific Islander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Oth- erwise, they are considered non-underrepresented minorities (NUR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In the time period considered here, the underrepre- sented student population was chiefly composed of Hispanic students (42%), students identifying as more than one ethnic- ity or race (30%), and Black students (27%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Practice Makes Better 4 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Demographic categories and sub-groups considered in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Category Sub-group Description Sex Male, Female Sex as catalogued in LARC Parental education ≤HS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' A,B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' M,D Highest degree earned: high school degree or less;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' associate’s or bachelor’s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' master’s or doctorate Parents at home Both, Single Whether the student comes from a single-parent household N/URMa URM, NUR, ITL ITL if student is international;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' else URM=underrepresented minority, NUR=non-URM Income bin 1st, 2nd, 3rd, 4th Median income of high school ZIP in near-quartile groups, divided by values of ${50, 75, 100}k a Nationality/URM status TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Fraction of course participants in each demographic subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Course Male Female ≤HS A,B M,D Single Both URM NUR ITL 1st 2nd 3rd 4th PHYSICS 140 64% 36% 10% 29% 60% 15% 85% 15% 80% 5% 22% 31% 18% 22% PHYSICS 240 74% 26% 9% 27% 62% 13% 87% 12% 83% 5% 22% 32% 19% 21% D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' ACT/SAT math score as a control condition The standard deviation of grades achieved in these courses is large, roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='8 on the standard 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0 scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' A particu- lar student’s grade is influenced by a host of factors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' we do not seek to (nor could we) capture all these factors simulta- neously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Instead, we opt to employ a proxy for mathematical proclivity using a student’s score on the ACT or SAT mathe- matics test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Recent work14 on a large sample of introductory physics students identifies ACT math score as a strong pre- dictor of final grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Of the 7417 students enrolled in this introductory physics sequence during the study period, only 498 have neither ACT nor SAT scores recorded, meaning that 93% of the total student sample have pre-college math test scores available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In our analysis, we map ACT to SAT test scores using a 2018 concordance table from Compass Prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='16 Though frac- tions of students with SAT and ACT scores is similar, be- tween PHYSICS 140 and 240, 3% more students have SAT than ACT scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We thus minimize imputation by convert- ing scores from ACT to SAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' (Imputing in the opposite di- rection has negligible effects on results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=') In cases where both scores were present, their average was taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In cases where the same test was submitted multiple times, the highest score was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Noting that the SAT minimum component score is 200 and maximum 800, we map SAT math scores to the unit interval by defining Ti = SATi −200 800−200 , (1) where SATi is the actual, imputed, or averaged SAT math score for the ith student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The distribution of T leans heavily toward high values in our datasets, especially so in the physics courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The two physics courses considered in our study have similar distri- butions of T, with medians near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='92, 10% & 90% quantile values near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='80 & 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0, respectively, and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='44 as their mini- mum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' A recent study on grade inflation17 at our univer- sity shows regular growth in ACT and SAT scores;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' rescaling SAT and ACT scores into the unit interval to match T, we see annual gains of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='006 in scores for both tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' While this in- crease reflects a heightened selectivity of the institution, it is an order of magnitude smaller than the scatter in T and there- fore has relatively little effect over our study period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0 T 1 2 3 4 GPE EECS 183 CHEM 130 STATS 250 PHYSICS 140 PHYSICS 240 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Quadratic fits to mean grade points earned, equation (2), as a function of normalized ACT/SAT math test score for multiple STEM courses, emphasizing the introductory physics sequence (see legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Fit parameters are listed in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Shaded regions show ±3σ uncertainties on the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The lower T limits displayed are set by a limit of ten students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' For each course, we find that mean grade point earned as a function of T is well-fit by a quadratic form µGPE(T) = a0 +a1(T −T0)+a2(T −T0)2 (2) with the pivot T0 set to the physics median value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='9 in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Fits for each course are illustrated in Figure 2 with parameters listed in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Higher-order polynomial terms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=', a3 and above) generally lacked significance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=', were consistent with zero), so we favor the simplicity of a quadratic fit over a cubic or higher-order polynomial fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We note that when we measured fit parameters using only those students who did not use PR for study, we found good consistency in slopes and curvatures (hinting that study trends don’t correlate strongly with T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' see lower panel of Figure 4 Practice Makes Better 5 TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Fit constants quadratic functions µGPE(T) (equation 2) for each course, giving the mean trend of GPE (course grade points earned) as a function of T (concordant ACT / SAT math subscores), determined using all students with ACT or SAT scores recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Given for each class individually as well as the merged set of PHYSICS 140 + 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Course a0 a1 a2 PHYSICS 140 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='2 +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='2 PHYSICS 240 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='2 +8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='7 MERGED 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='1 +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0 CHEM 130 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='53±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='1 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='5 EECS 183 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='63±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='7 STATS 250 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='1 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='5 and Figure 6), but intercepts were lower by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='097 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='035 grade points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' This is expected, as in total, those who didn’t study on PR in PHYSICS 140 had lower grades by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='14±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='04 points and in PHYSICS 240 by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='03 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' These sig- nificant shifts hint already at the benefits of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' For the two physics courses we found no significant dif- ferences in slope or curvature for any demographic distinc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Compared to the global parameterization of Table V, sub-population fits for curvature were consistent to < 2σ and slopes were consistent to < 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In contrast, there were sev- eral significant differences in offset a0 for three groups, with the ≤HS group falling ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='3 points below expectations while URM students and students of single-parent households both fell ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='1 points below expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' These differences in sub- group a0 do not detract from our analysis, as these significant vertical shifts are precisely what we set out to investigate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' As is visually apparent from Figure 2, student grades in physics follow a different form compared to the other subject areas represented there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Compared to the non-physics courses, the two physics courses have intercepts at T0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='9 lower by ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='7 grade points, the local slopes are nearly twice as large, and they have positive curvatures (a2 ∼ +7) whereas the other courses display negative curvatures (a2 ∼ −3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In essence, even students with the strong math abilities encounter a larger grade penalty in physics compared to these other subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Measuring study gains with KLLR We are now poised to address the question: “For a given student, how do PR study habits relate to their final course grade?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Our analysis focuses on the shift in grade earned by a given student, ∆GPE,i ≡ GPEi − µ(Ti), (3) from the expected mean given by equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We look to measure how mean shifts in student grade depend on some measure of study volume X, meaning we seek ⟨∆GPE⟩(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' To examine this mean relationship, we employ Kernel- Localized Linear Regression18,19 (KLLR), a method that de- termines parameters of a locally linear fit (mean, slope, and variance) within a sliding Gaussian window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' This approach to population modeling allows for more nuanced analysis than polynomial fitting as it does not enforce a particular global behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Below we primarily employ X = log10 NQ,tot as the study volume measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' As the center of the sliding Gaussian win- dow along the X dimension is varied, KLLR produces contin- uous fits to ⟨∆GPE⟩(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' A single parameter is required for the method, the width of the Gaussian filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The kernel widths for each study indicator are chosen to be roughly a fifth of their range, σKLLR = 1 5(q99% −q1%) (4) where qn% is the nth quantile of the logged data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' This ker- nel size minimizes noise in the fit while maximizing allowed internal variation, lying on the threshold of non-monotonic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' This results in a kernel width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='56 (factor of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='6) for the NQ,tot study measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' RESULTS Due to their similar structure, we analyze the combined sample of PHYSICS 140 and 240 student behavior here, be- ginning with deviations from mean expected GPE at a given math test score T as a function of study volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We then investigate differences in study volume and GPE among de- mographic sub-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The section concludes with remarks on using GPAO instead of T as a baseline condition for grade prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Evidence of grade gains from practice study Using KLLR, we measure considerable gains in mean final grade with increasing study volume shown in Figure 3 using log10(NQ,tot) as the independent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The top panel shows the mean behavior, which moves from −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='2 (a grade point penalty) to +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='6 (a grade point benefit) as the total number of questions attempted over the term increases from one to over one thousand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The middle panel shows the local slope of the grade gain, which increases nearly linearly in log(NQ,tot), and the lower panel shows the significance of that slope being non-zero (implying significant gains with increased study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The KLLR parameterization of study gains are largely in agreement with a simple quadratic trend, as seen in the near- linearity of the slope (middle panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Fitting the trend in Fig- ure 3 around the median value of NQ,tot = 40 we find mean behavior ⟨∆GPE⟩ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='00±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='01)+(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='22±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='02)ν +(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='02)ν2, (5) where ν ≡ log10 (NQ,tot/40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Grade gains from practice are thus quadratic in the log of number of questions attempted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' While doubling study from one question a day to two a day will benefit a student, doubling from two a day to four will, on average, yield even more incremental benefit to their final grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Practice Makes Better 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='5 GPE GPE(T) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='4 slope 100 101 102 103 NQ, tot 0 5 10 significance FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Grade gains as a function of practice effort for the com- bined sample of PHYSICS 140 and 240 students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Study volume is measured here by the total number of PR questions encountered over the term, NQ,tot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Vertical lines indicate quartile values of NQ,tot for students who use PR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' half practice between 14 and 108 problems, with a median of 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Statistics are computed with KLLR using a ker- nel width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='56 in log10 NQ,tot (a factor of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='6) and shaded regions are 1σ uncertainties from bootstrap resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Upper panel: Shift in mean grade relative to that expected from pre-college math score, equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Middle: Local slope of ∆GPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Lower: Significance level of a non-zero slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Note that students who didn’t engage with PR are not included in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The middle panel of Figure 3 shows that the slope is > 3σ significantly positive for NQ,tot ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The slope is roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='1 at this point, but four problems over the course of a full term is a very minimal amount of practice, and grade earned at this level is indeed lower than the T-conditioned mean by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='2 grade points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Gains in the top panel become evident above ten questions, and the slope continues to grow, reaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='3 at NQ,tot ≃ 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' This study volume represents roughly 25 ques- tions per week, or five questions per school day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The positive slope in grade gain continues up to the limit of our study vol- ume data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In our sample, 34 students, or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='5% of the popula- tion, attempted over 1000 questions within a semester, a vol- ume that slightly oversamples the number of questions avail- able and that corresponds to a rate of roughly 15 questions per school day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Table VI quantifies for each study volume indicator (NQ,tot, Nsess, and NQ,mean) the overall grade gain, max(∆GPE), de- fined as the maximum difference in KLLR mean values across TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Maximum study gains, max(∆GPE), between high and low PR study volume (see text) for each study measure along with the fractional reduction in grade variance after accounting for both T and study volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The latter is calculated as ∆Var0→i ≡ (σi2−σ02)/σ02, where σ0 is the scatter in GPE originally and σi is the scatter in grade point after removing trends in µGPE(T) (i = 1) and both that and ⟨∆GPE⟩(X) for study measure X (i = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Indicator max(∆GPE) ∆Var0→1 ∆Var0→2 NQ,tot +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='19±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='03 Nsess +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='62±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='18±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='03 NQ,mean +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='71±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='19±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='03 the study volume domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Uncertainties in the difference are propagated from the respective KLLR bootstrap errors on the minimum and maximum values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The table also quantifies the fractional reduction in variance ∆Var from the initial scatter in GPE σ0 to the scatter after accounting for µGPE(T) (∆Var0→1) and after additionally accounting for ⟨∆GPE⟩(X) in study mea- sure X (∆Var0→2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The reduction in variance from the middle to end stage is then ∆Var1→2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' For each study indicator, students with high study volume tend to do significantly (> 3σ) better than those with low study volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Across the three study indicators, we see simi- lar gains (within 1σ of each other) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='74±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='14 grade points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' This result is consistent with a two (minor) letter grade in- crement, for example, moving from a B- to a B+ or from a B+ to an A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' These results are also robust with varying ker- nel size: doubling or halving kernel width yields statistically identical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Furthermore, the results are robust with vary- ing student math ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Dividing the student sample into ter- ciles by T, we find similar overall grade gains: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='21, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='73±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='19, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='12 for the lowest, intermediate, and highest T terciles, respectively, averaged across study indica- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' All students benefit similarly from increased study vol- ume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Across each study indicator, we see similar reductions of variance, with ∼ 20% of the scatter in student grades ex- plained by the joint µGPE(T) and ⟨∆GPE⟩(X) trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The trend of mean grades running with T accounts for the ma- jority of this reduction, causing on its own a 16% ± 3% re- duction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The range of µGPE(T) is roughly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='2 grade points, so the running of ⟨∆GPE⟩(X) with maximal running of roughly half the range has less capacity to reduce variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The dis- tribution of NQ,tot also clusters strongly towards lower values, even further limiting its ability to reduce variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' However, on sampling evenly in logNQ,tot, we find a significant variance reduction on additionally including study volume in the model of ∆Var1→2 = (9 ± 3)%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' This suggests that our initial inabil- ity to measure significant reduction of variance on accounting for study trends is tied to the clustering of NQ,tot at relatively modest values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Below, we incorporate both math test score and study volume into a linear model of student grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We now turn our attention to several demographic subpopu- lations, investigating relationships to mean trends in the space of math test score, study volume, and final grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Practice Makes Better 7 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Demographic differences 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='2 GPE Female Male HS A,B M,D both single URM NUR ITL 1st 2nd 3rd 4th sex parental education parents at home N/URM income bins 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='96 T 50 60 70 80 90 100 110 NQ, tot Female Male HS A,B M,D both single URM NUR ITL 1st 2nd 3rd 4th FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Points show mean GPE, NQ,tot, and T for the various de- mographic subpopulations indicated in the legend, with error bars showing standard deviation of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Quadratic fits of GPE and NQ,tot as functions of T for the full student population, shown in blue with ±1, 2 and 3σ uncertainties, demonstrate starkly different trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Grades strongly correlate with T while study volume does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Differing educational experiences exist across lines of so- cioeconomic status, sex, race, ethnicity, home environment, and other factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='10–12,20,21 As detailed in §II C, we consider several of these demographic indicators (see Tables III & IV) to the extent and precision available to us;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' most indicators are self-reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We investigate the degree to which demo- graphic groups are stratified in the space of T, GPE, and NQ,tot, then quantify how each sub-group’s mean grade deviates from expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Figure 4 illustrates the relationship between mean measures for demographic groups, shown as points, and the overall trend for all students, shown as the line (with shaded regions indicating ±3σ uncertainty on the fit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' For this analysis, we measure mean NQ,tot values for all students for whom we have T scores, including those for whom NQ,tot = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Note that the mean value, ⟨NQ,tot⟩ ≃ 72, differs from the median log value of 40 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 3 because the distribution of NQ,tot is close to log-normal and has substantial width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' This level of effort cor- responds to attempting roughly five questions per week during the term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Stark differences exist between GPE versus NQ,tot in their trends with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Mean grade earned (top panel) correlates strongly with math test score (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='01), rising by half a grade point over just a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='1 increase in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In stark contrast, the volume of study by students (lower panel) is effectively flat (insignificantly correlated: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='02±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='01);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' low T students study very nearly as much as their high T counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We return to this issue when considering GPAO as an alternate to T below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Mean values of math test score T for demographic groups cover the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='1 range shown, with underrepresented minority (URM) students and students whose parents have the lowest education level (≤HS) possessing the lowest scores and in- ternational students the highest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' These groups lie somewhat off the overall trend in grade, with ITL students lying above and URM and ≤HS below the population mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Note that, because of the positive curvature in mean grade with T, mean values of sub-populations will tend to lie somewhat above the trend line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We find no global trend in study volume with T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' students across the spectrum of math test scores put in similar level of practice effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Differences between demographic groups are apparent in Figure 4, with URM and ≤HS populations lying below the trend while high income and international students rest above it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Do Math Score and Study Volume Explain Demographic Grade Shifts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The trends in the bottom panel of Figure 4 suggest an expla- nation for demographic trends for grade deviation from mean trends;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' that the demographic shifts in study volume often have similar sign to that of ∆GPE, the deviation of GPE from µGPE(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' However, these shifts sometimes misalign;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' for ex- ample, the highest income group tends to study more than av- erage yet achieves only average grades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' To address the degree to which study explains demographic deviations from mean trends, we employ a model that is a linear combination of the expected mean grade conditioned on T (equation (2)), com- bined with the grade shift as a function of study volume using the full student population (the KLLR fit of Figure 3, largely equivalent to the parameterized fit of equation (5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Figure 5 shows the mean deviation from these expectations for the demographic sub-populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The zero line here rep- resents the expected value based on our model of the full stu- dent population, and the shaded regions show median ±3σ uncertainties in the theory mean (essentially the combined un- certainties of the quadratic and KLLR fits for a median stu- dent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Most demographic sub-populations have mean devia- tions consistent with the model’s expectations, but significant outliers are seen toward lower earned grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Students whose parents had a high school degree or less lie significantly be- low expectations, at −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='04 grade points, as do URM students, at −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='04 grade points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In addition, students from households with only a single parent present as well as students from high schools in the lowest income regions dis- play a smaller, less significant deficit of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='04 grade points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' All other sub-populations were < 2σ deviant from ex- pectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Practice Makes Better 8 Female Male HS A,B M,D single both URM NUR ITL 1st 2nd 3rd 4th 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='1 deviation sex parental education parents at home N/URM income bins FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Mean GPE deviations of demographic groups from the over- all trend based on combining math score T and study volume NQ,tot for the full student population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The shaded region around the null line shows ±3σ uncertainties of the predictions based on combin- ing T and NQ,tot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The error bar for each demographic group is the ±1σ error in that group’s mean GPE deviation from the mean model prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Comparison to GPAO In this section, we investigate results when using GPAO as a baseline for comparison, rather than T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' First, we explain some limitations of GPAO (with more issues outlined in Ap- pendix B), Second, for PHYSICS 140 and 240 we show the µGPE(GPAO) plot analogous to Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Third, we show for the two courses combined a gains of study table, analogous to Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' GPAO correlation with study A core issue with using GPAO as a baseline for com- parison (as is done, for example, with the “grade anomaly” GPE − GPAO of W20) is its correlation with study habits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Averaged over log or non-logged versions of all three study volume indicators used in this study (a total of six possible metrics), correlations between study and GPAO were statisti- cally significant, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='132±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='013, whereas correlations between study and T were insignificant, only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='013±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' For each of the six possible metrics, study correlated with GPAO > 5σ significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' For T, while NQ,tot was consistent with null cor- relation (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='014 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='015), Nsess sightly favored a negative cor- relation (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='027 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='014), and NQ,mean favored a slight posi- tive correlation (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='054 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Regardless of study metric, GPAO consistently shows stronger and more significant cor- relations to study than T does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Because we seek to measure how study volume influences final grade, this coupling makes GPAO relatively untenable compared to T for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Figure 6 shows the interplay between PR study volume, GPAO, and composite test score T for PHYSICS 140 and 240 combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' KLLR fits to study volume as a function of GPAO show clear rising behavior for each individual T bin, while 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='00 GPAO 0 50 100 150 NQ, tot T|[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='8) T|[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='9] T|(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Study volume as a function of GPAO for students in bins of T shown in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Mean values are KLLR-derived using a kernel width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='40 and 1σ uncertainties are shown as shaded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Population means for each T group are shown as points with ±1σ error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Significant study trends with GPAO exist within each math score range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' the means (displayed as points with error bars) show no sig- nificant trend in study behavior with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The striation of trend lines shows that among students with the same GPAO, those with lower T scores tended to study more on PR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In contrast to the lack of correlation between T and study, at fixed GPAO there is a significant trend of decreasing PR study volume for students with higher math scores, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Though students with high T scores tend to have higher grades and though students who studied more tended to have higher grades, there was no significant correlation between T and study volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' This seeming inconsistency is resolved in the anticorrelation between study volume and T at fixed GPAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' If we take PR study volume as a proxy for typical stu- dent study habits, then this could perhaps be interpreted to reveal that students with higher T scores tended to not need to study as much to receive the same high grades as students with lower T scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Noting that this correlation between study volume and GPAO makes the latter a less reliable baseline for measuring grade gains due to study, we move forward in this direction with the expectation of finding a lower magnitude of grade gain due to study volume at fixed GPAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Fitting GPE as a function of GPAO We begin by fitting student grades earned in each physics class as a function of their end-of-term grade point average in all other courses (GPAO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' As in §II E, we use a quadratic fitting for ease of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Figure 7 shows the quadratic fit of mean GPE as a func- tion of GPAO for both physics courses individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' On split- ting between students with and without T scores available, we find identical fit parameterization besides normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Students with T scores available tended to have higher GPAO by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='044 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='017 grade points (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='6σ significant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' All other Practice Makes Better 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0 GPAO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0 GPE PHYSICS 140 PHYSICS 240 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' As Figure 2, but fitting mean GPE for the two physics courses as a function of GPAO rather than T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The GPAO distribu- tion was similar between courses, with means around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='4, standard deviation of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='5, and GPAO ≳ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='45 for 95% of students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' fit parameters were ≲ 2σ different, showing good agreement between population dynamics of those with and without T scores recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' As with µGPE(T), we see > 5σ signifi- cantly positive curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The bulk of the population has GPE < GPAO (diagonal line), meaning that these physics courses tend to be the lower grades on their transcripts, es- pecially so for students with low GPAO already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' For exam- ple, a student with otherwise straight As (GPAO = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0) would likely get an A- while a student with otherwise straight Bs (GPAO = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0) would likely get a C+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Deviations from GPAO mean due to study volume After calculating a deviation from expected grade at a given GPAO, ∆ = GPE − µGPE(GPAO), we investigate whether a trend exists in this deviation as a function of PR study vol- ume, using KLLR fitting, analogous to that done in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We present gains of study in Table VII, analogous to Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Those with highest study volume tended to do better than ex- pected by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='07 grade points (averaging over the three study indicators) as compared to those with low study volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' TABLE VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' As Table VI, but measuring gains and variance reduc- tion with GPAO as a baseline for grade expectations (rather than T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Indicator max(∆GPE) ∆Var0→1 ∆Var0→2 NQ,tot +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='46±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='01 Nsess +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='46±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='01 NQ,mean +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='33±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='46±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='01 This is less than half the grade gain measured in Table VI, where T was used as a baseline for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The stark dif- ference in grade gains measured suggests that the significant correlation between GPAO and study volume (see §III C 1) has subsumed a large portion of the grade gains measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' That is, because GPAO reflects in part an individual’s study volume, subtracting it out in the initial baseline grade esti- mate µGPE(GPAO) removes a significant fraction of the mea- surable grade gains due to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Despite GPAO predicting grades better than T (reducing variance by 47% rather than only 19%), GPAO has less utility in quantifying the grade ben- efits of study, washing out over half of the measurable grade benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' DISCUSSION Our findings help demonstrate to educators and learners the benefits of practice study in introductory physics courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' While this finding is neither original nor surprising, the overall magnitude in grade gain for students classified by pre-college math ability is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We acknowledge that there is a good match between the nature of the PR practice, which constitutes low-stakes for- mative assessment, and the summative midterm and final ex- aminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' After all, PR consists of nearly 1000 problems used on past examinations in each course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Given the me- dian response time per problem of roughly 90 seconds3, we do not think that memorization per se plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Rather, other factors, such as the similarity of problem con- struction and structure, the mix of quantitative and qualita- tive questions, and particular “tricky” problem styles involv- ing multiple physics concepts, is likely to be similar between old and new exams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Timed, multiple choice examinations have been criticized as an obstacle to higher-level thinking22 and as a poten- tial causal factor in gender inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='23 Mixed forms of assessment24 may improve inequity and promote higher-level thinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Our findings with respect to demographic characteristics motivate more study of how better to support physics learn- ing for first-generation (our ≤HS category) and underrepre- sented minority students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Students from single-parent and low income households also earn lower grades than expected (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' These categories overlap, inviting future work to understand how students with multiple intersecting identities perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='25 Our comparison to GPAO study trends reveals far more grade benefit measured when conditioning on T than on GPAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Because GPAO significantly correlates with study vol- ume (while T does not), subtracting out a GPAO baseline from grades subtracts out the effects of study behaviors to some ex- tent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' While GPAO may be the most accurate predictor of GPE, it is a flawed baseline for the purposes of our analysis (see also Appendix B) as it washes away the majority the the study benefit trend measured in §III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In essence, we find utility in using T as a baseline for student comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The utility of T could in part be due to its correlation with general mental ability g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='26–29 As g is largely intrinsic (polygenic, yet ≳ 50% heritable and ≳ 90% consistent with age30–33) to each student, it is largely orthogonal to a stu- dent’s external, personal choice of study habits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In contrast, GPAO represents a mix of both internal ability and external choice, and thus subsumes a portion of the study signal we attempt to measure in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Regardless of interpretation, Practice Makes Better 10 this difference in grade gains (measured with GPAO vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' T baselines) warrants further investigation and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' As more colleges move to “test-optional” admissions in the fu- ture, we may in the process be limiting the potential of future investigations to quantify student learning gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Another potential reason for the strong utility of T could lie in the similarity of assessment styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Both standardized tests and the assessment methods used in many large introductory STEM courses take the form of high-stakes multiple-choice questions in a timed setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' CONCLUDING SUMMARY Using a large sample of practice study from the Prob- lem Roulette service, we find that student final grades scale quadratically with the logarithm of the term-aggregated num- ber of questions encountered, with an overall gain of nearly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='8 grade points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' By comparison, the largest demographic de- viation we find is ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='3 grade points, so with roughly a ques- tion per day on average, this deficit can be overcome purely through study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We summarize our findings in the following advice to stu- dents and teachers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Our advice to students is: Do a problem for every time you brush your teeth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Doing at least one or two problems a day each school day of the term will likely give you a quar- ter letter grade raise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Quadratic gains with log study mean that doubling study volume from one to two questions per week benefits you, but dou- bling from two to four questions per week benefits your grade even more!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Our advice to teachers is: Differences between demographic groups, once accounting for T and study, were ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='3 grade points, with the largest outlier by far being first- generation students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Study can benefit students by up to three-quarters of a letter grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Encour- aging students to do at least one problem a day puts them above median study habits, on average putting them above expectations at a given T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' This study only scratches the surface of PR data avail- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Besides only working with physics courses, our model was relatively simple, treating T and GPE separately and us- ing other demographic, academic, and study-related variables only tangentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' A more nuanced model such as multi-level modeling or simultaneous fitting of many variables at once (such as with the machine learning tool SHAP34) could help disentangle which factors are more or less causal (though it could not determine absolute causality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Future analyses should also treat nationality, ethnicity, and race more care- fully, rather than using the broad categories of nationality and binary URM status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We also would like to investigate the effects of study session length—early results suggest that working for more than about 50 minutes tends to yield mini- mal gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Finally, we wish to compare findings between all courses, observing which trends persist cross-subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' We should never lose sight that the purpose of this work is to help individual students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Teachers and students alike can benefit from understanding how T and study affect their grades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Teaching is a handshake, requiring earnest participa- tion of both parties for best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' (We can’t fall into the trap of subscribing to either a student deficit model or a teacher deficit model alone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' both parties need to improve their habits and grow as individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=') As students improve their scholastic habits and teachers improve course structure and learn how to best reach struggling individuals, we can grow together and improve the education system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Appendix A: Problem Roulette structure After logging in and selecting a course, students can choose from various modes of study: Individual, Group, or Practice Exam (student-generated or faculty-generated as formal prac- tice exams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Problem Roulette also collects questions from topics that students struggle with most, which students can study from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' In the Individual or Group study modes, students select which topics to study (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=', “Vector Algebra”, “Relative Mo- tion”), whether or not to use a timer for the session, and how many questions to pull (defaults: 10, 25, all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' They are then presented with an exam-like set of questions one at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Af- ter the session, they are presented with a review of the ques- tions attempted with correct answers indicated, along with an overall accuracy score and the session duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Instructors can view course analytics for a given term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' At the instructor dashboard, they see student activity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=', which days saw more PR use), student accuracy versus questions an- swered for each topic, the accuracy of answers by topic, and the accuracy of individual questions answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' This infor- mation can reveal which topics or types of questions students tend to struggle with most or least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Appendix B: Additional issues with GPAO Though GPAO correlates more with course grade earned (GPE) than any other indicator we investigated, it has a sev- eral issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The most crucial issue of its correlation with study is detailed in §III C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' here we delineate several additional is- sues with using GPAO as a baseline for GPE comparison (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' T as a baseline, as used in this paper), that is, using grade anomaly GPE − GPAO instead of the metric GPE − µGPE(T) used herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Relation to personality traits Because GPAO depends on study, significant differences exist between different demographic populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' As dis- cussed in Richardson, Abraham, and Bond 35, male students tend to study less than female students, and they tend to have a lower GPAO than female students by roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='07 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='14 Practice Makes Better 11 points, despite having significantly higher T scores on aver- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' This means that in a class where each student got iden- tical grades, the grade anomaly for males and females could still differ for causes extrinsic to the course in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Differing course selection GPAO depends on course selection of individual students, but course selection differs drastically by demographics, and not all course loads are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' If all students took similar courses regardless of demographic group, then GPAO would be a fair measure of “anomaly”—how differently the course was graded compared to expectations from other courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' However, different groups tend to take different courses, lead- ing to systemic shifts between different demographic sub- groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' If a student tends to take easier courses, they could have a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0 GPAO, then this would allow the student more free time to study on PR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' However, a similar signal comes from a student who, despite taking incredibly challenging courses, works very hard and still has a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content='0 GPAO, yet then has very little free time to study on PR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' While this exemplifies why GPAO has power in predicting GPE, it also shows that inter- pretation of GPAO is clouded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Deletion of campus-wide trends Grade anomaly GPE − GPAO wipes out campus-wide bi- ases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Expressing this numerically: if every course in the school consistently awarded lower grades by a bias b to a given group g versus the rest of the population p, then the grade anomaly of that group would be (grade anomaly)g = GPEg −GPAOg (B1) = (GPEp −b)−(GPAOp −b) = GPEp −GPAOp = (grade anomaly)p, so the group then has identical grade anomaly to the rest of the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' This then erases the effects of campus-wide biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' ACKNOWLEDGMENTS WKB was partially funded by an Academic Innovation Fund award from the UM Center for Academic Innovation (CAI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' The data were obtained from CAI and the UM Learn- ing Analytics Data Architecture (LARC) database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' BIBLIOGRAPHY 1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE1T4oBgHgl3EQfHgN2/content/2301.02927v1.pdf'} +page_content=' Wong, Y.' metadata={'source': 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+2Universit´e Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France +We study the continuum limit of two-dimensional chiral magnets in which Dzyaloshinskii-Moriya +interaction (DMI) is due to the interplay between a smooth magnetic texture and spin-orbit coupling. +The resulting free energy density of the system contains linear terms in the spatial gradient of +the magnetic texture, which mark an instability of the system towards the formation of nontrivial +magnetic orders such as skyrmions or chiral domain walls. We perform a microscopic analysis of +DMI tensors responsible for this contribution to free energy based on a Berry phase formulation in +the mixed space of momentum and position, and reveal that they exhibit non-Lifshitz invariants +features. In particular, a perturbation theory shows in the case of Rashba spin-orbit interactions the +presence of non-Lifshitz invariants to third order in the small spin-orbit interaction and fourth order +in the small exchange coupling. The higher order terms may even lead to an enhancement of DMI +interaction at strong spin-orbit coupling due to divergences in the density of states at the bottom of +the conduction band. Finally, we also study the DMI free energy generated from Rashba spin-orbit +interaction in different symmetry groups. +I. +INTRODUCTION +Chiral symmetry-breaking in magnetic materials re- +sults in an antisymmetric exchange coupling called the +Dzyaloshinskii-Moriya interaction (DMI) [1–3], which +tends to cant neighboring spins such that noncollinear +magnetic orders are favored in the system [4–6]. As a +consequence of DMI, nontrivial magnetic structures such +as chiral domain walls [7–9] and skyrmions [10–14] be- +come stable. The latter are excitations in the form of +magnetization vortices, which are topologically robust, +and have been the subject of intense research in recent +years [15–18]. The controlled creation and annihilation +of skyrmions with spin-polarized currents [19, 20], gate +voltages [21, 22], or lasers [23] feeds the driving goal to re- +alize energy-efficient spintronic devices operating at room +temperature for memory storage [24–27]. +A promising platform for probing such physics is in +effectively two-dimensional systems where interfacial DMI +develops [28]. In thin ferromagnetic films or in multilayers +with alternating magnetic and nonmagnetic layers, the +inversion symmetry is broken at the interfaces and thus +a strong spin-orbit coupling (SOC) is generated. In a +long wavelength approach, the magnetic texture below +the Curie temperature is described by the continuous +magnetic density vector m(r) of unit amplitude, with +position r in the plane of the magnetic layer (x, y). The +effect of SOC is to generate in the free energy linear terms +in the texture gradient which are characteristic for the +DMI. The micromagnetic DMI free energy follows from +Ω1 = +� +α∈{x,y,z} +j∈{x,y} +wjα +∂mα +∂rj +. +(1) +∗ doru.sticlet@itim-cj.ro +† frederic.piechon@universite-paris-saclay.fr +The most common approach [2] is to consider that energies +wjα are linear in magnetization m, +Ω1 ≃ 1 +2Dαβ,j(mα∂rjmβ − mβ∂rjmα), +(2) +which amounts to take into account only the well-known +Lifshitz invariant (LI) contribution to Ω1 (see A 1). The +microscopic analytical calculation of DMI tensor Dαβ,j +in the continuum limit was only recently performed for +the first time in topological insulators [29, 30] and a two- +dimensional (2D) Rashba thin film [31]. Notably, these +were preceded by different approaches where DMI was +explained in the vein of Ruderman-Kittel-Kasuya-Yosida +theory as due to spin interactions mediated by conduc- +tion electrons [32, 33]. An analysis of effects beyond the +Lifshitz invariant correction was performed in Ref. [34] in +order to get a more general description of DMI in chiral +magnets, and it has established that such corrections can +be consequential. It was soon shown that indeed there are +cases as in tetrahedral magnets where the conventional +LI contribution vanishes by symmetry while the remain- +ing non-LI contributions lead to a non-collinear magnetic +structure [35, 36]. +In this paper, we revisit the issue of non-Lifshitz invari- +ant contribution to DMI from a different point of view, +in which DMI is due to Berry curvature in phase space. +Our approach assumes that the magnetization m varies +smoothly in space such that the effect of m on the pe- +riodic Bloch wave functions is considered perturbatively. +In this sense, the Bloch wave vectors depend on position +through m, |n, k, m(r)⟩. In such cases it is naturally to +consider an approach based on a generalized Berry phase +in the space defined by position and momentum [37]. In- +deed, it was shown that the dynamics of electrons in the +thin layer is determined by the Berry curvature in the +phase space [38, 39]. +Here we analyze the generic case of two-band systems +with crossings near the Γ point. In such cases it is an- +alytically tractable to obtain the form of DMI tensors. +arXiv:2301.11855v1 [cond-mat.mes-hall] 27 Jan 2023 + +2 +These are determined from the corresponding DMI en- +ergy Ω1, which follows from an expansion of the total +grand-canonical thermodynamic free energy Ω in texture +gradients . This generates a contribution that is propor- +tional to the Berry curvature in phase space [38]. The +expansion in gradient is also further refined with addi- +tional expansions in SOC or exchange amplitude. This +allowed us to determine the order at which non-LI contri- +butions might become relevant. +As an application, the present study focuses on Rashba +SOC which occurs naturally in effective 2D systems due +to the large variation in the electrostatic potential normal +to the layer. Usually the structure of the LI and non-LI +invariants may be determined by symmetry analysis [2, 4, +34]. Here we perform instead a microscopic analysis where +such the structure is emergent from effective two-band +Hamiltonians. Such models are based on the specific form +of the Bloch bands at the Γ point, as constrained by +symmetries of the magnetic point groups. The form of +Rashba SOC to cubic order in momentum was classified +for 2D materials in Refs. [40, 41], and constitutes for us +a starting point in determining microscopically the DMI. +Our analysis reveals in the Berry curvature formulation +of the problem, that the DMI free energy decomposes +into two distincts parts Ω1 = Ω(0) +1 ++ Ω(1) +1 . Usually only +the first part has been a subject of investigation. The +second contribution, Ω(1) +1 , is higher order in SOC, but +nonetheless it is of the same order in the magnetization +m, and also enters to the same order in the exchange +coupling strength. For example, the second contribution +in its lowest order in SOC is responsible for symmetric +DMI tensors, which are usually discarded in the bulk, +but may generate some nontrivial edge spin texture [42]. +The term Ω(1) +1 +contains also antisymmetric DMI tensors, +which renormalize the Ω(0) +1 +contribution, and additionally, +we show that they can lead to divergences in the free +energy since they contain Fermi surface contributions +which diverge at low temperature due to singularities in +the density of states. All our investigations are made +concrete in the study of effective models with Rashba +SOC in different symmetry groups. +The article is organized as follows. Sec. II introduces +the class of two-band Hamiltonian models in which DMI +develops. The section also reviews the generic structure +of the energy density Ω1 that is linear in a smooth spatial +gradient of the magnetization, using a Berry phase for- +mulation in the mixed space of momentum and position. +Sec. III develops a perturbation theory which uncovers +the non-Lifshitz invariants corrections to Ω1. The section +expresses the general form of all DMI tensors, and devel- +ops further expansions in the small and large SOC limit, +relative to the exchange energy. Sec. IV particularizes the +analysis to the case of Rashba SOC in the C∞v group. +Sec. V looks briefly at the DMI contribution from Rashba +SOC in different symmetry groups. Appendix A details +several of the points in the main paper such as: a deter- +mination of DMI constants for the conventional Rashba +SOC in C∞v group, an analysis of group D3 where the +SOC exhibits an out-of-plane component, a table with +LI and Ω1 in all 10 2D groups obtained in the limit of +small SOC, etc. Sec. VI summarizes the main points in +the paper. +II. +PHASE SPACE BERRY CURVATURE +FORMULATION OF SPIN ORBIT INDUCED +FREE ENERGY TERMS, LINEAR IN SPATIAL +MAGNETIZATION GRADIENT +This section briefly recalls the derivation of the correc- +tion Ω1 to free energy density that is linear in the gradient +of the magnetic texture. Starting from generic two-band +Hamiltonian models, it is shown that the correction Ω1 +writes as an average over occupied states of the momen- +tum and position-dependent skyrmion-like density of a +vector field h(k, r) that combines the spin-orbit coupling +and the exchange coupling to the magnetic texture. Com- +plementary to previous works, we show that this skyrmion +density entails two distinct contributions that appear at +different order in spin-orbit coupling, but nonetheless +both contributing to same order in the magnetization m +and exchange coupling. +Model Hamiltonian. +In the following, we focus on +generic two-band Hamiltonian models of the form +H(k, r) = ξ(k)σ0 + h(k, r) · σ, +h(k, r) = ∆soγ(k) + ∆sdm(r), +(3) +with σ the vector of Pauli matrices, and σ0 the identity +matrix. The first term is the energy dispersion of elec- +trons in the absence of spin-orbit coupling which is an +even function of momentum ξ(−k) = ξ(k). The second +contribution is a momentum and position dependent vec- +tor field h(k, r) that combines the spin-orbit coupling +(SOC) and the exchange coupling to the magnetic texture. +The SOC is described by an antisymmetric spin-orbit +vector γ(−k) = −γ(k) and a coupling strength ∆so. The +magnetic exchange is characterized by a coupling strength +∆sd = JsdS, with S, the magnitude of spins in the mag- +netic layer, and Jsd, the exchange coupling. The magnetic +texture is modeled by a (unit length) vector m(r) which +varies smoothly in space. As it appears below, the cou- +pling strengths ∆so and ∆sd are useful parameters to +keep track of the order in a perturbation theory in weak +spin-orbit or weak exchange coupling limits. +Free energy density. +The free energy density is ob- +tained from the local density of states ρ(ε, r), +Ω(r) = +� +dερ(ε, r)g(ε), +(4) +with g(ε) the primitive of the Fermi-Dirac distribution +function f(ε) = g′(ε), f(ε) = 1/(1 + eβ(ε−µ)). The local +density of states, expressed using a Wigner representation +of Green’s function, is obtained from a Moyal gradient +expansion in the smooth magnetic texture as ρ = ρ0 + +ρ1 + . . . [31, 38, 39]. + +3 +The effective density of states to linear order in the +magnetization gradient reads +ρ(ε, r) = ⟨(1 + Bjj +s,k)δ(ε − εs,k − shBjj +s,k)⟩ +(5) +with the shorthand notation +⟨. . .⟩ ≡ +� +s=± +� +ddk +(2π)d . . . , +(6) +and where summation over repeated indices j is assumed. +In Eq. (5), εs,k is the semiclassical energy spectrum of +the Hamiltonian in Eq. (3) +εs,k(r) = ξ(k) + sh(k, r), +h(k, r) = +� +∆2 +sd + ∆2soγ2 + 2∆so∆sdγ · m, +(7) +with s = ±, the band index, and h ≡ |h|. Lastly, +Bij +s,k(r) = −s1 +2 +h · (∂rih × ∂kjh) +|h|3 +, +(8) +denotes the element (ij) of the intraband phase space +Berry curvature tensor. +The expression Eq. (5) illustrates two qualitatively dis- +tinct effects resulting from the gradient corrections. On +the one hand, there is a momentum-position dependent +shift of the band spectrum, and, on the other hand, there +is also a modification of the spectral weight [38]. To lin- +ear order in the gradient, both effects are proportional +to the phase-space Berry curvature Eq. (8). Using this +effective density of states, the free energy density is de- +composed as Ω = Ω0 +Ω1, with a zero-order contribution +describing the uniform state, Ω0(r) = +� +dερ0(ε)g(ε), and +a contribution Ω1(r), linear in the gradient of m, which +reads +Ω1(r) = ⟨h · (∂rjh × ∂kjh) +2 +Fs,k⟩, +(9) +with +Fs,k(r) = sg(εs,k) − hf(εs,k) +h3 +. +(10) +Using the explicit expression of h from Eq. (3), it follows +that the correction to free energy density Ω1 has two +distinct contributions +Ω1 = Ω(0) +1 ++ Ω(1) +1 , +(11) +with +Ω(0) +1 (r) = ∆so∆2 +sd +2 +⟨∂kjγ · (m × ∂rjm)Fs,k⟩, +Ω(1) +1 (r) = ∆2 +so∆sd +2 +⟨∂rjm · (∂kjγ × γ)Fs,k⟩. +(12) +These contributions Ω(0,1) +1 +(r) have a structure similar to +the one in Eq. (1) and as detailed in the next section, +both generate Lifshitz invariant and non-Lifshitz invariant +contributions to the DMI interaction. +At this point, a few remarks are in order. The possibility +to express the linear gradient corrections in Eq. (5) and +Eq. (9) solely in terms of the intraband phase-space Berry +curvature is specific to two-band models. Likewise, the +possibility to express the phase-space Berry curvature +directly in terms of a phase-space skyrmion-like density +of the vector field h(k, r) is also specific to two-band +models. However, Eq. (5) and Eq. (9) are valid for any +two-band model (in any dimension) of the form given +by Eq. (3). +Importantly, the expression Eq. (9) and +Eq. (12) contain full non-perturbative dependencies in +the coupling strengths ∆so and ∆sd and also full non- +linear dependencies in the magnetization vector m(r) +since all these parameters appear implicitly in h and +Fs,k. +III. +GENERAL EXPANSION OF NON-LIFSHITZ +INVARIANT CONTRIBUTIONS +In the following, the free energy density contributions +Ω(0,1) +1 +are expressed as Ginzburg-Landau-like expansions +in m when considering ∆so∆sdγ · m/λ2 as a small pa- +rameter, with +λ = +� +∆2 +sd + ∆2soγ2. +(13) +Generically, the free energy densities are expanded as +Ω(i) +1 += +∞ +� +n=0 +Ω(i) +1,n, +(14) +with +Ω(i) +1,n = D(i) +αβµ1...µ2n,j(mα∂rjmβ)mµ1· · · mµ2n, +(15) +where D(i) are the DMI tensors of odd rank. The low- +est order term n = 0 is quadratic in m and yields the +Lifshitz invariant contributions D(i) +αβ,j of the DMI tensor. +The higher terms n > 0 yield the non-Lifshitz-invariant +contributions D(i) +αβµ1...µ2n,j of the DMI tensor. +More quantitatively (see appendix A 2 for details), the +expansion of eigenenergies leads to an expansion of Fs,k. +Note that since γ is antisymmetric in k, only the sym- +metric part of Fs,k in k contributes to Ω(0) +1 , and only the +antisymmetric part of Fs,k contributes to Ω(1) +1 , +Ω(0) +1 += +∞ +� +n=0 +∆2n+1 +so +∆2n+2 +sd +2 +⟨∂kjγ · (m × ∂rjm)(γ · m)2n +× F(2n) +s,k (λ)⟩, +(16) +Ω(1) +1 += +∞ +� +n=0 +∆2n+3 +so +∆2n+2 +sd +2 +⟨∂rjm · (∂kjγ × γ)(γ · m)2n+1 +× F(2n+1) +s,k +(λ)⟩, + +4 +where the coefficients F(n) +s,k are even in k, and are deter- +mined iteratively +F(0) +s,k(λ) = Fs,k +���� +γ·m=0 +, F(n) +s,k (λ) = 1 +nλ +∂F(n−1) +s,k +(λ) +∂λ +, (17) +for n > 0. The use of argument λ in previous expressions +implies that all dependence on energy εs,k simplifies to +one on ε(0) +s,k = ξ + sλ. +The Eqs. (15) and (16) readily yield the general form +of DMI tensors +D(0) +αβµ1...µ2n,j = 1 +2∆2n+1 +so +∆2n+2 +sd +ϵαβδ⟨(∂kjγδ)γµ1· · · γµ2n +× F(2n) +s,k (λ)⟩, +(18) +D(1) +αβµ1...µ2n,j = 1 +2∆2n+3 +so +∆2n+2 +sd +ϵνβδ⟨γαγν(∂kjγδ) +× γµ1· · · γµ2nF(2n+1) +s,k +(λ)⟩, +(19) +with ϵαβδ, the Levi-Civita symbol. +The usual Lifshitz invariants contribution to the en- +ergy is contained in Ω(i) +1,0, and the related DMI tensors are +D(i) +αβ,j. The expansion beyond the first order is responsible +for non-Lifshitz invariants. Note that even to first order, +there is a marked difference between the two tensors. The +first tensor D(0) +αβ,j is antisymmetric in α and β indices, +while there is no such constraint on D(1) +αβ,j. The sym- +metric part of the latter tensor is usually neglected since +it multiplies a total derivative ∂rj(mαmβ), and vanishes +when integrating over the entire sample. It was shown +however that it has physical effects in generating specific +magnetic textures at the sample boundary [42]. Since we +treat here the case of an infinite system, we consider only +the antisymmetric part. +It is particularly revealing to truncate the free energy +density expansion to the first term where non-LI contribu- +tions are present. This is done either in the limit of small +spin-orbit coupling, or small exchange coupling. From +Eqs. (18) and (19), it follows that at weak SOC the free +energy is approximated +Ω1 = Ω(0) +1,0 + Ω(1) +1,0 + Ω(0) +1,1 + O(∆5 +so/∆5 +sd), +(20) +≃ (D(0) +αβ,j + D(1) +αβ,j + D(0) +αβµ1µ2,jmµ1mµ2)mα∂rjmβ. +Similarly, in the case of weak exchange coupling (or large +SOC) ∆so ≫ ∆sd, +Ω1 = Ω(0) +1,0 + Ω(1) +1,0 + Ω(0) +1,1 + Ω(1) +1,1 + O(∆6 +sd/∆6 +so), (21) +≃ +� +i=0,1 +(D(i) +αβ,j + D(i) +αβµ1µ2,jmµ1mµ2)mα∂rjmβ. +Note that the power counting in the two expansions is +different. At small SOC, the linear order in ∆so is con- +tained in Ω(0) +1,0 alone. This contribution to free energy +and all conventional LI invariants are therefore deter- +mined exactly in this limit from the analysis of Ω(0) +1,0. In +the Appendix A 5 we have microscopically obtained the +LI invariants in all 10 two-dimensional point groups by +considering the symmetry-allowed spin-orbit coupling to +cubic order in momentum. In contrast, in the limit of +large SOC or small exchange, both tensors D(0,1) +αβ,j already +contribute at the lowest order ∆2 +sd, such that both Ω(0) +1,0 +and Ω(1) +1,0 are needed. Finally, the explicit expression of +F(n) +s,k coefficients (17) up to n = 4, necessary to give the +dominant non-Lifshitz invariants in both limit cases of +Eqs. (20) and (21) are given in App. A 2. +IV. +APPLICATION TO RASBHA SPIN-ORBIT +COUPLING +The general theory from above is instantiated now in +the important case of Rashba spin-orbit interactions. The +simplest case is that of the C∞v group with a rotationally +symmetric Rashba coupling ∆soγ = αR(−ky, kx, 0) for +electrons with a parabolic spectrum, +H = (ℏ2k2 +2m − µ)σ0 + αR(k × σ)z + ∆sdm · σ, +(22) +with αR, the amplitude of Rashba SOC. +Small SOC expansion. +The limit of weak spin-orbit +coupling relative to the exchange coupling ∆sd is relevant +in experiment and is the focus of the following. To ob- +tain the first non-LI invariant contribution to free energy +density it is necessary to expand Ω1 to cubic order in +αR as shown in Eq. (20). That requires determining the +tensors D(0) +αβ,j, D(1) +αβ,j, and D(0) +αβµ1µ2,j (see App. A 3 for +details about the DMI tensors involved beyond the weak +SOC approximation). +Using the rotational symmetry of the Rashba SOC +allows one to readily show that all nonzero tensor elements +of D(0) +αβ,j are equal in amplitude, such that there is a single +DMI constant characterizing the free energy density +Ω(0) +1,0 = D(0) +0 Ljz,j, +(23) +with the DMI constant D(0) +0 += D(0) +xz,x, +D(0) +0 += −αR∆2 +sd +4π +� +s +� dkk +λ3 +� +sg0,s − λf0,s +� +, +(24) +and Lifshitz invariant +Lαβ,j = mα∂rjmβ − mβ∂rjmα. +(25) +To first order in αR, λ = ∆sd in Eq. (13), recovering the +result in Ref. [31]. +The tensors D(1) +αβ,j, and D(0) +αβµ1µ2,j are analyzed simi- +larly, yielding the free energy contributions Ω(1) +1,0 and Ω(0) +1,1, +respectively. Since again, in each tensor, the components +are equal in amplitude, it is possible to factor out a single +DMI constant in the free energies. +Ω(1) +1,0 = D(1) +0 Ljz,j, +Ω(0) +1,1 = D(0) +1 (1 − m2 +z)Ljz,j. +(26) + +5 +−2 +−1 +0 +1 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +D(0) +0 +(a) +ER/∆sd +0th +0.1 +0.2 +0.4 +−2 +−1 +0 +1 +2 +−0.1 +0.0 +0.1 +0.2 +0.3 +D(1) +0 +(b) +ER/∆sd +0th +0.1 +0.2 +0.4 +−2 +−1 +0 +1 +2 +−0.2 +0.0 +0.2 +0.4 +0.6 +D(0) +1 +(c) +ER/∆sd +0th +0.1 +0.2 +−2 +−1 +0 +1 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +D(0) +0 +(d) +T/∆sd +0.01 +0.1 +0.2 +0.4 +−2 +−1 +0 +1 +2 +µ/∆sd +−0.05 +0.00 +0.05 +0.10 +D(1) +0 +(e) +T/∆sd +0.01 +0.1 +0.2 +0.4 +−2 +−1 +0 +1 +2 +−0.2 +0.0 +0.2 +0.4 +0.6 +D(0) +1 +(f) +T/∆sd +0.01 +0.1 +0.2 +0.4 +FIG. 1. The DMI constants D(0) +0 , D(1) +0 , and D(0) +1 +in units of kR∆sd/8π as a function of chemical potential in the limit of small +spin-orbit coupling. Panels (a, c, e) show the three DMI constants at different ER and kBT = 0.01∆sd, with the red line +represents denoting the zeroth order approximation where the DMI constants are linear in αR. Panels (b, d, f) present the same +DMI constants’ behavior at different temperatures and at fixed ER = 0.2∆sd [kB = 1]. +The constant D(1) +0 += D(1) +xz,x/2 is obtained by extracting +out the antisymmetric contribution in D(1) +αβ,j. To cubic +order in αR it reads +D(1) +0 += −α3 +R∆2 +sd +16π +� +s +� +dkk3F(1) +s,k(∆sd). +(27) +Finally, the DMI constant D(0) +1 += D(0) +xzxx,x from D(0) +αβµ1µ2,j +has the expression to O(α3 +R): +D(0) +1 += −α3 +R∆4 +sd +8π +� +s +� +dkk3F(2) +s,k(∆sd). +(28) +Therefore, the free energy density Ω1 in this approxima- +tion is determined by all the three contributions +Ω1 ≃ [D(0) +0 ++ D(1) +0 ++ (1 − m2 +z)D(0) +1 ]Ljz,j. +(29) +Already, to cubic order in αR there are now non-LI invari- +ants in the free energy, m2 +zLjz,j. The additional depen- +dence on m2 +z is a property due to the rotational symmetry +of the problem and was already predicted [34]. +Using natural momentum and energy scales character- +izing the Rashba SOC, +kR = mαR +ℏ2 , and ER = mα2 +R +2ℏ2 , +(30) +respectively, yields simple analytical formulas for the con- +stants in the zero-temperature approximation to O(α3 +R): +D(0) +0 +≃ kR∆sd +8π +� +1 − 2ERµ +∆2 +sd +�� +1 − µ2 +∆2 +sd +� +Θ +� +1 − µ2 +∆2 +sd +� +, +D(1) +0 +≃ −kRERµ +8π∆sd +(1 − µ2 +∆2 +sd +)Θ(1 − µ2 +∆2 +sd +), +(31) +D(0) +1 +≃ 3kRERµ +4π∆sd +(1 − 5µ2 +3∆2 +sd +)Θ(1 − µ2 +∆2 +sd +), +with Θ the Heaviside function. The results for D(0) +0 +from +Ref. [31] are recovered by eliminating the cubic depen- +dence on SOC by formally setting ER to 0. In the zero +temperature limit it follows that the DMI energy is non- +vanishing only when the Fermi surface determined by µ +is inside the exchange gap. In this case there is a single +circular Fermi surface at k ≃ +� +2m(∆sd + µ)/ℏ. Out- +side the exchange gap µ > ∆sd, there are always two +Fermi surfaces, with equal contribution and opposite sign, +canceling in the sum over the bands. +The behavior of DMI constants for different relative +strengths ER/∆sd and at different temperatures are shown +in Fig. 1. The zero-temperature approximation recovers +the numerical behavior at low temperature and weak +SOC ER/∆sd ≪ 1. +At weak SOC the constant D(0) +0 +is symmetric in µ, while D(1) +0 +and D(0) +1 , antisymmetric +around the middle of exchange gap µ/∆sd = 0. Such +symmetry is quickly lost at larger SOC and generally the +constants have a higher value near the bottom of the gap, +as explained below. Larger corrections in αR also lead to +increasing the number of zeros in the free energy in their +respective contribution at µ = 0. +With decreasing temperature and increasing ER/∆sd, +the constants develop divergences at the bottom of the + +6 +−10 +−5 +0 +5 +10 +0.025 +0.050 +0.075 +0.100 +DOS +(a) +−10.0 +−7.5 +−5.0 +−2.5 +0.0 +µ/∆sd +0.0 +0.1 +0.2 +0.3 +D(0) +0 +(b) +−10.0 +−7.5 +−5.0 +−2.5 +0.0 +0.00 +0.25 +0.50 +0.75 +D(1) +0 +(c) +ER/∆sd +6 +8 +10 +FIG. 2. (a) Density of states at small ∆sd (large SOC, ER/∆sd > 1) presents divergences at the bottom of the lowest band. +Comparison between numerics (solid line) at kBT/∆sd = 0.1 [kB = 1] zero-temperature analytic approximation (dashed line) for +(b) D(0) +0 +and (c) D(1) +0 +DMI constants in the Rashba C∞v case in units of kR∆sd/8π. +band. This is visible in Fig. 1(b), (c), (e) and (f) and it +is due to the presence of derivatives of the Fermi-Dirac +distribution in coefficients F(1) +s,k and F(2) +s,k (A12). This +effect cannot be captured analytically in the weak SOC +expansion since αR enters only as an overall prefactor, +and the effective energy bands are determined by ∆sd +alone. The divergence is however readily understood when +considering αR non-perturbatively. +Large SOC expansion. +It is telling to analyze this +effect quantitatively in the opposite limit ER/∆sd ≫ 1, +although the effect is visible beyond this limit. At large +SOC, the two energy parabolas ℏ2k2/2m for spin up +and down are shifted, creating a degenerate manifold of +momentum states with zero group velocity at k ≃ kR +at the bottom of the lower band µ ≃ −ER. Since the +density of states is effectively one dimensional (1D) there, +the total density of states will exhibit the usual inverse +square root energy singularity [Fig. 2(a)]. +More quantitatively, at zero temperature and in the +limit of ER ≫ ∆sd, the leading approximation involves +both D(0) +0 +and D(1) +0 +to O(∆2 +sd), +D(0) +0 ++ D(1) +0 +≃ kR∆2 +sd +16πER +× +� +(1 + +µ +ER ) +1 +2 + (1 + +µ +ER )− 1 +2 , +µ ∈ (−ER, −∆sd), +1 − +µ +∆sd , +µ ∈ (−∆sd, ∆sd). +(32) +At µ = −∆sd, the two asymptotic expressions match to +leading order in ∆sd/ER. More importantly, near the +band minimum at µ ≃ −ER, the constant D(1) +0 +displays +the typical 1D singularity in the density of states D(1) +0 +∼ +(1 + µ/ER)−1/2 ∼ 1/√ε, where ε is the energy calculated +from −ER. The analytical results are corroborated with +the numerical calculation of DMI constants presented in +Fig. 2(a) and (c), where the typical divergences in the 1D +density of states are accompanied by the divergence in +D(1) +0 . Similar results are expected for higher-order terms +in the expansion that contribute to order ∆4 +sd such as +D(0) +1 +[see Fig. 1 (c) and (f)] since they contain a stronger +divergence generated by Fermi surface terms such as f ′′(ε) +that occur in F(3) +s,k in Eqs. (A12). +The above considerations explain the divergences devel- +oping in DMI constants of higher order in αR (see details +in App. A 3). This effect could be used as an exploit to +single out non-LI contributions, with the provision that +it would be seen only in the low-temperature regime, at +strong SOC, with a chemical potential finely tuned near +the lower band bottom. +V. +RASBA SPIN-ORBIT COUPLING IN +DIFFERENT SYMMETRY GROUPS +In order to analyze microscopically the DMI free energy +in all 10 two-dimensional point groups, we consider effec- +tive SOC derived to cubic order in momentum in Ref. [41]. +Notably, in such cases, the rotational symmetry of C∞v +may be lost, and the SOC vector may develop out-of-plane +components. The latter is true in point groups where a +π rotation around z axis is not a group element: C1, C3, +D1, and D3. In the remaining 6 groups, symmetry under +a π rotation and antisymmetry of γz, imposes γz = 0. +Consequently, in these groups there is a drastic reduction +in the number of linearly independent components of the +DMI tensors. Namely, from Eqs. (18) and (19), it follows +that in the DMI energy mz enters only once, and the DMI +tensors are reduced to D(0,1) +izl1...l2n,j, with Latin indices in +the (x, y) plane. +Let us briefly analyze the example of group D3 in Γ4 +bands where the SOC vector develops an out-of-plane +component, +∆soγ = (−α1ky, α1kx, α2ky(3k2 +x − k2 +y)). +(33) +The SOC in this group is relevant for topological surface +states of Bi2Te3 and Bi2Se3 [43, 44], BiTeI [45], hole gases +in quasi-2D semiconductors [46], (001) surface states of +oxide SrTiO3 [47], etc. + +7 +Note that the spin-orbit vector is identical in the x +and y components to the case explored in the previous +section, and therefore one expects to recover some of the +same structure of DMI tensors from C∞v case. However, +there is an additional cubic dependence on momentum +in the z component of the SOC vector. The analysis +in App. A 4 shows that to first order in a perturbation +theory Ω(0) +1,0 +Ω(1) +1,0 there is no contribution from the cubic +term, and the expected LI invariant follows, i.e. Ljz,j +generated by γx and γy. +The effect of cubic Rashba +term γz is visible only at the level of non-LI invariants +present in the material. +To cubic order in the SOC, +there are now two non-LI invariants generated in the free- +energy expansion. One is identical to the previous C∞v +case, and represents a quartic interaction of spins of the +form m2 +zLjz,j. Additionally, there is a new invariant that +involves only in-plane interactions between the spins, +2mxmyLyx,x + (m2 +x − m2 +y)Lyx,y. +(34) +This non-LI is proportional (up to total derivatives that +vanish in the bulk) to the invariant mx(m2 +x − 3m2 +y)∂imi +that was analyzed in detail in Ref. [48] for the group D3h. +The difference being that in our case this contribution to +DMI appears alongside the conventional LIs and non-LI +m2 +zLjz,j. +The cubic terms in momentum in the SOC, such as +those in γz for D3, are not reflected at the level of LI +invariants, and generally may only contribute to higher- +orders in the perturbation theory, to non-LI invariants. +To the fifth order in SOC, our calculations show that these +terms only contribute to non-LI invariants only when the +linear contribution in SOC coupling is present. For the +above case, that means the DMI energy will contain to this +order only terms of type αn +1αm +2 with n > 0, (where α1 is +the strength of the Rasbha coupling linear in momentum). +Conversely, the Rashba coupling in Γ5 and Γ6 bands in +D3 group has no linear terms in momentum (see. Tab. I), +and yields no contribution to DMI energy to the lowest +orders in spin-orbit coupling strength. +As a byproduct of the present theory, we also deter- +mine the conventional LI invariants which follow in a first +order perturbation theory in weak SOC. A table of mi- +croscopically calculated DMI constants and LI-invariants +in all symmetry groups is shown in Tab. I in App. A 5, +and recovers the conventional invariants obtained in a +standard symmetry analysis [4, 49]. +VI. +CONCLUSIONS +In this article, we investigated generic two-dimensional, +two-band continuum models where Dzyaloshinskii-Moriya +interaction is generated in the interplay between spin-orbit +coupling and a magnetic texture. The DMI micromag- +netic free energy, proportional to the first derivative in +the gradient of a smooth magnetic texture, was analyzed +in detailed to reveal its structure beyond the Lifshitz +invariants corrections. A second expansion in weak SOC +or weak exchange coupling allows to pinpoint the ex- +act order at which non-Lifshitz invariants are manifest, +namely to third order in small SOC and fourth order in +small exchange coupling. The calculation of DMI tensors +was performed in these limits explicitly for the case of +rotation-symmetric C∞v Rashba spin-orbit coupling. A +signature of higher order terms is revealed in divergences +in the DMI energy due to singularities in the electronic +bands. In the case of Rashba interactions this occurs due +to the effective one-dimensional density of states near the +bottom of the band at larger spin-orbit coupling, which +generates an inverse square root singularity in energy. +Thus, a signature of non-Lifshitz invariants might be vis- +ible in measurements of the DMI constants, provided a +strong SOC, a low temperature regime kBT ≪ ∆sd, with +chemical potential tuned near the bottom of the band. +We have also shown how effective models for spin-orbit +coupling in different point groups may be used to deter- +mine microscopically the DMI energy. The lower sym- +metry of the Rashba vector compared to the continuum +model with rotational symmetry induces new non-Lifshitz +invariants. This approach is checked also by deducing +the conventional LI invariants when taking only the first +order in a weak SOC expansion. +A nontrivial extension to the present work is the investi- +gation of multi-band effects in systems hosting skyrmions. +The free energy linear in the gradient of the magnetization +is still expressed as a function of the Berry phase [38], but +a simple decomposition as in Eq. (11) is not readily avail- +able. Another open venue is the analysis of free-energy +contributions that depend on higher-order gradients of +the magnetization, which play a role in the stabilization +of the skyrmion textures. +ACKNOWLEDGMENTS +The authors thank A. Thiaville for enlightening discus- +sions on the topic. This work is supported by “Investisse- +ments d’Avenir” LabEx (ANR-10-LABX-0039-PALM). +D.S. also acknowledges financial support from the Ro- +manian National Authority for Scientific Research and +Innovation, UEFISCDI through the contract ERANET- +QUANTERA QuCos 120/16.09.2019, and through Core +Program 27N/03.01.2023, project No. PN 23 24 01 04. + +8 +Appendix A: Detailed DMI constant determination in Rashba models +1. +Notation +The convention used in the article is that Greek letters denote indices that can take values in {x, y, z}, while Roman +ones, only in the two-dimensional plane {x, y} of the layer. Einstein notation, where repeated indices are summed over, +is also employed throughout the paper. Spatial derivatives are denoted as ∂rj and act in the 2D plane of the material. +The conventional DMI tensor notation is related the one used in this paper as follows: +D(i) +αjβµ1...µ2n ≡ D(i) +αβµ1...µ2n,j +(A1) +where the j index is separated out since it corresponds in the free energy Ω1 to a spatial derivative ∂x or ∂y of the +magnetization m. +The free energy is expressed conveniently with the aid of LI invariants +Lαβ,j = mα∂rjmβ − mβ∂rjmα. +(A2) +Such invariants are also denoted in the literature as L(j) +αβ. +2. +Free energy expansion in two-dimensional, two-band models with spin-orbit interactions +Here we present in more detail the model and the expansion of the free energy density from Sec. II and III. To +improve readability, some equations in the main text are restated. +The continuum model from the main text in Eq. (3) +H = ξ(k)σ0 + h · σ, +h = ∆soγ(k) + ∆sdm(r), +(A3) +with local energy eigenvalues +εs,k(r) = ξ + s +� +∆2 +sd + ∆2soγ2 + 2∆so∆sdγ · m(r). +(A4) +The correction to the first contribution to the gradient expansion in the free-energy density Ω1 uses an expansion of +energy eigenstates +εs,k = ξ + sλ +� +1 + η, +λ = +� +∆2 +sd + ∆2soγ2, +η = 2∆so∆sd +λ2 +γ · m, +(A5) +with η the small parameter. This procedure generates in the free energy density a Ginzburg-Landau expansion in the +magnetization m. +The free energy density in Eq. (12) are +Ω(0) +1 (r) = ∆so∆2 +sd +2 +⟨∂kjγ · (m × ∂rjm)Fs,k⟩, +Ω(1) +1 (r) = ∆2 +so∆sd +2 +⟨∂rjm · (∂kjγ × γ)Fs,k⟩. +(A6) +The functions Fs,k and implicitly the free energy density Ω1 are expanded in powers of magnetization field +Fs,k = +∞ +� +n=0 +F(n) +s,k (λ)∆n +so∆n +sd(γ · m)n, +Ω(i) +1 += +∞ +� +n=0 +Ω(i) +1,n. +(A7) +The spin-orbit vector γ is antisymmetric in k, while the expansion coefficients F are symmetric in k. Therefore, only +the symmetric in k part of Fs,k contributes to Ω(0) +1 , and only the antisymmetric part of Fs,k, to Ω(1) +1 , such that the +following simplified expressions follow. Each order in the expansion is related to the rank of a corresponding DMI +tensor D in the following way, +Ω(i) +1,n = D(i) +αβµ1...µ2n,j(mα∂rjmβ)mµ1· · · mµ2n, +(A8) + +9 +with DMI tensors +D(0) +αβµ1...µ2n,j = 1 +2∆2n+1 +so +∆2n+2 +sd +ϵαβδ⟨(∂kjγδ)γµ1· · · γµ2nF(2n) +s,k (λ)⟩, +(A9) +D(1) +αβµ1...µ2n,j = 1 +2∆2n+3 +so +∆2n+2 +sd +ϵνβδ⟨γαγν(∂kjγδ)γµ1· · · γµ2nF(2n+1) +s,k +(λ)⟩. +(A10) +In calculations, it is profitable to perform a decomposition of the tensors in symmetric and antisymmetric parts, thus +revealing a reduction in the number of linearly independent components: +D(0) +αβµ1...µ2n,j = D(0) +[αβ](µ1...µ2n),j, +(A11) +D(1) +αβµ1...µ2n,j = D(1) +[αβ](µ1...µ2n),j + D(1) +(αβ)(µ1...µ2n),j. +Here [. . . ] and (. . . ) denote antisymmetric and symmetric tensor in those indices, respectively. Further reductions are +apparent only by considering specific point groups under which DMI tensors transform. +For practical purposes the free energy density may be analyzed analytically in the weak SOC or weak exchange +coupling limits to identify the leading non-LI contributions. This leads to the truncated expansions in the ∆so ≪ ∆sd +limit, +Ω1 = Ω(0) +1,0 + Ω(1) +1,0 + Ω(0) +1,1 + O(∆5 +so/∆5 +sd) ≃ (D(0) +αβ,j + D(1) +αβ,j + D(0) +αβµ1µ2,jmµ1mµ2)mα∂rjmβ, +or ∆so ≫ ∆sd limit, +Ω1 = Ω(0) +1,0 + Ω(1) +1,0 + Ω(0) +1,1 + Ω(1) +1,1 + O(∆6 +sd/∆6 +so) ≃ +� +i=0,1 +(D(i) +αβ,j + D(i) +αβµ1µ2,jmµ1mµ2)mα∂rjmβ. +Computing the first terms in the free energy density expansions above [or Eqs. (20) and (21) in the main text] to +obtain the non-LI invariants requires the first four coefficients determined from Eq. (17): +F(0) +s,k(λ) = 1 +λ3 (sg0,s − λf0,s) +F(1) +s,k(λ) = 1 +λ5 (−3sg0,s + 3λf0,s − sλ2f ′ +0,s), +(A12) +F(2) +s,k(λ) = +1 +2λ7 (15sg0,s − 15λf0,s + 6sλ2f ′ +0,s − λ3f ′′ +0,s), +F(3) +s,k(λ) = +1 +6λ9 (−105sg0,s + 105λf0,s − 45sλ2f ′ +0,s + 10λ3f ′′ +0,s − sλ4f ′′′ +0,s). +The primes denote derivatives with respect to the energy argument of the Fermi-Dirac functions. Also, g0,s ≡ g(ε(0) +s,k), +f0,s ≡ f(ε(0) +s,k), and derivatives are evaluated at ε(0) +s,k = ξ + sλ at vanishing γ · m. Higher order coefficients F contain +derivatives of the Fermi-Dirac distribution function which capture mainly Fermi surface contribution to the DMI +tensor elements. Although such terms are small in a perturbation theory in either small SOC or small exchange, they +can yield divergences in the free energy at small temperature when the density of states diverges such as for flat bands, +van Hove singularities etc. +3. +Group C∞v +This subsection details the calculation of DMI tensors and constants in the Rashba model from Eq. (22) in group +C∞v. The spin-orbit coupling vector in this group is given by +∆soγ = αR(−ky, kx, 0), +(A13) +and it is identical to the spin-orbit coupling in D4: Γ6 and Γ7, and D6: Γ7 and Γ8. The γ expression determines the +DMI tensor elements when using Eq. (18). We derive in the following the general form of the first four DMI tensors +which capture the dominant contribution to non-LI invariants. Later in the subsection we perform a perturbation +theory in either weak SOC or weak exchange coupling to get explicit forms for the DMI constants. +The DMI tensor in Ω(0) +1,0 is sparse with only 4 nonzero elements which are equal in amplitude: +D(0) +jz,j = −D(0) +zj,j = −αR∆2 +sd +2 +⟨F(0) +s,k(λ)⟩. +(A14) + +10 +In these tensor elements the repeated indices are not summed. This convention also applies below and in the next +sections whenever discussing a given DMI tensor element. Factoring out one of the elements determines the DMI +constant +Ω(0) +1,0 = D(0) +0 (mj∂rjmz − mz∂rjmj) = D(0) +0 Ljz,j +(A15) +with D(0) +0 += D(0) +xz,x or, explicitly +D(0) +0 += −αR∆2 +sd +4π +� +s +� +dkkF(0) +s,k(λ). +(A16) +The next contribution is from the 3-rank tensor D(1) +αβ,j. The nontrivial tensor elements are +D(1) +xz,x = −α3 +R∆2 +sd +2 +⟨k2 +yF(1) +s,k(λ)⟩, +D(1) +yz,y = −α3 +R∆2 +sd +2 +⟨k2 +xF(1) +s,k(λ)⟩, +D(1) +xz,y = D(1) +yz,x = α3 +R∆2 +sd +2 +⟨kxkyF(1) +s,k(λ)⟩. +(A17) +The last two elements vanish by using the spherical symmetry of the problem in the integrals over momentum in +⟨. . . ⟩, and the remaining tensor elements read +D(1) +xz,x = D(1) +yz,y = −α3 +R∆2 +sd +4 +⟨k2F(1) +s,k(λ)⟩, +k2 = k2 +x + k2 +y. +(A18) +The symmetric part of the tensor integrates to zero over the bulk as it multiplies a total derivative ∂rj(mjmz). +Therefore, the nonvanishing part of the free-energy density has only the antisymmetric part +Ω(1) +1,0 = D(1) +0 Ljz,j, +D(1) +0 += D(1) +xz,x/2, +(A19) +with explicit DMI constant +D(1) +0 += −α3 +R∆2 +sd +16π +� +s +� +dkk3F(1) +s,k(λ). +(A20) +The DMI tensor in Ω(0) +1,1 has 8 nonvanishing tensor components: +D(0) +jzii,j = −D(0) +zjii,j = −α3 +R∆4 +sd +4 +⟨k2F(2) +s,k(λ)⟩. +(A21) +Therefore, the free energy correction reads +Ω(0) +1,1 = D(0) +1 (mx∂xmz − mz∂xmx)(m2 +x + m2 +y) + (x ↔ y) = D(0) +1 (1 − m2 +z)Ljz,j, +(A22) +using m2 = 1 in the second equality, with DMI constant D(0) +1 += D(0) +xzxx,x, or +D(0) +1 += −α3 +R∆4 +sd +8π +� +s +� +dkk3F(2) +s,k(λ). +(A23) +The final DMI tensor considered here has 16 non-vanishing components (not explicit here), leading to a free energy +density contribution: +Ω(1) +1,1 = 2D(1) +1 (1 − m2 +z)(mx∂xmz + my∂ymz). +(A24) +The free energy density, after eliminating total derivatives ∂rj(mjmz) and ∂rj(mjm3 +z), also reads +Ω(1) +1,1 = D(1) +1 (1 − m2 +z +2 )Ljz,j, +(A25) +which presents the same m2 +z correction to Lifshitz invariants. The DMI constant reads +D(1) +1 += −3α5 +R∆4 +sd +64π +� +s +� +dkk5F(3) +s,k(λ). +(A26) +In the following we introduce the characteristic Rashba momentum and energy scales +kR = mαR +ℏ2 , +ER = mα2 +R +2ℏ2 , +(A27) +and perform a perturbation theory either in the small or large SOC limit. + +11 +a. +Small SOC +We consider now the expansion in αR/∆sd as in Eq. (20). In Figs. (1) we have shown the exact behavior of the +DMI constants by numerical integration over bands and momentum, including the limit of small SOC. Analytically, we +also compute the DMI constants in the zero temperature limit, to the lowest orders in αR. +To O(α3 +R), the zero temperature DMI constants read as follows, +D(0) +0 +≃ kR∆sd +8π +� +1 − 2ERµ +∆2 +sd +�� +1 − µ2 +∆2 +sd +� +Θ +� +1 − µ2 +∆2 +sd +� +, +(A28) +D(1) +0 +≃ −kRERµ +8π∆sd +� +1 − µ2 +∆2 +sd +� +Θ +� +1 − µ2 +∆2 +sd +� +, +(A29) +and +D(0) +1 +≃ 3kRERµ +4π∆sd +� +1 − 5µ2 +3∆2 +sd +� +Θ +� +1 − µ2 +∆2 +sd +� +, +(A30) +with Heaviside step function Θ. To obtain D(1) +0 +it was necessary to expand F (0) +s,k(λ) from Eq. (A16) to α3 +R, hence +the term proportional to αRER. Note that to linear order in αR, only D(1) +0 +survives by formally setting in the +expression ER = 0, such that it reproduces the results in Ref. [31]. The zero temperature results are obtained by +performing the sums and integrals in Eqs. (A16), (A20), and (A26) using Fermi-Dirac formulas at zero temperature +g(ε) = (ε − µ)Θ(µ − ε), f(ε) = Θ(µ − ε), and f ′(ε) = −δ(µ − ε). +b. +Large SOC +In the limit of ER ≫ ∆sd we obtain from Eq. (A16) the leading zero-temperature approximation to D(0) +0 , +D(0) +0 +≃ kR∆2 +sd +8πER +× +� +2 +� +1 + µ/ER, +µ ∈ (−ER, −∆sd), +1 − µ/∆sd, +µ ∈ (−∆sd, ∆sd). +(A31) +To same order in ∆sd there is the additional contribution from D(1) +0 +from Eq. (A20), +D(1) +0 += kR∆2 +sd +16πER +× +� +−3(1 + µ/ER)1/2 + (1 + µ/ER)−1/2, +µ ∈ (−ER, −∆sd), +µ/∆sd − 1, +µ ∈ (−∆sd, ∆sd). +(A32) +Therefore, the DMI constant at large SOC (small exchange coupling) D(0) +0 ++ D(1) +0 +is the one in Eq. (32). The D(1) +0 +constant has a divergence at the bottom of the band in the limit of small exchange coupling (or large SOC) due to the +Fermi surface contribution to the free energy density. This is a consequence of the large density of states that develops +at the bottom of the band, where the minimum occurs on a circle of constant energy at k = kR, such that the density +of states there has a characteristic divergence of a 1D model. The DMI constant has a similar divergence near the +band minimum at µ ≃ −ER, +D(1) +0 +∼ (1 + µ/ER)−1/2 ∼ 1/√ε, +(A33) +where ε is the energy calculated from −ER. Such effects start to be visible at low temperature even at ER < ∆sd in +Fig. 1(b) and (e), and more so at ER > ∆sd, where the divergences in the DOS are accompanied by the divergence in +D(1) +0 , respectively in at Fig. 2(a) and (c). This situation becomes more visible for higher terms in the expansion that +contribute to order ∆4 +sd such as D(0) +1 +[see Fig. 1 (c) and (f)] since they contain a stronger divergence generated by +Fermi surface terms such as f ′′(ε) that occur in F (3) +s,k in Eqs. (17). +4. +Group D3 +There are cases where the SOC vector develops nonzero out-of-plane components where one could expect qualitatively +different results. This occurs for groups where rotation symmetry by π is absent: C1,3 and D1,3. This subsection +details the calculation of DMI tensors and constants in the Rashba model from Sec. V in group D3 for Γ4 bands. In +this case, the spin-orbit vector reads +∆soγ = (−α1ky, α1kx, α2ky(3k2 +x − k2 +y)), +(A34) + +12 +where for convenience α2 is defined as half of α2 from Tab. (I). The cubic term breaks the rotational symmetry of the +spectrum. +In the following, we will analyze the first terms, Ω(i) +1,0 and Ω(i) +1,1, in the Ginzburg-Landau expansion in magnetization +m without assuming either relative small SOC, or small exchange coupling. At the end of the section, the small SOC +will be treated in more detail since it allows analytical resolution for the DMI constants. +The non-vanishing components of D(0) +αβ,j tensor read +D(0) +jz,j = −D(0) +zj,j = −α1∆2 +sd +2 +⟨F(0) +s,k(λ)⟩, +D(0) +xy,x = −D(0) +yx,x = 6α2∆2 +sd +2 +⟨kxkyF(0) +s,k(λ)⟩, +(A35) +D(0) +xy,y = −D(0) +yx,y = 3α2∆2 +sd +2 +⟨(k2 +x − k2 +y)F(0) +s,k(λ)⟩, +with +λ = +� +∆2 +sd + α2 +1k2 + α2 +2k2y(3k2x − k2y)2. +(A36) +The last two equations in (A35) vanish by symmetry and therefore the free energy contribution reads +Ω(0) +1,0 = −α1∆2 +sd +2 +� +s +� d2k +4π2 F(0) +s,k(λ)Ljz,j. +(A37) +It exhibits the usual LI structure (see Tab. I). +There are 8 components of the D(1) +αβ,j tensor that do not vanish under the constraint D3 group imposes on the +angular integral, +D(1) +iz,i = −α3 +1∆2 +sd +2 +⟨k2 +¯i F(1) +s,k(λ)⟩ +D(1) +xx,x = α2 +1α2∆2 +sd +2 +⟨k2 +y(3k2 +x + k2 +y)F(1) +s,k(λ)⟩, +D(1) +xy,y = −α2 +1α2∆2 +sd⟨k4 +yF(1) +s,k(λ)⟩, +D(1) +yy,x = −3α2 +1α2∆2 +sd⟨k2 +xk2 +yF(1) +s,k(λ)⟩, +D(1) +yx,y = 3α2 +1α2∆2 +sd +2 +⟨k2 +x(k2 +y − k2 +x)F(1) +s,k(λ)⟩, +(A38) +D(1) +zx,x = α1α2 +2∆2 +sd +2 +⟨k2 +y(k4 +y − 9k4 +x)F(1) +s,k(λ)⟩, +D(1) +zy,y = α1α2 +2∆2 +sd⟨k4 +y(3k2 +x − k2 +y)F(1) +s,k(λ)⟩. +Due to the lack of rotation symmetry it is not immediate to resolve these integrals as was the case in C∞v. Using +polar coordinates and adding the contribution from all tensors as in Eq. (15) yields the energy density: +Ω(1) +1,0 = −∆2 +sd +2 +� dkk +4π2 +�α3 +1k2 +2 +(mx∂xmz + my∂ymz)r(1) +0 (k) +(A39) ++ 3α2 +1α2k4 +4 +(my∂xmy − mx∂xmx + my∂ymx + mx∂ymy)r(1) +0 (k) + α1α2 +2k6 +2 +(mz∂xmx + mz∂ymy)r(1) +1 (k) +� +, +where the angular integral acts inside functions r(1) +0,1. These are generally defined for following use, +r(m) +n +(k) = +� +s +� 2π +0 +dθ sin(3θ)2nF(m) +s,k (λ). +(A40) +The second term O(α2 +1α2) vanishes since it contains only total derivatives over products of magnetization components. +Then after factoring out the symmetric part of the rest of the components, which integrates to zero in the bulk, one +obtains +Ω(1) +1,0 = D(1) +0 Ljz,j +D(1) +0 += −α1∆2 +sd +32π2 +� +dkk3[α2 +1r(1) +0 (k) − k4α2 +2r(1) +1 (k)]. +(A41) +Thus, the usual LI invariant is indeed recovered to this order and a unique DMI constant is defined. +Higher-order tensors are expected to yield the non-LI contributions. There are 28 nonvanishing components to +tensors D(0) +αβµ1µ2,j. Adding the respective energy contribution from each of them yields +Ω(0) +1,1 = ∆4 +sd +2 +� ∞ +0 +dkk +4π2 +� +− α3 +1k2 +2 +(1 − m2 +z)r(2) +0 (k)Ljz,j + 3α2 +1α2k4 +4 +r(2) +0 (k)[2mxmyLyx,x + (m2 +x − m2 +y)Lyx,y] ++ 3α1α2 +2k6 +2 +(1 − 8 +3m2 +z)r(2) +1 (k)Ljz,j +� +, +(A42) + +13 +with the functions r(2) +0,1 defined as in Eq. (A40). At this order, it is practical to define 3 DMI constants to quantitatively +describe the free energy, +Ω(0) +1,1 = D(0) +1a (1 − m2 +z)Ljz,j + D(0) +1b [2mxmyLyx,x + (m2 +x − m2 +y)Lyx,y] + D(0) +1c [(1 − 8 +3m2 +z)Ljz,j]. +(A43) +Finally, there are 80 non-zero components to D(1) +αβµ1µ2,j. Adding the contributions from each one yields the free +energy +Ω(1) +1,1 = ∆4 +sd +2 +� +dkk +(2π)2 +� +− 3α5 +1k4 +16 +r(3) +0 (1 − m2 +z +2 )Ljz,j + 9α4 +1a2k6 +16 +(r(3) +0 +− r(3) +1 )[2mxmyLyx,x + (m2 +x − m2 +y)Lyx,y] +(A44) ++ 15α3 +1α2 +2k8 +16 +r(3) +1 (1 − 5 +2m2 +z)Ljz,j − 9α2 +1α3 +2k10 +8 +r(3) +1 [2mxmyLyx,x + (m2 +x − m2 +y)Lyx,y] + 3α1α4 +2k12 +8 +r(3) +2 (k)m2 +zLjz,j +� +. +The non-LI invariants that are present in the free-energy expansion to higher order are characterized by qualitatively +new invariants of the type 2mxmyLyx,x + (m2 +x − m2 +y)Lyx,y. These are identical to the non-LI invariant in the D3h +group analyzed in Ref. [48]. Modulo total derivatives which vanish in the bulk, they are related as +mx(m2 +x − 3m2 +y) = −3 +4(2mxmyLyx,x + (m2 +x − m2 +y)Lyx,y). +(A45) +Small SOC +Several simplifications are possible in the small SOC limit, where the rotation symmetry breaking SOC distortion to +the energy spectrum is treated perturbatively. The first order corrections require knowledge of tensors D(0) +αβ,j, D(1) +αβ,j, +and D(0) +αβµ1µ2,j. In the small SOC limit, at each order one recovers in the integral the rotational symmetry such that +the expression for DMI constants is further simplified. +Computed to cubic order in spin-orbit coupling, the nonvanishing components are from (A35) +D(0) +iz,i ≃ −α1∆2 +sd +2 +� +s +� dkk +2π +� +F(0) +s,k(∆sd) + k2 +2 +� +α2 +1 + α2 +2k4 +2 +� +F(1) +s,k(∆sd) +� +. +(A46) +This exhibits the same structure as in the C∞v case. Defining D(0) +0 += D(0) +xz,x, the free energy density reads +Ω(0) +1,0 = D(0) +0 Ljz,j. +(A47) +Thus working to linear order in SOC yields the conventional LI invariant characterizing the D3 (or C3v) point group +in 2D (e.g. see Ref. [4]). +The contribution from D(1) +αβ,j also simplifies since to cubic order in SOC F(1) +s,k(λ) = F(1) +s,k(∆sd) and the angular +integral is trivial. Therefore, it readily follows from Eq. (A41) that +Ω(1) +1,0 = D(1) +0 Ljz,j, +D(1) +0 += −α1∆2 +sd +16π +� +s +� +dkk3� +α2 +1 − α2 +2k4 +2 +� +F(1) +s,k(∆sd), +(A48) +which renormalizes the previous term (A47). +Finally, the last term to cubic order in SOC is the contribution from D(0) +αβµ1µ2,j. From Eq. (A42) it follows directly +that the 3 DMI constants are determined after performing the angular integral in r(2) +n +functions: +D(0) +1a = −α3 +1∆4 +sd +8π +� +s +� +dkk3F(2) +s,k(∆sd), +D(0) +1b = 3α2 +1α2∆4 +sd +16π +� +s +� +dkk5F(2) +s,k(∆sd), +(A49) +D(0) +1c = 3α1α2 +2∆4 +sd +16π +� +s +� +dkk7F(2) +s,k(∆sd). +In the zero-temperature approximation the DMI constants reveal that there is a nonvanishing contribution due +to cubic terms in momentum. Since both D(0) +0 +and D(1) +0 +contribute to the conventional LI, we add them to yield +D(+) +0 += D(0) +0 ++ D(1) +0 , +D(+) +0 += −α1∆2 +sd +4π +� +s +� +dkk +� +F (0) +s,k(∆sd) + k2 +4 +� +3α2 +1 + α2 +2k4 +2 +� +F (1) +s,k(∆sd) +� +, +(A50) + +14 +where the k6 term contributes at zero-temperature above the gap, +D(+) +0 += +�α1∆sdm +8πℏ2 +(1 − µ2 +∆2 +sd +) − 3α3 +1 +16π +µ +∆sd +� m +ℏ2 +�2 +(1 − µ2 +∆2 +sd +) + α1α2 +2 +80π +� m +ℏ2 +�4 +(1 + +µ +∆sd +)3(8 − 9 µ +∆sd ++ 3 µ2 +∆2 +sd +) +� +Θ(1 − µ2 +∆2 +sd +) ++ α1α2 +2∆2 +sd +5π +� m +ℏ2 +�4 +Θ(µ − ∆sd). +(A51) +Finally, from Eqs. (A49) we obtain +D(0) +1a = +3α3 +1µ +16π∆sd +� m +ℏ2 +�2� +1 − 5µ2 +3∆2 +sd +� +Θ +� +1 − µ2 +∆2 +sd +� +, +D(0) +1b = 3α2 +1α2∆sd +32π +� m +ℏ2 +�3� +1 − µ2 +∆2 +sd +�� +1 − 5µ2 +∆2 +sd +� +Θ +� +1 − µ2 +∆2 +sd +� +, +(A52) +D(0) +1c = 9α1α2 +2µ∆sd +16π +� m +ℏ2 +�4� +1 − µ2 +∆2 +sd +�2 +Θ +� +1 − µ2 +∆2 +sd +� +. +The SOC in Γ5 and Γ6 bands in point group D3 reads [see Tab. I] +∆soγ = (iα1(k3 ++ − k3 +−), α2(k3 ++ + k3 +−), iα3(k3 ++ − k3 +−)), +(A53) +with k± = kx ± iky. Working at small SOC, the integrals are expanded term by term, and we find zero contribution to +O(α5 +i ). +5. +DMI constants for LI invariants in all 2D symmetry groups +TABLE I. Lifshitz invariants for all the two-dimensional groups obtained to linear order in SOC as contained in the free energy +density Ω(1) +1,0. The parameters αi are real and βi are complex, k± = kx ± iky. The spin-orbit interaction ∆soγ · σ is determined +by the vector ∆soγ as derived in Ref. [41] (here modulo an eventual overall sign change). +Group +Γ +∆soγ +Ω(1) +1,0 +C1 +Γ2 +(α1kx + α2ky, α3kx + α4ky, α5kx + α6ky) +I1Lyz,x + I2Lyz,y + I3Lzx,x + I4Lzx,y ++ I5Lxy,x + I6Lxy,y +C2 +Γ3,4 +(α1kx + α2ky, α3kx + α4ky, 0) +I1Lyz,x + I2Lyz,y + I3Lzx,x + I4Lzx,y +C3 +Γ4,5 +(α1kx + α2ky, −α2kx + α1ky, β1k3 ++ + β∗ +1k3 +−) I1(Lyz,x + Lzx,y) + I2(Lxz,x + Lyz,y) +Γ6 +(β1k3 ++ + β∗ +1k3 +−, β2k3 ++ + β∗ +2k3 +−, β3k3 ++ + β∗ +3k3 +−) +0 +C4 +Γ5,6,7,8 +(α1kx + α2ky, −α2kx + α1ky, 0) +I1(Lyz,x + Lzx,y) + I2(Lxz,x + Lyz,y) +C6 +Γ7,8,9,10 +(α1kx + α2ky, −α2kx + α1ky, 0) +I1(Lyz,x + Lzx,y) + I2(Lxz,x + Lyz,y) +Γ11,12 +(β1k3 ++ + β∗ +1k3 +−, β2k3 ++ + β∗ +2k3 +−, 0) +0 +D1 +Γ3,4 +(α1ky, α2kx, α3ky) +I1Lyz,y + I2Lzx,x + I3Lxy,y +D2 +Γ5 +(α1ky, α2kx, 0) +I1Lyz,y + I2Lzx,x +D3 +Γ4 +(−α1ky, α1kx, −iα2(k3 ++ − k3 +−)) +I1(Lzx,x + Lzy,y) +Γ5,6 +(iα1(k3 ++ − k3 +−), α2(k3 ++ + k3 +−), iα3(k3 ++ − k3 +−)) +0 +D4 +Γ6,7 +(α1ky, −α1kx, 0) +I1(Lxz,x + Lyz,y) +D6 +Γ7,8 +(α1ky, −α1kx, 0) +I1(Lxz,x + Lyz,y) +Γ9 +(iα1(k3 ++ − k3 +−), α2(k3 ++ + k3 +−), 0) +0 +Here we determine the DMI constants for LIs obtained in the approximation of relatively weak SOC ∆so ≪ ∆sd, +extracted from Ω(0) +1,0. The results are presented in Table I for all symmetry groups. Generically we see that cubic terms +in momentum are irrelevant to first order in spin-orbit coupling. This readily yields the DMI constants determined by +a single integral, +Ii = αi∆2 +sd +4π +� +s +� +dkkF(0) +s,k(∆sd). +(A54) +In the zero temperature limit it reads +Ii = −αim∆sd +8πℏ2 +� +1 − µ2 +∆2 +sd +� +Θ +� +1 − µ2 +∆2 +sd +� +. +(A55) + +15 +The Ii coefficients are formally the same (up the value of αi) with the one analyzed in detail the C∞v case, i.e., D(0) +0 +from Eq. (A16). +6. +Gapped Dirac model +An important limit with application to topological materials is that of a Dirac model with Rashba spin-orbit +interactions and gapped by the exchange coupling. The Hamiltonian for a C∞v model reads +H = −µσ0 + αR(k × σ)z + ∆sdm(r) · σ. +(A56) +In this case, the first DMI constants are computed exactly to all orders at zero temperature. +The first DMI coefficient from Eq. 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(North-Holland +Pub.Co., 1966). + diff --git a/UtFKT4oBgHgl3EQfly6I/content/tmp_files/load_file.txt b/UtFKT4oBgHgl3EQfly6I/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d426857fcbf92afe019c731b73ae682a92a9649 --- /dev/null +++ b/UtFKT4oBgHgl3EQfly6I/content/tmp_files/load_file.txt @@ -0,0 +1,1160 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf,len=1159 +page_content='Non-Lifshitz invariants corrections to Dzyaloshinskii-Moriya interaction energy Doru Sticlet1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' ∗ and Fr´ed´eric Pi´echon2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' † 1National Institute for R&D of Isotopic and Molecular Technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 67-103 Donat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 400293 Cluj-Napoca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Romania 2Universit´e Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Laboratoire de Physique des Solides,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 91405 Orsay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' France We study the continuum limit of two-dimensional chiral magnets in which Dzyaloshinskii-Moriya interaction (DMI) is due to the interplay between a smooth magnetic texture and spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The resulting free energy density of the system contains linear terms in the spatial gradient of the magnetic texture, which mark an instability of the system towards the formation of nontrivial magnetic orders such as skyrmions or chiral domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' We perform a microscopic analysis of DMI tensors responsible for this contribution to free energy based on a Berry phase formulation in the mixed space of momentum and position, and reveal that they exhibit non-Lifshitz invariants features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In particular, a perturbation theory shows in the case of Rashba spin-orbit interactions the presence of non-Lifshitz invariants to third order in the small spin-orbit interaction and fourth order in the small exchange coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The higher order terms may even lead to an enhancement of DMI interaction at strong spin-orbit coupling due to divergences in the density of states at the bottom of the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Finally, we also study the DMI free energy generated from Rashba spin-orbit interaction in different symmetry groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' INTRODUCTION Chiral symmetry-breaking in magnetic materials re- sults in an antisymmetric exchange coupling called the Dzyaloshinskii-Moriya interaction (DMI) [1–3], which tends to cant neighboring spins such that noncollinear magnetic orders are favored in the system [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' As a consequence of DMI, nontrivial magnetic structures such as chiral domain walls [7–9] and skyrmions [10–14] be- come stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The latter are excitations in the form of magnetization vortices, which are topologically robust, and have been the subject of intense research in recent years [15–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The controlled creation and annihilation of skyrmions with spin-polarized currents [19, 20], gate voltages [21, 22], or lasers [23] feeds the driving goal to re- alize energy-efficient spintronic devices operating at room temperature for memory storage [24–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' A promising platform for probing such physics is in effectively two-dimensional systems where interfacial DMI develops [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In thin ferromagnetic films or in multilayers with alternating magnetic and nonmagnetic layers, the inversion symmetry is broken at the interfaces and thus a strong spin-orbit coupling (SOC) is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In a long wavelength approach, the magnetic texture below the Curie temperature is described by the continuous magnetic density vector m(r) of unit amplitude, with position r in the plane of the magnetic layer (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The effect of SOC is to generate in the free energy linear terms in the texture gradient which are characteristic for the DMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The micromagnetic DMI free energy follows from Ω1 = � α∈{x,y,z} j∈{x,y} wjα ∂mα ∂rj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (1) ∗ doru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='sticlet@itim-cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='ro † frederic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='piechon@universite-paris-saclay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='fr The most common approach [2] is to consider that energies wjα are linear in magnetization m, Ω1 ≃ 1 2Dαβ,j(mα∂rjmβ − mβ∂rjmα), (2) which amounts to take into account only the well-known Lifshitz invariant (LI) contribution to Ω1 (see A 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The microscopic analytical calculation of DMI tensor Dαβ,j in the continuum limit was only recently performed for the first time in topological insulators [29, 30] and a two- dimensional (2D) Rashba thin film [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Notably, these were preceded by different approaches where DMI was explained in the vein of Ruderman-Kittel-Kasuya-Yosida theory as due to spin interactions mediated by conduc- tion electrons [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' An analysis of effects beyond the Lifshitz invariant correction was performed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [34] in order to get a more general description of DMI in chiral magnets, and it has established that such corrections can be consequential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' It was soon shown that indeed there are cases as in tetrahedral magnets where the conventional LI contribution vanishes by symmetry while the remain- ing non-LI contributions lead to a non-collinear magnetic structure [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In this paper, we revisit the issue of non-Lifshitz invari- ant contribution to DMI from a different point of view, in which DMI is due to Berry curvature in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Our approach assumes that the magnetization m varies smoothly in space such that the effect of m on the pe- riodic Bloch wave functions is considered perturbatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In this sense, the Bloch wave vectors depend on position through m, |n, k, m(r)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In such cases it is naturally to consider an approach based on a generalized Berry phase in the space defined by position and momentum [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In- deed, it was shown that the dynamics of electrons in the thin layer is determined by the Berry curvature in the phase space [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Here we analyze the generic case of two-band systems with crossings near the Γ point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In such cases it is an- alytically tractable to obtain the form of DMI tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='11855v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='mes-hall] 27 Jan 2023 2 These are determined from the corresponding DMI en- ergy Ω1, which follows from an expansion of the total grand-canonical thermodynamic free energy Ω in texture gradients .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' This generates a contribution that is propor- tional to the Berry curvature in phase space [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The expansion in gradient is also further refined with addi- tional expansions in SOC or exchange amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' This allowed us to determine the order at which non-LI contri- butions might become relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' As an application, the present study focuses on Rashba SOC which occurs naturally in effective 2D systems due to the large variation in the electrostatic potential normal to the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Usually the structure of the LI and non-LI invariants may be determined by symmetry analysis [2, 4, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Here we perform instead a microscopic analysis where such the structure is emergent from effective two-band Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Such models are based on the specific form of the Bloch bands at the Γ point, as constrained by symmetries of the magnetic point groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The form of Rashba SOC to cubic order in momentum was classified for 2D materials in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [40, 41], and constitutes for us a starting point in determining microscopically the DMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Our analysis reveals in the Berry curvature formulation of the problem, that the DMI free energy decomposes into two distincts parts Ω1 = Ω(0) 1 + Ω(1) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Usually only the first part has been a subject of investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The second contribution, Ω(1) 1 , is higher order in SOC, but nonetheless it is of the same order in the magnetization m, and also enters to the same order in the exchange coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' For example, the second contribution in its lowest order in SOC is responsible for symmetric DMI tensors, which are usually discarded in the bulk, but may generate some nontrivial edge spin texture [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The term Ω(1) 1 contains also antisymmetric DMI tensors, which renormalize the Ω(0) 1 contribution, and additionally, we show that they can lead to divergences in the free energy since they contain Fermi surface contributions which diverge at low temperature due to singularities in the density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' All our investigations are made concrete in the study of effective models with Rashba SOC in different symmetry groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' II introduces the class of two-band Hamiltonian models in which DMI develops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The section also reviews the generic structure of the energy density Ω1 that is linear in a smooth spatial gradient of the magnetization, using a Berry phase for- mulation in the mixed space of momentum and position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' III develops a perturbation theory which uncovers the non-Lifshitz invariants corrections to Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The section expresses the general form of all DMI tensors, and devel- ops further expansions in the small and large SOC limit, relative to the exchange energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' IV particularizes the analysis to the case of Rashba SOC in the C∞v group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' V looks briefly at the DMI contribution from Rashba SOC in different symmetry groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Appendix A details several of the points in the main paper such as: a deter- mination of DMI constants for the conventional Rashba SOC in C∞v group, an analysis of group D3 where the SOC exhibits an out-of-plane component, a table with LI and Ω1 in all 10 2D groups obtained in the limit of small SOC, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' VI summarizes the main points in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' PHASE SPACE BERRY CURVATURE FORMULATION OF SPIN ORBIT INDUCED FREE ENERGY TERMS, LINEAR IN SPATIAL MAGNETIZATION GRADIENT This section briefly recalls the derivation of the correc- tion Ω1 to free energy density that is linear in the gradient of the magnetic texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Starting from generic two-band Hamiltonian models, it is shown that the correction Ω1 writes as an average over occupied states of the momen- tum and position-dependent skyrmion-like density of a vector field h(k, r) that combines the spin-orbit coupling and the exchange coupling to the magnetic texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Com- plementary to previous works, we show that this skyrmion density entails two distinct contributions that appear at different order in spin-orbit coupling, but nonetheless both contributing to same order in the magnetization m and exchange coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Model Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In the following, we focus on generic two-band Hamiltonian models of the form H(k, r) = ξ(k)σ0 + h(k, r) · σ, h(k, r) = ∆soγ(k) + ∆sdm(r), (3) with σ the vector of Pauli matrices, and σ0 the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The first term is the energy dispersion of elec- trons in the absence of spin-orbit coupling which is an even function of momentum ξ(−k) = ξ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The second contribution is a momentum and position dependent vec- tor field h(k, r) that combines the spin-orbit coupling (SOC) and the exchange coupling to the magnetic texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The SOC is described by an antisymmetric spin-orbit vector γ(−k) = −γ(k) and a coupling strength ∆so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The magnetic exchange is characterized by a coupling strength ∆sd = JsdS, with S, the magnitude of spins in the mag- netic layer, and Jsd, the exchange coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The magnetic texture is modeled by a (unit length) vector m(r) which varies smoothly in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' As it appears below, the cou- pling strengths ∆so and ∆sd are useful parameters to keep track of the order in a perturbation theory in weak spin-orbit or weak exchange coupling limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Free energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The free energy density is ob- tained from the local density of states ρ(ε, r), Ω(r) = � dερ(ε, r)g(ε), (4) with g(ε) the primitive of the Fermi-Dirac distribution function f(ε) = g′(ε), f(ε) = 1/(1 + eβ(ε−µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The local density of states, expressed using a Wigner representation of Green’s function, is obtained from a Moyal gradient expansion in the smooth magnetic texture as ρ = ρ0 + ρ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [31, 38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 3 The effective density of states to linear order in the magnetization gradient reads ρ(ε, r) = ⟨(1 + Bjj s,k)δ(ε − εs,k − shBjj s,k)⟩ (5) with the shorthand notation ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='⟩ ≡ � s=± � ddk (2π)d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' , (6) and where summation over repeated indices j is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (5), εs,k is the semiclassical energy spectrum of the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (3) εs,k(r) = ξ(k) + sh(k, r), h(k, r) = � ∆2 sd + ∆2soγ2 + 2∆so∆sdγ · m, (7) with s = ±, the band index, and h ≡ |h|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Lastly, Bij s,k(r) = −s1 2 h · (∂rih × ∂kjh) |h|3 , (8) denotes the element (ij) of the intraband phase space Berry curvature tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The expression Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (5) illustrates two qualitatively dis- tinct effects resulting from the gradient corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' On the one hand, there is a momentum-position dependent shift of the band spectrum, and, on the other hand, there is also a modification of the spectral weight [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' To lin- ear order in the gradient, both effects are proportional to the phase-space Berry curvature Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Using this effective density of states, the free energy density is de- composed as Ω = Ω0 +Ω1, with a zero-order contribution describing the uniform state, Ω0(r) = � dερ0(ε)g(ε), and a contribution Ω1(r), linear in the gradient of m, which reads Ω1(r) = ⟨h · (∂rjh × ∂kjh) 2 Fs,k⟩, (9) with Fs,k(r) = sg(εs,k) − hf(εs,k) h3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (10) Using the explicit expression of h from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (3), it follows that the correction to free energy density Ω1 has two distinct contributions Ω1 = Ω(0) 1 + Ω(1) 1 , (11) with Ω(0) 1 (r) = ∆so∆2 sd 2 ⟨∂kjγ · (m × ∂rjm)Fs,k⟩, Ω(1) 1 (r) = ∆2 so∆sd 2 ⟨∂rjm · (∂kjγ × γ)Fs,k⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (12) These contributions Ω(0,1) 1 (r) have a structure similar to the one in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (1) and as detailed in the next section, both generate Lifshitz invariant and non-Lifshitz invariant contributions to the DMI interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' At this point, a few remarks are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The possibility to express the linear gradient corrections in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (9) solely in terms of the intraband phase-space Berry curvature is specific to two-band models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Likewise, the possibility to express the phase-space Berry curvature directly in terms of a phase-space skyrmion-like density of the vector field h(k, r) is also specific to two-band models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' However, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (9) are valid for any two-band model (in any dimension) of the form given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Importantly, the expression Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (9) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (12) contain full non-perturbative dependencies in the coupling strengths ∆so and ∆sd and also full non- linear dependencies in the magnetization vector m(r) since all these parameters appear implicitly in h and Fs,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' GENERAL EXPANSION OF NON-LIFSHITZ INVARIANT CONTRIBUTIONS In the following, the free energy density contributions Ω(0,1) 1 are expressed as Ginzburg-Landau-like expansions in m when considering ∆so∆sdγ · m/λ2 as a small pa- rameter, with λ = � ∆2 sd + ∆2soγ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (13) Generically, the free energy densities are expanded as Ω(i) 1 = ∞ � n=0 Ω(i) 1,n, (14) with Ω(i) 1,n = D(i) αβµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='µ2n,j(mα∂rjmβ)mµ1· · · mµ2n, (15) where D(i) are the DMI tensors of odd rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The low- est order term n = 0 is quadratic in m and yields the Lifshitz invariant contributions D(i) αβ,j of the DMI tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The higher terms n > 0 yield the non-Lifshitz-invariant contributions D(i) αβµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='µ2n,j of the DMI tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' More quantitatively (see appendix A 2 for details), the expansion of eigenenergies leads to an expansion of Fs,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Note that since γ is antisymmetric in k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' only the sym- metric part of Fs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k in k contributes to Ω(0) 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' and only the antisymmetric part of Fs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k contributes to Ω(1) 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Ω(0) 1 = ∞ � n=0 ∆2n+1 so ∆2n+2 sd 2 ⟨∂kjγ · (m × ∂rjm)(γ · m)2n × F(2n) s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k (λ)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (16) Ω(1) 1 = ∞ � n=0 ∆2n+3 so ∆2n+2 sd 2 ⟨∂rjm · (∂kjγ × γ)(γ · m)2n+1 × F(2n+1) s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k (λ)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 4 where the coefficients F(n) s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k are even in k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' and are deter- mined iteratively F(0) s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k(λ) = Fs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k ���� γ·m=0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' F(n) s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k (λ) = 1 nλ ∂F(n−1) s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k (λ) ∂λ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (17) for n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The use of argument λ in previous expressions implies that all dependence on energy εs,k simplifies to one on ε(0) s,k = ξ + sλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (15) and (16) readily yield the general form of DMI tensors D(0) αβµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='µ2n,j = 1 2∆2n+1 so ∆2n+2 sd ϵαβδ⟨(∂kjγδ)γµ1· · · γµ2n × F(2n) s,k (λ)⟩, (18) D(1) αβµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='µ2n,j = 1 2∆2n+3 so ∆2n+2 sd ϵνβδ⟨γαγν(∂kjγδ) × γµ1· · · γµ2nF(2n+1) s,k (λ)⟩, (19) with ϵαβδ, the Levi-Civita symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The usual Lifshitz invariants contribution to the en- ergy is contained in Ω(i) 1,0, and the related DMI tensors are D(i) αβ,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The expansion beyond the first order is responsible for non-Lifshitz invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Note that even to first order, there is a marked difference between the two tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The first tensor D(0) αβ,j is antisymmetric in α and β indices, while there is no such constraint on D(1) αβ,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The sym- metric part of the latter tensor is usually neglected since it multiplies a total derivative ∂rj(mαmβ), and vanishes when integrating over the entire sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' It was shown however that it has physical effects in generating specific magnetic textures at the sample boundary [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Since we treat here the case of an infinite system, we consider only the antisymmetric part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' It is particularly revealing to truncate the free energy density expansion to the first term where non-LI contribu- tions are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' This is done either in the limit of small spin-orbit coupling, or small exchange coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (18) and (19), it follows that at weak SOC the free energy is approximated Ω1 = Ω(0) 1,0 + Ω(1) 1,0 + Ω(0) 1,1 + O(∆5 so/∆5 sd), (20) ≃ (D(0) αβ,j + D(1) αβ,j + D(0) αβµ1µ2,jmµ1mµ2)mα∂rjmβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Similarly, in the case of weak exchange coupling (or large SOC) ∆so ≫ ∆sd, Ω1 = Ω(0) 1,0 + Ω(1) 1,0 + Ω(0) 1,1 + Ω(1) 1,1 + O(∆6 sd/∆6 so), (21) ≃ � i=0,1 (D(i) αβ,j + D(i) αβµ1µ2,jmµ1mµ2)mα∂rjmβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Note that the power counting in the two expansions is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' At small SOC, the linear order in ∆so is con- tained in Ω(0) 1,0 alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' This contribution to free energy and all conventional LI invariants are therefore deter- mined exactly in this limit from the analysis of Ω(0) 1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In the Appendix A 5 we have microscopically obtained the LI invariants in all 10 two-dimensional point groups by considering the symmetry-allowed spin-orbit coupling to cubic order in momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In contrast, in the limit of large SOC or small exchange, both tensors D(0,1) αβ,j already contribute at the lowest order ∆2 sd, such that both Ω(0) 1,0 and Ω(1) 1,0 are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Finally, the explicit expression of F(n) s,k coefficients (17) up to n = 4, necessary to give the dominant non-Lifshitz invariants in both limit cases of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (20) and (21) are given in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' APPLICATION TO RASBHA SPIN-ORBIT COUPLING The general theory from above is instantiated now in the important case of Rashba spin-orbit interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The simplest case is that of the C∞v group with a rotationally symmetric Rashba coupling ∆soγ = αR(−ky, kx, 0) for electrons with a parabolic spectrum, H = (ℏ2k2 2m − µ)σ0 + αR(k × σ)z + ∆sdm · σ, (22) with αR, the amplitude of Rashba SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Small SOC expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The limit of weak spin-orbit coupling relative to the exchange coupling ∆sd is relevant in experiment and is the focus of the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' To ob- tain the first non-LI invariant contribution to free energy density it is necessary to expand Ω1 to cubic order in αR as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' That requires determining the tensors D(0) αβ,j, D(1) αβ,j, and D(0) αβµ1µ2,j (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' A 3 for details about the DMI tensors involved beyond the weak SOC approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Using the rotational symmetry of the Rashba SOC allows one to readily show that all nonzero tensor elements of D(0) αβ,j are equal in amplitude, such that there is a single DMI constant characterizing the free energy density Ω(0) 1,0 = D(0) 0 Ljz,j, (23) with the DMI constant D(0) 0 = D(0) xz,x, D(0) 0 = −αR∆2 sd 4π � s � dkk λ3 � sg0,s − λf0,s � , (24) and Lifshitz invariant Lαβ,j = mα∂rjmβ − mβ∂rjmα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (25) To first order in αR, λ = ∆sd in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (13), recovering the result in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The tensors D(1) αβ,j, and D(0) αβµ1µ2,j are analyzed simi- larly, yielding the free energy contributions Ω(1) 1,0 and Ω(0) 1,1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Since again, in each tensor, the components are equal in amplitude, it is possible to factor out a single DMI constant in the free energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Ω(1) 1,0 = D(1) 0 Ljz,j, Ω(0) 1,1 = D(0) 1 (1 − m2 z)Ljz,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (26) 5 −2 −1 0 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='8 D(0) 0 (a) ER/∆sd 0th 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='4 −2 −1 0 1 2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='3 D(1) 0 (b) ER/∆sd 0th 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='4 −2 −1 0 1 2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='6 D(0) 1 (c) ER/∆sd 0th 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='2 −2 −1 0 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='8 D(0) 0 (d) T/∆sd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='4 −2 −1 0 1 2 µ/∆sd −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='10 D(1) 0 (e) T/∆sd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='4 −2 −1 0 1 2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='6 D(0) 1 (f) T/∆sd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The DMI constants D(0) 0 , D(1) 0 , and D(0) 1 in units of kR∆sd/8π as a function of chemical potential in the limit of small spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Panels (a, c, e) show the three DMI constants at different ER and kBT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='01∆sd, with the red line represents denoting the zeroth order approximation where the DMI constants are linear in αR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Panels (b, d, f) present the same DMI constants’ behavior at different temperatures and at fixed ER = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='2∆sd [kB = 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The constant D(1) 0 = D(1) xz,x/2 is obtained by extracting out the antisymmetric contribution in D(1) αβ,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' To cubic order in αR it reads D(1) 0 = −α3 R∆2 sd 16π � s � dkk3F(1) s,k(∆sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (27) Finally, the DMI constant D(0) 1 = D(0) xzxx,x from D(0) αβµ1µ2,j has the expression to O(α3 R): D(0) 1 = −α3 R∆4 sd 8π � s � dkk3F(2) s,k(∆sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (28) Therefore, the free energy density Ω1 in this approxima- tion is determined by all the three contributions Ω1 ≃ [D(0) 0 + D(1) 0 + (1 − m2 z)D(0) 1 ]Ljz,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (29) Already, to cubic order in αR there are now non-LI invari- ants in the free energy, m2 zLjz,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The additional depen- dence on m2 z is a property due to the rotational symmetry of the problem and was already predicted [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Using natural momentum and energy scales character- izing the Rashba SOC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' kR = mαR ℏ2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' and ER = mα2 R 2ℏ2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (30) respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' yields simple analytical formulas for the con- stants in the zero-temperature approximation to O(α3 R): D(0) 0 ≃ kR∆sd 8π � 1 − 2ERµ ∆2 sd �� 1 − µ2 ∆2 sd � Θ � 1 − µ2 ∆2 sd � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' D(1) 0 ≃ −kRERµ 8π∆sd (1 − µ2 ∆2 sd )Θ(1 − µ2 ∆2 sd ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (31) D(0) 1 ≃ 3kRERµ 4π∆sd (1 − 5µ2 3∆2 sd )Θ(1 − µ2 ∆2 sd ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' with Θ the Heaviside function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The results for D(0) 0 from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [31] are recovered by eliminating the cubic depen- dence on SOC by formally setting ER to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In the zero temperature limit it follows that the DMI energy is non- vanishing only when the Fermi surface determined by µ is inside the exchange gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In this case there is a single circular Fermi surface at k ≃ � 2m(∆sd + µ)/ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Out- side the exchange gap µ > ∆sd, there are always two Fermi surfaces, with equal contribution and opposite sign, canceling in the sum over the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The behavior of DMI constants for different relative strengths ER/∆sd and at different temperatures are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The zero-temperature approximation recovers the numerical behavior at low temperature and weak SOC ER/∆sd ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' At weak SOC the constant D(0) 0 is symmetric in µ, while D(1) 0 and D(0) 1 , antisymmetric around the middle of exchange gap µ/∆sd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Such symmetry is quickly lost at larger SOC and generally the constants have a higher value near the bottom of the gap, as explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Larger corrections in αR also lead to increasing the number of zeros in the free energy in their respective contribution at µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' With decreasing temperature and increasing ER/∆sd, the constants develop divergences at the bottom of the 6 −10 −5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='100 DOS (a) −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='0 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='0 µ/∆sd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='3 D(0) 0 (b) −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='0 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='75 D(1) 0 (c) ER/∆sd 6 8 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (a) Density of states at small ∆sd (large SOC, ER/∆sd > 1) presents divergences at the bottom of the lowest band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Comparison between numerics (solid line) at kBT/∆sd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='1 [kB = 1] zero-temperature analytic approximation (dashed line) for (b) D(0) 0 and (c) D(1) 0 DMI constants in the Rashba C∞v case in units of kR∆sd/8π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' This is visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 1(b), (c), (e) and (f) and it is due to the presence of derivatives of the Fermi-Dirac distribution in coefficients F(1) s,k and F(2) s,k (A12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' This effect cannot be captured analytically in the weak SOC expansion since αR enters only as an overall prefactor, and the effective energy bands are determined by ∆sd alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The divergence is however readily understood when considering αR non-perturbatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Large SOC expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' It is telling to analyze this effect quantitatively in the opposite limit ER/∆sd ≫ 1, although the effect is visible beyond this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' At large SOC, the two energy parabolas ℏ2k2/2m for spin up and down are shifted, creating a degenerate manifold of momentum states with zero group velocity at k ≃ kR at the bottom of the lower band µ ≃ −ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Since the density of states is effectively one dimensional (1D) there, the total density of states will exhibit the usual inverse square root energy singularity [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 2(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' More quantitatively, at zero temperature and in the limit of ER ≫ ∆sd, the leading approximation involves both D(0) 0 and D(1) 0 to O(∆2 sd), D(0) 0 + D(1) 0 ≃ kR∆2 sd 16πER × � (1 + µ ER ) 1 2 + (1 + µ ER )− 1 2 , µ ∈ (−ER, −∆sd), 1 − µ ∆sd , µ ∈ (−∆sd, ∆sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (32) At µ = −∆sd, the two asymptotic expressions match to leading order in ∆sd/ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' More importantly, near the band minimum at µ ≃ −ER, the constant D(1) 0 displays the typical 1D singularity in the density of states D(1) 0 ∼ (1 + µ/ER)−1/2 ∼ 1/√ε, where ε is the energy calculated from −ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The analytical results are corroborated with the numerical calculation of DMI constants presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 2(a) and (c), where the typical divergences in the 1D density of states are accompanied by the divergence in D(1) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Similar results are expected for higher-order terms in the expansion that contribute to order ∆4 sd such as D(0) 1 [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 1 (c) and (f)] since they contain a stronger divergence generated by Fermi surface terms such as f ′′(ε) that occur in F(3) s,k in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The above considerations explain the divergences devel- oping in DMI constants of higher order in αR (see details in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' A 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' This effect could be used as an exploit to single out non-LI contributions, with the provision that it would be seen only in the low-temperature regime, at strong SOC, with a chemical potential finely tuned near the lower band bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' RASBA SPIN-ORBIT COUPLING IN DIFFERENT SYMMETRY GROUPS In order to analyze microscopically the DMI free energy in all 10 two-dimensional point groups, we consider effec- tive SOC derived to cubic order in momentum in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Notably, in such cases, the rotational symmetry of C∞v may be lost, and the SOC vector may develop out-of-plane components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The latter is true in point groups where a π rotation around z axis is not a group element: C1, C3, D1, and D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In the remaining 6 groups, symmetry under a π rotation and antisymmetry of γz, imposes γz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Consequently, in these groups there is a drastic reduction in the number of linearly independent components of the DMI tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Namely, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (18) and (19), it follows that in the DMI energy mz enters only once, and the DMI tensors are reduced to D(0,1) izl1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='l2n,j, with Latin indices in the (x, y) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Let us briefly analyze the example of group D3 in Γ4 bands where the SOC vector develops an out-of-plane component, ∆soγ = (−α1ky, α1kx, α2ky(3k2 x − k2 y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (33) The SOC in this group is relevant for topological surface states of Bi2Te3 and Bi2Se3 [43, 44], BiTeI [45], hole gases in quasi-2D semiconductors [46], (001) surface states of oxide SrTiO3 [47], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 7 Note that the spin-orbit vector is identical in the x and y components to the case explored in the previous section, and therefore one expects to recover some of the same structure of DMI tensors from C∞v case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' However, there is an additional cubic dependence on momentum in the z component of the SOC vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The analysis in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' A 4 shows that to first order in a perturbation theory Ω(0) 1,0 +Ω(1) 1,0 there is no contribution from the cubic term, and the expected LI invariant follows, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Ljz,j generated by γx and γy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The effect of cubic Rashba term γz is visible only at the level of non-LI invariants present in the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' To cubic order in the SOC, there are now two non-LI invariants generated in the free- energy expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' One is identical to the previous C∞v case, and represents a quartic interaction of spins of the form m2 zLjz,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Additionally, there is a new invariant that involves only in-plane interactions between the spins, 2mxmyLyx,x + (m2 x − m2 y)Lyx,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (34) This non-LI is proportional (up to total derivatives that vanish in the bulk) to the invariant mx(m2 x − 3m2 y)∂imi that was analyzed in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [48] for the group D3h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The difference being that in our case this contribution to DMI appears alongside the conventional LIs and non-LI m2 zLjz,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The cubic terms in momentum in the SOC, such as those in γz for D3, are not reflected at the level of LI invariants, and generally may only contribute to higher- orders in the perturbation theory, to non-LI invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' To the fifth order in SOC, our calculations show that these terms only contribute to non-LI invariants only when the linear contribution in SOC coupling is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' For the above case, that means the DMI energy will contain to this order only terms of type αn 1αm 2 with n > 0, (where α1 is the strength of the Rasbha coupling linear in momentum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Conversely, the Rashba coupling in Γ5 and Γ6 bands in D3 group has no linear terms in momentum (see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' I), and yields no contribution to DMI energy to the lowest orders in spin-orbit coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' As a byproduct of the present theory, we also deter- mine the conventional LI invariants which follow in a first order perturbation theory in weak SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' A table of mi- croscopically calculated DMI constants and LI-invariants in all symmetry groups is shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' I in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' A 5, and recovers the conventional invariants obtained in a standard symmetry analysis [4, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' CONCLUSIONS In this article, we investigated generic two-dimensional, two-band continuum models where Dzyaloshinskii-Moriya interaction is generated in the interplay between spin-orbit coupling and a magnetic texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The DMI micromag- netic free energy, proportional to the first derivative in the gradient of a smooth magnetic texture, was analyzed in detailed to reveal its structure beyond the Lifshitz invariants corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' A second expansion in weak SOC or weak exchange coupling allows to pinpoint the ex- act order at which non-Lifshitz invariants are manifest, namely to third order in small SOC and fourth order in small exchange coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The calculation of DMI tensors was performed in these limits explicitly for the case of rotation-symmetric C∞v Rashba spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' A signature of higher order terms is revealed in divergences in the DMI energy due to singularities in the electronic bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In the case of Rashba interactions this occurs due to the effective one-dimensional density of states near the bottom of the band at larger spin-orbit coupling, which generates an inverse square root singularity in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Thus, a signature of non-Lifshitz invariants might be vis- ible in measurements of the DMI constants, provided a strong SOC, a low temperature regime kBT ≪ ∆sd, with chemical potential tuned near the bottom of the band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' We have also shown how effective models for spin-orbit coupling in different point groups may be used to deter- mine microscopically the DMI energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The lower sym- metry of the Rashba vector compared to the continuum model with rotational symmetry induces new non-Lifshitz invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' This approach is checked also by deducing the conventional LI invariants when taking only the first order in a weak SOC expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' A nontrivial extension to the present work is the investi- gation of multi-band effects in systems hosting skyrmions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The free energy linear in the gradient of the magnetization is still expressed as a function of the Berry phase [38], but a simple decomposition as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (11) is not readily avail- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Another open venue is the analysis of free-energy contributions that depend on higher-order gradients of the magnetization, which play a role in the stabilization of the skyrmion textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Thiaville for enlightening discus- sions on the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' This work is supported by “Investisse- ments d’Avenir” LabEx (ANR-10-LABX-0039-PALM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' also acknowledges financial support from the Ro- manian National Authority for Scientific Research and Innovation, UEFISCDI through the contract ERANET- QUANTERA QuCos 120/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='2019, and through Core Program 27N/03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='2023, project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' PN 23 24 01 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 8 Appendix A: Detailed DMI constant determination in Rashba models 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Notation The convention used in the article is that Greek letters denote indices that can take values in {x, y, z}, while Roman ones, only in the two-dimensional plane {x, y} of the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Einstein notation, where repeated indices are summed over, is also employed throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Spatial derivatives are denoted as ∂rj and act in the 2D plane of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The conventional DMI tensor notation is related the one used in this paper as follows: D(i) αjβµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='µ2n ≡ D(i) αβµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='µ2n,j (A1) where the j index is separated out since it corresponds in the free energy Ω1 to a spatial derivative ∂x or ∂y of the magnetization m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The free energy is expressed conveniently with the aid of LI invariants Lαβ,j = mα∂rjmβ − mβ∂rjmα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A2) Such invariants are also denoted in the literature as L(j) αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Free energy expansion in two-dimensional, two-band models with spin-orbit interactions Here we present in more detail the model and the expansion of the free energy density from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' II and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' To improve readability, some equations in the main text are restated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The continuum model from the main text in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (3) H = ξ(k)σ0 + h · σ, h = ∆soγ(k) + ∆sdm(r), (A3) with local energy eigenvalues εs,k(r) = ξ + s � ∆2 sd + ∆2soγ2 + 2∆so∆sdγ · m(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A4) The correction to the first contribution to the gradient expansion in the free-energy density Ω1 uses an expansion of energy eigenstates εs,k = ξ + sλ � 1 + η, λ = � ∆2 sd + ∆2soγ2, η = 2∆so∆sd λ2 γ · m, (A5) with η the small parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' This procedure generates in the free energy density a Ginzburg-Landau expansion in the magnetization m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The free energy density in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (12) are Ω(0) 1 (r) = ∆so∆2 sd 2 ⟨∂kjγ · (m × ∂rjm)Fs,k⟩, Ω(1) 1 (r) = ∆2 so∆sd 2 ⟨∂rjm · (∂kjγ × γ)Fs,k⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A6) The functions Fs,k and implicitly the free energy density Ω1 are expanded in powers of magnetization field Fs,k = ∞ � n=0 F(n) s,k (λ)∆n so∆n sd(γ · m)n, Ω(i) 1 = ∞ � n=0 Ω(i) 1,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A7) The spin-orbit vector γ is antisymmetric in k, while the expansion coefficients F are symmetric in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Therefore, only the symmetric in k part of Fs,k contributes to Ω(0) 1 , and only the antisymmetric part of Fs,k, to Ω(1) 1 , such that the following simplified expressions follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Each order in the expansion is related to the rank of a corresponding DMI tensor D in the following way, Ω(i) 1,n = D(i) αβµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='µ2n,j(mα∂rjmβ)mµ1· · · mµ2n, (A8) 9 with DMI tensors D(0) αβµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='µ2n,j = 1 2∆2n+1 so ∆2n+2 sd ϵαβδ⟨(∂kjγδ)γµ1· · · γµ2nF(2n) s,k (λ)⟩, (A9) D(1) αβµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='µ2n,j = 1 2∆2n+3 so ∆2n+2 sd ϵνβδ⟨γαγν(∂kjγδ)γµ1· · · γµ2nF(2n+1) s,k (λ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A10) In calculations, it is profitable to perform a decomposition of the tensors in symmetric and antisymmetric parts, thus revealing a reduction in the number of linearly independent components: D(0) αβµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='µ2n,j = D(0) [αβ](µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='µ2n),j, (A11) D(1) αβµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='µ2n,j = D(1) [αβ](µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='µ2n),j + D(1) (αβ)(µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='µ2n),j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Here [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' ] and (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' ) denote antisymmetric and symmetric tensor in those indices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Further reductions are apparent only by considering specific point groups under which DMI tensors transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' For practical purposes the free energy density may be analyzed analytically in the weak SOC or weak exchange coupling limits to identify the leading non-LI contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' This leads to the truncated expansions in the ∆so ≪ ∆sd limit, Ω1 = Ω(0) 1,0 + Ω(1) 1,0 + Ω(0) 1,1 + O(∆5 so/∆5 sd) ≃ (D(0) αβ,j + D(1) αβ,j + D(0) αβµ1µ2,jmµ1mµ2)mα∂rjmβ, or ∆so ≫ ∆sd limit, Ω1 = Ω(0) 1,0 + Ω(1) 1,0 + Ω(0) 1,1 + Ω(1) 1,1 + O(∆6 sd/∆6 so) ≃ � i=0,1 (D(i) αβ,j + D(i) αβµ1µ2,jmµ1mµ2)mα∂rjmβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Computing the first terms in the free energy density expansions above [or Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (20) and (21) in the main text] to obtain the non-LI invariants requires the first four coefficients determined from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (17): F(0) s,k(λ) = 1 λ3 (sg0,s − λf0,s) F(1) s,k(λ) = 1 λ5 (−3sg0,s + 3λf0,s − sλ2f ′ 0,s), (A12) F(2) s,k(λ) = 1 2λ7 (15sg0,s − 15λf0,s + 6sλ2f ′ 0,s − λ3f ′′ 0,s), F(3) s,k(λ) = 1 6λ9 (−105sg0,s + 105λf0,s − 45sλ2f ′ 0,s + 10λ3f ′′ 0,s − sλ4f ′′′ 0,s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The primes denote derivatives with respect to the energy argument of the Fermi-Dirac functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Also, g0,s ≡ g(ε(0) s,k), f0,s ≡ f(ε(0) s,k), and derivatives are evaluated at ε(0) s,k = ξ + sλ at vanishing γ · m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Higher order coefficients F contain derivatives of the Fermi-Dirac distribution function which capture mainly Fermi surface contribution to the DMI tensor elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Although such terms are small in a perturbation theory in either small SOC or small exchange, they can yield divergences in the free energy at small temperature when the density of states diverges such as for flat bands, van Hove singularities etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Group C∞v This subsection details the calculation of DMI tensors and constants in the Rashba model from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (22) in group C∞v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The spin-orbit coupling vector in this group is given by ∆soγ = αR(−ky, kx, 0), (A13) and it is identical to the spin-orbit coupling in D4: Γ6 and Γ7, and D6: Γ7 and Γ8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The γ expression determines the DMI tensor elements when using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' We derive in the following the general form of the first four DMI tensors which capture the dominant contribution to non-LI invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Later in the subsection we perform a perturbation theory in either weak SOC or weak exchange coupling to get explicit forms for the DMI constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The DMI tensor in Ω(0) 1,0 is sparse with only 4 nonzero elements which are equal in amplitude: D(0) jz,j = −D(0) zj,j = −αR∆2 sd 2 ⟨F(0) s,k(λ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A14) 10 In these tensor elements the repeated indices are not summed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' This convention also applies below and in the next sections whenever discussing a given DMI tensor element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Factoring out one of the elements determines the DMI constant Ω(0) 1,0 = D(0) 0 (mj∂rjmz − mz∂rjmj) = D(0) 0 Ljz,j (A15) with D(0) 0 = D(0) xz,x or, explicitly D(0) 0 = −αR∆2 sd 4π � s � dkkF(0) s,k(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A16) The next contribution is from the 3-rank tensor D(1) αβ,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The nontrivial tensor elements are D(1) xz,x = −α3 R∆2 sd 2 ⟨k2 yF(1) s,k(λ)⟩, D(1) yz,y = −α3 R∆2 sd 2 ⟨k2 xF(1) s,k(λ)⟩, D(1) xz,y = D(1) yz,x = α3 R∆2 sd 2 ⟨kxkyF(1) s,k(λ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A17) The last two elements vanish by using the spherical symmetry of the problem in the integrals over momentum in ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' ⟩, and the remaining tensor elements read D(1) xz,x = D(1) yz,y = −α3 R∆2 sd 4 ⟨k2F(1) s,k(λ)⟩, k2 = k2 x + k2 y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A18) The symmetric part of the tensor integrates to zero over the bulk as it multiplies a total derivative ∂rj(mjmz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Therefore, the nonvanishing part of the free-energy density has only the antisymmetric part Ω(1) 1,0 = D(1) 0 Ljz,j, D(1) 0 = D(1) xz,x/2, (A19) with explicit DMI constant D(1) 0 = −α3 R∆2 sd 16π � s � dkk3F(1) s,k(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A20) The DMI tensor in Ω(0) 1,1 has 8 nonvanishing tensor components: D(0) jzii,j = −D(0) zjii,j = −α3 R∆4 sd 4 ⟨k2F(2) s,k(λ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A21) Therefore, the free energy correction reads Ω(0) 1,1 = D(0) 1 (mx∂xmz − mz∂xmx)(m2 x + m2 y) + (x ↔ y) = D(0) 1 (1 − m2 z)Ljz,j, (A22) using m2 = 1 in the second equality, with DMI constant D(0) 1 = D(0) xzxx,x, or D(0) 1 = −α3 R∆4 sd 8π � s � dkk3F(2) s,k(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A23) The final DMI tensor considered here has 16 non-vanishing components (not explicit here), leading to a free energy density contribution: Ω(1) 1,1 = 2D(1) 1 (1 − m2 z)(mx∂xmz + my∂ymz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A24) The free energy density, after eliminating total derivatives ∂rj(mjmz) and ∂rj(mjm3 z), also reads Ω(1) 1,1 = D(1) 1 (1 − m2 z 2 )Ljz,j, (A25) which presents the same m2 z correction to Lifshitz invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The DMI constant reads D(1) 1 = −3α5 R∆4 sd 64π � s � dkk5F(3) s,k(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A26) In the following we introduce the characteristic Rashba momentum and energy scales kR = mαR ℏ2 , ER = mα2 R 2ℏ2 , (A27) and perform a perturbation theory either in the small or large SOC limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 11 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Small SOC We consider now the expansion in αR/∆sd as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (1) we have shown the exact behavior of the DMI constants by numerical integration over bands and momentum, including the limit of small SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Analytically, we also compute the DMI constants in the zero temperature limit, to the lowest orders in αR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' To O(α3 R), the zero temperature DMI constants read as follows, D(0) 0 ≃ kR∆sd 8π � 1 − 2ERµ ∆2 sd �� 1 − µ2 ∆2 sd � Θ � 1 − µ2 ∆2 sd � , (A28) D(1) 0 ≃ −kRERµ 8π∆sd � 1 − µ2 ∆2 sd � Θ � 1 − µ2 ∆2 sd � , (A29) and D(0) 1 ≃ 3kRERµ 4π∆sd � 1 − 5µ2 3∆2 sd � Θ � 1 − µ2 ∆2 sd � , (A30) with Heaviside step function Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' To obtain D(1) 0 it was necessary to expand F (0) s,k(λ) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A16) to α3 R, hence the term proportional to αRER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Note that to linear order in αR, only D(1) 0 survives by formally setting in the expression ER = 0, such that it reproduces the results in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The zero temperature results are obtained by performing the sums and integrals in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A16), (A20), and (A26) using Fermi-Dirac formulas at zero temperature g(ε) = (ε − µ)Θ(µ − ε), f(ε) = Θ(µ − ε), and f ′(ε) = −δ(µ − ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Large SOC In the limit of ER ≫ ∆sd we obtain from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A16) the leading zero-temperature approximation to D(0) 0 , D(0) 0 ≃ kR∆2 sd 8πER × � 2 � 1 + µ/ER, µ ∈ (−ER, −∆sd), 1 − µ/∆sd, µ ∈ (−∆sd, ∆sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A31) To same order in ∆sd there is the additional contribution from D(1) 0 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A20), D(1) 0 = kR∆2 sd 16πER × � −3(1 + µ/ER)1/2 + (1 + µ/ER)−1/2, µ ∈ (−ER, −∆sd), µ/∆sd − 1, µ ∈ (−∆sd, ∆sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A32) Therefore, the DMI constant at large SOC (small exchange coupling) D(0) 0 + D(1) 0 is the one in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The D(1) 0 constant has a divergence at the bottom of the band in the limit of small exchange coupling (or large SOC) due to the Fermi surface contribution to the free energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' This is a consequence of the large density of states that develops at the bottom of the band, where the minimum occurs on a circle of constant energy at k = kR, such that the density of states there has a characteristic divergence of a 1D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The DMI constant has a similar divergence near the band minimum at µ ≃ −ER, D(1) 0 ∼ (1 + µ/ER)−1/2 ∼ 1/√ε, (A33) where ε is the energy calculated from −ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Such effects start to be visible at low temperature even at ER < ∆sd in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 1(b) and (e), and more so at ER > ∆sd, where the divergences in the DOS are accompanied by the divergence in D(1) 0 , respectively in at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 2(a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' This situation becomes more visible for higher terms in the expansion that contribute to order ∆4 sd such as D(0) 1 [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 1 (c) and (f)] since they contain a stronger divergence generated by Fermi surface terms such as f ′′(ε) that occur in F (3) s,k in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Group D3 There are cases where the SOC vector develops nonzero out-of-plane components where one could expect qualitatively different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' This occurs for groups where rotation symmetry by π is absent: C1,3 and D1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' This subsection details the calculation of DMI tensors and constants in the Rashba model from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' V in group D3 for Γ4 bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In this case, the spin-orbit vector reads ∆soγ = (−α1ky, α1kx, α2ky(3k2 x − k2 y)), (A34) 12 where for convenience α2 is defined as half of α2 from Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The cubic term breaks the rotational symmetry of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In the following, we will analyze the first terms, Ω(i) 1,0 and Ω(i) 1,1, in the Ginzburg-Landau expansion in magnetization m without assuming either relative small SOC, or small exchange coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' At the end of the section, the small SOC will be treated in more detail since it allows analytical resolution for the DMI constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The non-vanishing components of D(0) αβ,j tensor read D(0) jz,j = −D(0) zj,j = −α1∆2 sd 2 ⟨F(0) s,k(λ)⟩, D(0) xy,x = −D(0) yx,x = 6α2∆2 sd 2 ⟨kxkyF(0) s,k(λ)⟩, (A35) D(0) xy,y = −D(0) yx,y = 3α2∆2 sd 2 ⟨(k2 x − k2 y)F(0) s,k(λ)⟩, with λ = � ∆2 sd + α2 1k2 + α2 2k2y(3k2x − k2y)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A36) The last two equations in (A35) vanish by symmetry and therefore the free energy contribution reads Ω(0) 1,0 = −α1∆2 sd 2 � s � d2k 4π2 F(0) s,k(λ)Ljz,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A37) It exhibits the usual LI structure (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' There are 8 components of the D(1) αβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='j tensor that do not vanish under the constraint D3 group imposes on the angular integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' D(1) iz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='i = −α3 1∆2 sd 2 ⟨k2 ¯i F(1) s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k(λ)⟩ D(1) xx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x = α2 1α2∆2 sd 2 ⟨k2 y(3k2 x + k2 y)F(1) s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k(λ)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' D(1) xy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y = −α2 1α2∆2 sd⟨k4 yF(1) s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k(λ)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' D(1) yy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x = −3α2 1α2∆2 sd⟨k2 xk2 yF(1) s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k(λ)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' D(1) yx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y = 3α2 1α2∆2 sd 2 ⟨k2 x(k2 y − k2 x)F(1) s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k(λ)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A38) D(1) zx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x = α1α2 2∆2 sd 2 ⟨k2 y(k4 y − 9k4 x)F(1) s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k(λ)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' D(1) zy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y = α1α2 2∆2 sd⟨k4 y(3k2 x − k2 y)F(1) s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k(λ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Due to the lack of rotation symmetry it is not immediate to resolve these integrals as was the case in C∞v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Using polar coordinates and adding the contribution from all tensors as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (15) yields the energy density: Ω(1) 1,0 = −∆2 sd 2 � dkk 4π2 �α3 1k2 2 (mx∂xmz + my∂ymz)r(1) 0 (k) (A39) + 3α2 1α2k4 4 (my∂xmy − mx∂xmx + my∂ymx + mx∂ymy)r(1) 0 (k) + α1α2 2k6 2 (mz∂xmx + mz∂ymy)r(1) 1 (k) � , where the angular integral acts inside functions r(1) 0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' These are generally defined for following use, r(m) n (k) = � s � 2π 0 dθ sin(3θ)2nF(m) s,k (λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A40) The second term O(α2 1α2) vanishes since it contains only total derivatives over products of magnetization components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Then after factoring out the symmetric part of the rest of the components, which integrates to zero in the bulk, one obtains Ω(1) 1,0 = D(1) 0 Ljz,j D(1) 0 = −α1∆2 sd 32π2 � dkk3[α2 1r(1) 0 (k) − k4α2 2r(1) 1 (k)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A41) Thus, the usual LI invariant is indeed recovered to this order and a unique DMI constant is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Higher-order tensors are expected to yield the non-LI contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' There are 28 nonvanishing components to tensors D(0) αβµ1µ2,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Adding the respective energy contribution from each of them yields Ω(0) 1,1 = ∆4 sd 2 � ∞ 0 dkk 4π2 � − α3 1k2 2 (1 − m2 z)r(2) 0 (k)Ljz,j + 3α2 1α2k4 4 r(2) 0 (k)[2mxmyLyx,x + (m2 x − m2 y)Lyx,y] + 3α1α2 2k6 2 (1 − 8 3m2 z)r(2) 1 (k)Ljz,j � , (A42) 13 with the functions r(2) 0,1 defined as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' At this order, it is practical to define 3 DMI constants to quantitatively describe the free energy, Ω(0) 1,1 = D(0) 1a (1 − m2 z)Ljz,j + D(0) 1b [2mxmyLyx,x + (m2 x − m2 y)Lyx,y] + D(0) 1c [(1 − 8 3m2 z)Ljz,j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A43) Finally, there are 80 non-zero components to D(1) αβµ1µ2,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Adding the contributions from each one yields the free energy Ω(1) 1,1 = ∆4 sd 2 � dkk (2π)2 � − 3α5 1k4 16 r(3) 0 (1 − m2 z 2 )Ljz,j + 9α4 1a2k6 16 (r(3) 0 − r(3) 1 )[2mxmyLyx,x + (m2 x − m2 y)Lyx,y] (A44) + 15α3 1α2 2k8 16 r(3) 1 (1 − 5 2m2 z)Ljz,j − 9α2 1α3 2k10 8 r(3) 1 [2mxmyLyx,x + (m2 x − m2 y)Lyx,y] + 3α1α4 2k12 8 r(3) 2 (k)m2 zLjz,j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The non-LI invariants that are present in the free-energy expansion to higher order are characterized by qualitatively new invariants of the type 2mxmyLyx,x + (m2 x − m2 y)Lyx,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' These are identical to the non-LI invariant in the D3h group analyzed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Modulo total derivatives which vanish in the bulk, they are related as mx(m2 x − 3m2 y) = −3 4(2mxmyLyx,x + (m2 x − m2 y)Lyx,y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A45) Small SOC Several simplifications are possible in the small SOC limit, where the rotation symmetry breaking SOC distortion to the energy spectrum is treated perturbatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The first order corrections require knowledge of tensors D(0) αβ,j, D(1) αβ,j, and D(0) αβµ1µ2,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In the small SOC limit, at each order one recovers in the integral the rotational symmetry such that the expression for DMI constants is further simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Computed to cubic order in spin-orbit coupling, the nonvanishing components are from (A35) D(0) iz,i ≃ −α1∆2 sd 2 � s � dkk 2π � F(0) s,k(∆sd) + k2 2 � α2 1 + α2 2k4 2 � F(1) s,k(∆sd) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A46) This exhibits the same structure as in the C∞v case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Defining D(0) 0 = D(0) xz,x, the free energy density reads Ω(0) 1,0 = D(0) 0 Ljz,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A47) Thus working to linear order in SOC yields the conventional LI invariant characterizing the D3 (or C3v) point group in 2D (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The contribution from D(1) αβ,j also simplifies since to cubic order in SOC F(1) s,k(λ) = F(1) s,k(∆sd) and the angular integral is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Therefore, it readily follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A41) that Ω(1) 1,0 = D(1) 0 Ljz,j, D(1) 0 = −α1∆2 sd 16π � s � dkk3� α2 1 − α2 2k4 2 � F(1) s,k(∆sd), (A48) which renormalizes the previous term (A47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Finally, the last term to cubic order in SOC is the contribution from D(0) αβµ1µ2,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A42) it follows directly that the 3 DMI constants are determined after performing the angular integral in r(2) n functions: D(0) 1a = −α3 1∆4 sd 8π � s � dkk3F(2) s,k(∆sd), D(0) 1b = 3α2 1α2∆4 sd 16π � s � dkk5F(2) s,k(∆sd), (A49) D(0) 1c = 3α1α2 2∆4 sd 16π � s � dkk7F(2) s,k(∆sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' In the zero-temperature approximation the DMI constants reveal that there is a nonvanishing contribution due to cubic terms in momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Since both D(0) 0 and D(1) 0 contribute to the conventional LI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' we add them to yield D(+) 0 = D(0) 0 + D(1) 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' D(+) 0 = −α1∆2 sd 4π � s � dkk � F (0) s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k(∆sd) + k2 4 � 3α2 1 + α2 2k4 2 � F (1) s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='k(∆sd) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A50) 14 where the k6 term contributes at zero-temperature above the gap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' D(+) 0 = �α1∆sdm 8πℏ2 (1 − µ2 ∆2 sd ) − 3α3 1 16π µ ∆sd � m ℏ2 �2 (1 − µ2 ∆2 sd ) + α1α2 2 80π � m ℏ2 �4 (1 + µ ∆sd )3(8 − 9 µ ∆sd + 3 µ2 ∆2 sd ) � Θ(1 − µ2 ∆2 sd ) + α1α2 2∆2 sd 5π � m ℏ2 �4 Θ(µ − ∆sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A51) Finally, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A49) we obtain D(0) 1a = 3α3 1µ 16π∆sd � m ℏ2 �2� 1 − 5µ2 3∆2 sd � Θ � 1 − µ2 ∆2 sd � , D(0) 1b = 3α2 1α2∆sd 32π � m ℏ2 �3� 1 − µ2 ∆2 sd �� 1 − 5µ2 ∆2 sd � Θ � 1 − µ2 ∆2 sd � , (A52) D(0) 1c = 9α1α2 2µ∆sd 16π � m ℏ2 �4� 1 − µ2 ∆2 sd �2 Θ � 1 − µ2 ∆2 sd � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The SOC in Γ5 and Γ6 bands in point group D3 reads [see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' I] ∆soγ = (iα1(k3 + − k3 −), α2(k3 + + k3 −), iα3(k3 + − k3 −)), (A53) with k± = kx ± iky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Working at small SOC, the integrals are expanded term by term, and we find zero contribution to O(α5 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' DMI constants for LI invariants in all 2D symmetry groups TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Lifshitz invariants for all the two-dimensional groups obtained to linear order in SOC as contained in the free energy density Ω(1) 1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The parameters αi are real and βi are complex, k± = kx ± iky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The spin-orbit interaction ∆soγ · σ is determined by the vector ∆soγ as derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [41] (here modulo an eventual overall sign change).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Group Γ ∆soγ Ω(1) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='0 C1 Γ2 (α1kx + α2ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' α3kx + α4ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' α5kx + α6ky) I1Lyz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x + I2Lyz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y + I3Lzx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x + I4Lzx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y + I5Lxy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x + I6Lxy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y C2 Γ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='4 (α1kx + α2ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' α3kx + α4ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 0) I1Lyz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x + I2Lyz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y + I3Lzx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x + I4Lzx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y C3 Γ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='5 (α1kx + α2ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' −α2kx + α1ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' β1k3 + + β∗ 1k3 −) I1(Lyz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x + Lzx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y) + I2(Lxz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x + Lyz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y) Γ6 (β1k3 + + β∗ 1k3 −,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' β2k3 + + β∗ 2k3 −,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' β3k3 + + β∗ 3k3 −) 0 C4 Γ5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='8 (α1kx + α2ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' −α2kx + α1ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 0) I1(Lyz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x + Lzx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y) + I2(Lxz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x + Lyz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y) C6 Γ7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='10 (α1kx + α2ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' −α2kx + α1ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 0) I1(Lyz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x + Lzx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y) + I2(Lxz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x + Lyz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y) Γ11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='12 (β1k3 + + β∗ 1k3 −,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' β2k3 + + β∗ 2k3 −,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 0) 0 D1 Γ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='4 (α1ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' α2kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' α3ky) I1Lyz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y + I2Lzx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x + I3Lxy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y D2 Γ5 (α1ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' α2kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 0) I1Lyz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y + I2Lzx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x D3 Γ4 (−α1ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' α1kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' −iα2(k3 + − k3 −)) I1(Lzx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x + Lzy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y) Γ5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='6 (iα1(k3 + − k3 −),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' α2(k3 + + k3 −),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' iα3(k3 + − k3 −)) 0 D4 Γ6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='7 (α1ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' −α1kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 0) I1(Lxz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x + Lyz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y) D6 Γ7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='8 (α1ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' −α1kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 0) I1(Lxz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='x + Lyz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='y) Γ9 (iα1(k3 + − k3 −),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' α2(k3 + + k3 −),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 0) 0 Here we determine the DMI constants for LIs obtained in the approximation of relatively weak SOC ∆so ≪ ∆sd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' extracted from Ω(0) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The results are presented in Table I for all symmetry groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Generically we see that cubic terms in momentum are irrelevant to first order in spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' This readily yields the DMI constants determined by a single integral, Ii = αi∆2 sd 4π � s � dkkF(0) s,k(∆sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A54) In the zero temperature limit it reads Ii = −αim∆sd 8πℏ2 � 1 − µ2 ∆2 sd � Θ � 1 − µ2 ∆2 sd � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A55) 15 The Ii coefficients are formally the same (up the value of αi) with the one analyzed in detail the C∞v case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=', D(0) 0 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Gapped Dirac model An important limit with application to topological materials is that of a Dirac model with Rashba spin-orbit interactions and gapped by the exchange coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The Hamiltonian for a C∞v model reads H = −µσ0 + αR(k × σ)z + ∆sdm(r) · σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A56) In this case, the first DMI constants are computed exactly to all orders at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' The first DMI coefficient from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A16) in the zero-temperature limit is D(0) 0 = � − ∆sdµ 4παR , µ ∈ (−∆sd, ∆sd), − ∆2 sd 4παR sign(µ), µ /∈ (−∆sd, ∆sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A57) Note that the zero-order perturbation theory in small spin-orbit coupling would be divergent due to flat bands for the zero-order energy at ±∆sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Nevertheless, summation of all orders gives a dispersion to the bands, which returns a finite DMI constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' To the same order in magnetic texture, D(2) 0 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A20) is half the amplitude of D(1) 0 , such that the total contribution reads D(1) 0 = −1 2D(0) 0 , D(0) 0 + D(1) 0 = 1 2D(0) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A58) The first non-Lifshitz invariant, in the zero temperature approximation, is non-zero only in the gap and the related DMI constant from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A23) reads D(0) 1 = − µ∆sd 8παR Θ(1 − µ2/∆2 sd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A59) The remaining contribution to O(m4) reads from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A23) D(1) 1 = −1 2D(0) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' (A60) [1] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Dzyaloshinskii, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 4, 241 (1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [2] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Dzyaloshinskii, Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' JETP 19, 960 (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Moriya, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 120, 91 (1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Bogdanov and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Yablonskii, Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' JETP 68, 101 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Bogdanov and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' R¨oßler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 87, 037203 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [6] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' R¨oßler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Bogdanov, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Pfleiderer, Nature 442, 797 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Thiaville, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Rohart, ´E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Ju´e, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Cros, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Fert, EPL 100, 57002 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Emori, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Bauer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Ahn, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Martinez, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Beach, Nat.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Leighton, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' MacDonald, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Ralph, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Arena, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' D¨urr, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Fischer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Grollier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Heremans, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Jungwirth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Kimel, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Koopmans, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Krivorotov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 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Schuller, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Slavin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Stiles, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Tchernyshyov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Thiaville, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Ezawa, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Nagaosa, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' 5, 13638 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [31] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Ado, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Qaiumzadeh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Duine, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Brataas, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Titov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Babaev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' B 104, L020406 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' [37] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Xiao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} +page_content=' Shi, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFKT4oBgHgl3EQfly6I/content/2301.11855v1.pdf'} 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b/VNFOT4oBgHgl3EQf6jQA/content/tmp_files/2301.12958v1.pdf.txt @@ -0,0 +1,1629 @@ +Programmable phase behavior in fluids with designable interactions +Fan Chen and William M. Jacobs∗ +Department of Chemistry, Princeton University, Princeton, NJ 08544, USA +(Dated: January 31, 2023) +Intracellular fluids can self-organize into phase-separated condensates as a result of evolved +biomolecular interactions. However, the extent to which the equilibrium phase behavior of a com- +plex fluid can be controlled by tuning such interactions remains unclear. Here we apply convex +optimization to design fluids that demix into phases with prescribed compositions at thermody- +namic equilibrium. +We then show how multicomponent phase diagrams can differ qualitatively +from those of simple fluids. Our approach enables the rational design of multicomponent fluids with +phase diagrams of arbitrary complexity. +Intracellular mixtures of biopolymers can demix to +form “biomolecular condensates” via the mechanism of +liquid–liquid phase separation (LLPS) [1–3]. Since phase +separation occurs spontaneously in vitro using naturally +occurring or engineered biopolymers, biomolecular LLPS +is widely believed to be governed primarily by equilib- +rium thermodynamics [4–7], even though nonequilibrium +processes may affect the phase behavior in vivo [8, 9]. Al- +though it is not surprising that heteropolymers can phase +separate at high concentrations, it is remarkable that +LLPS can establish coexisting condensates with distinct +molecular compositions required for carrying out specific +biological functions [10, 11]. Despite recent progress in +modeling multicomponent phase separation [12–19], the +relationship between the “design space” of such specific +interactions and the capacity for biological fluids to self- +organize into chemically diverse droplets via LLPS re- +mains poorly understood. +Exploration of the design space for tunable LLPS is +challenging due to the combinatorial complexity associ- +ated with multicomponent phase coexistence. In order +to identify coexisting phases in a model multicomponent +mixture, it is first necessary to locate all the candidate +phases in a high-dimensional concentration space. This +constitutes a search problem whose complexity scales +exponentially with the dimension of the concentration +space. As a result, any algorithm for predicting or de- +signing multicomponent phase diagrams that relies on +computing phase coexistence from a prescribed set of in- +teractions must either be limited to mixtures with a small +number of components or employ additional assumptions +in the search problem. +Such assumptions can qualita- +tively affect the predictions of the algorithm. +This combinatorial complexity can be avoided by solv- +ing the inverse problem of designing interactions to +achieve a target phase diagram [20]. By instead speci- +fying the compositions of the coexisting phases as design +criteria, this approach eliminates the need to search for +candidate phases. +Furthermore, if the thermodynamic +constraints on bulk phase coexistence can be cast as a +convex optimization problem, then suitable interactions +∗ wjacobs@princeton.edu +can be found using efficient algorithms [21]. Solving the +inverse problem therefore allows us to associate an equi- +librium phase diagram with a set of designed interactions, +leading to a more complete picture of multicomponent +phase behavior. +Here we show that an inverse-problem approach can +be applied to design equilibrium phase diagrams with ar- +bitrary condensed-phase compositions. We first explain +how this strategy can be applied to generic mean-field +models with pairwise intermolecular interactions. +We +then show that this approach reveals unexpected fea- +tures of multicomponent phase diagrams, which differ +qualitatively from the intuitive phase behavior of sim- +ple fluids. Finally, we perform molecular simulations and +free-energy calculations to demonstrate that our insights +apply beyond mean-field theoretical models. +We begin by assuming that the intermolecular inter- +actions in a multicomponent solution are pairwise addi- +tive and that the elements of the symmetric interaction +matrix, ϵ, are independently tunable. +With these as- +sumptions, the vector of excess chemical potentials for +all molecular species can be written in the form +⃗µex(⃗φ; ϵ,⃗v) = ⃗µv(⃗φ;⃗v) + ϵ⃗φ, +(1) +where ⃗φ and ⃗v represent the volume fractions and molec- +ular volumes of the species, respectively, and ⃗µv is inde- +pendent of ϵ. For simplicity, we assume that the ther- +mal energy kBT = 1. Eq. (1) describes the mean-field +Flory–Huggins [22] and van der Waals [23] models, as well +as approximate field-theoretic treatments of sequence- +dependent heteropolymer mixtures [24], with appropriate +choices of ⃗µv. The osmotic pressure, P(⃗φ; ϵ,⃗v), which can +be determined from Eq. (1) via the Gibbs–Duhem rela- +tion, is also linear with respect to ϵ. +Our objective is to find an N × N interaction matrix, +ϵ, and an N-dimensional chemical potential vector, ⃗µ, +that lead to equilibrium phase coexistence among a di- +lute phase and K condensed phases in a solution with N +solute species (“components”). The inverse design prob- +lem is defined by the target volume fractions of each of +the condensed phases, ⃗φ(α), indexed by α = 1, . . . , K +(Fig. 1a). In general, each target condensed phase con- +sists of M (α) “enriched” components, which comprise the +majority of the volume fraction of phase α, and N −M (α) +arXiv:2301.12958v1 [cond-mat.soft] 30 Jan 2023 + +2 +A +B +C +D +convex +relaxation +feasible +region +coexistence +coexistence +regularization +off-target phases +FIG. 1. Inverse design approach to multicomponent phase coexistence. (a) Schematic of the design problem. Each +condensed-phase droplet (gray circle) has a distinct composition of the five molecular components (colors). The enriched-species +compositions are indicated by pie charts. (b) Depiction of the solution space for a 3-component, 2-phase problem (see SI for +details). Inequality constraints (blue line) and minimum eigenvalue constraints (contours) delineate the feasible region (dashed +white line). (c) Illustration of the common-hyperplane construction for two phases, α and β, of the convex relaxation solution +shown in panel b. (d) Schematic of the regularization heuristic for eliminating stable (∆Ω ≤ 0) off-target phases γ and δ. +“depleted” components, which are found at negligible +concentrations in phase α. Bulk phase coexistence oc- +curs when all K + 1 phases have equal osmotic pressures +and each molecular species has the same chemical poten- +tial in each of the K + 1 phases. Furthermore, all K + 1 +phases must be stable with respect to concentration fluc- +tuations, such that ∂⃗µ(⃗φ)/∂⃗φ is positive definite. To find +an ϵ and ⃗µ that satisfy these thermodynamic constraints, +we perform a convex relaxation by making three minor +approximations. First, since the depleted components in +each condensed phase have an insignificant effect on the +phase diagram, we write the volume-fraction constraint +for each depleted component j in every phase as an in- +equality, such that φ(α) +j +< φ(α) +depl ≡ φ(α) +T /M (α)(N −M (α)), +where φ(α) +T +≡ �N +i=1 φ(α) +i +. Second, we assume that the +contributions of the depleted components to ⃗µex in each +condensed phase are negligible and can thus be ignored. +Finally, we assume that the total volume fraction in the +dilute phase, φ(0) +T , is very small, so that the osmotic pres- +sure at coexistence is near zero; this approximation is +valid far from a critical manifold when every component +is enriched in at least one target condensed phase. +Taken together, these conditions define a semidefinite +program (SDP) that is convex with respect to ϵ and ⃗µ +(Fig. 1b): +µid,i(⃗φ(α);⃗v) + µex,i(⃗φ(α); ϵ,⃗v) ≥ µi ∀i, α +(2a) +P(⃗φ(α); ϵ,⃗v) = 0 ∀α +(2b) +∂[⃗µid(⃗φ(α);⃗v) + ⃗µex(⃗φ(α); ϵ,⃗v)]/∂⃗φ ≻ λminI ∀α +(2c) +φ(0) +T (⃗µ;⃗v) < φ∗ +T(⃗v), +(2d) +where µid,i = v−1 +i +log φ(α) +i +for any component i that is en- +riched in the α phase, µid,i = v−1 +i +log φ(α) +depl for any com- +ponent i that is depleted in the α phase, and the equal- +ity(inequality) in Eq. (2a) applies to enriched(depleted) +components. In Eq. (2c), the parameter λmin ≥ 0 places +a lower bound on the smallest eigenvalue of the second- +derivative matrix to guarantee thermodynamic stability. +The final constraint, Eq. (2d), ensures that the volume +fraction in the dilute phase, φ(0) +T , is less than the criti- +cal volume fraction, φ∗ +T(⃗v); this condition is independent +of ϵ due to the zero-osmotic-pressure assumption. This +program is straightforward to solve using modern convex +optimization tools [25, 26]. Moreover, it is possible to +prove whether this convex relaxation is infeasible, mean- +ing that no solution (ϵ, ⃗µ) exists for the target condensed- +phase volume fractions {⃗φ(α)}. Because the approxima- +tions required to establish this convex relaxation are well +controlled, we expect that there is a close correspondence +between the feasible domain of {⃗φ(α)} and the domain on +which thermodynamic coexistence can be established. +To confirm that the precise thermodynamic conditions +for bulk-phase coexistence can be satisfied, we next per- +form a multicomponent generalization of the common- +tangent construction. +Starting from the SDP solution +(ϵ, ⃗µ), we adjust ⃗µ in order to fit a common tangent plane +to the local minima of the grand potential, Ω(⃗φ; ⃗µ, ϵ,⃗v) ≡ +�N +i=1 +� +dφi[v−1 +i +log φi + µex,i(⃗φ) − µi] (Fig. 1c). The con- +ditions specified in Eq. (2) imply that the grand potential +evaluated at the SDP solution has local minima close to +the prescribed target-phase and dilute-phase volume frac- +tions. Therefore, we can fit a common tangent plane by +minimizing the norm of ⃗∆Ω(⃗µ), where ∆Ω(α)(⃗µ) is the +difference between Ω(⃗φ; ⃗µ) evaluated at the local mini- +mum near the dilute phase and at the local minimum +near the α condensed phase. This procedure converges +rapidly using standard numerical methods [27], since the +convex relaxation is a good approximation of this nonlin- +ear hyperplane-fitting problem. In the extensive numeri- +cal tests described below, we indeed find that a solution +to the convex relaxation typically implies that the condi- +tions for coexistence can be satisfied for the target phases +to numerical precision. +This algorithm provides a scalable and highly accurate +means for predicting whether prescribed target phases +can be in simultaneous thermodynamic coexistence and, +if so, for determining a coexistence point (ϵ, ⃗µ). In gen- + +3 +eral, the convex relaxation specified by Eq. (2) defines +a continuous space of interaction matrices that solve the +inverse design problem, with a unique ⃗µ corresponding +to each point in this space. However, we have not yet +considered the possibility that other “off-target” con- +densed phases may be equally or even more stable than +the target phases at the calculated coexistence point, +meaning that the target phases are only in marginal or +metastable coexistence. This possibility can be addressed +by introducing a regularization heuristic that attempts +to maximize Ω(φ; ⃗µ, ϵ,⃗v) away from the target phases +(Fig. 1d). Specifically, based on the form of Eq. (1), we +seek to minimize both the norm of ϵ − ¯µ/⟨φ(α) +T ⟩, where +¯µij = (µi +µj)/2 and ⟨φ(α) +T ⟩ is the mean target-phase to- +tal volume fraction, and the variance of the elements of +⃗µ (see Supplementary Information for details). Regular- +izing the SDP in this way tends to destabilize off-target +phases while guaranteeing that the solution to our convex +relaxation is unique. +Our approach reveals unexpected features of multicom- +ponent phase behavior, which we demonstrate in the case +of a Flory–Huggins polymer model [22] with degree of +polymerization ranging from L = 1 to 100. (See the Sup- +plementary Information for the model and SDP defini- +tions.) Here we describe results for mixtures with N = 6 +species, a sufficient number to uncover qualitative differ- +ences with simple fluids while still permitting exhaustive +searches for off-target phases. For simplicity, we choose +the same total volume fraction, φα +T = φ(cond) +T +, for each +condensed phase. We begin by designing phase diagrams +with “equimolar” target phases, in which every enriched +component within a target phase has the same volume +fraction ≃ φ(cond) +T +/M (α). Interestingly, we find that the +feasibility of the SDP for any particular target phase di- +agram is independent of L and φ(cond) +T +. +However, the +probability that a solution to the regularized SDP re- +sults in phase coexistence tends to increase with L and +φ(cond) +T +≫ φ∗ +T (Fig. 2a), since the convex relaxation be- +comes a more accurate approximation as the coexistence +pressure decreases. +Intuition based on the phase behavior of simple mix- +tures suggests that small changes in {⃗φ(α)} should result +in small changes in ϵ, and vice versa, unless the mixture +is near a critical point where two or more of the ⃗φ(α) +merge. For example, small changes in the dimensionless +interaction parameter in an incompressible binary mix- +ture energy perturb the binodal but do not change the +coexistence region qualitatively, as long as φT ≫ φ∗ +T [22]. +Furthermore, the Gibbs Phase Rule (GPR) [28], which +relates the number of coexisting phases to the number +of thermodynamic degrees of freedom, should be ex- +pected to limit the number of condensed phases, K, for +which our method can find a coexistence point. For the +N-component fluids that we study here, this expected +bound is K ≤ N. +Our inverse approach reveals multiple ways in which +such intuition can mislead in multicomponent fluids. Sur- +〈p(coex)〉equimolar +〈p(coex)〉equimolar +〈p(global)〉equimolar +A +C +B +condensed phase count +p(coex|feas) +zero noise +noise +std. dev. = 10−3 +-space +convex relaxation +regularized solution +solution space for +phase coexistence +coexisting phases +are globally stable +target phases +: +FIG. 2. +Features of multicomponent phase coexis- +tence. +(a) Schematic of ϵ-space and validation that SDP +solutions (feas) result in coexistence (coex). This probabil- +ity, p(coex|feas), approaches one as the degree of polymer- +ization, L, increases. Here we consider phase diagrams with +N = 6 species and equimolar condensed-phase compositions +(see text). Each class of isomorphic phase diagrams, which are +equivalent under permutation of component and target-phase +indices, is considered once in these calculations. (b) Sensi- +tivity of phase coexistence to perturbations in the interaction +matrix. +We add zero-mean Gaussian noise to the interac- +tion matrices that produce phase coexistence among target +phases with equimolar compositions, and then attempt to re- +establish coexistence. +The probability of success, averaged +over many trials, is ⟨p(coex)⟩equimolar. +(c) The probability +that coexistence can be achieved for condensed phases with +arbitrary compositions. Starting from feasible phase diagrams +with equimolar condensed phases, we construct target phases +by randomly scaling the enriched component compositions in +each condensed phase. We also show whether the SDP so- +lution leads to global phase coexistence, meaning no stable +off-target phases. In panels b–c, φ(cond) +T += 0.95 and L = 100. +prisingly, we can design coexistence points where the +condensed-phase count, K, is greater than the number of +distinct species, N (Fig. 2b). At first glance, these exam- +ples might appear to conflict with the GPR. Furthermore, +in terms of bulk phase coexistence, these examples imply +that the lever rule, which relates the total concentrations +of the various species in a mixture to the mole fractions +of the coexisting phases, does not have a unique solu- +tion. However, in mixtures with designed interactions, +we can end up with coexistence equations that are lin- +early dependent, allowing us to perform a common tan- +gent plane construction when K > N. This behavior can +be understood by realizing that the design problem, with +N(N+1)/2 tunable interaction-matrix parameters, is not +overdetermined, and that convex optimization identifies +interaction matrices that result in linearly dependent co- + +4 +existence equations at the prescribed {⃗φ(α)}. These un- +usual coexistence conditions do not occur in mixtures +with fewer than N = 5 species, but become increasingly +common as the number of components increases (see Sup- +plementary Information for further discussion). +Consistent with this explanation, we find that small, +random perturbations to the designed interaction matri- +ces almost always preclude phase coexistence of the tar- +get phases when K > N (Fig. 2b). Specifically, we add +zero-mean Gaussian noise to the designed matrix ϵ, and +then attempt to perform a common tangent plane con- +struction for phases close to the original target phases +by tuning ⃗µ. +After perturbation, only a subset of the +original K condensed phases can be brought into coexis- +tence with the dilute phase, while the remaining phases +become metastable. Similarly, we can make random per- +turbations to the initially equimolar compositions of the +enriched components in each target phase. If we then at- +tempt to solve the convex relaxation for these perturbed +target phases, we almost always find that the SDP is +infeasible when K > N (Fig. 2c). This indicates that +a particular relationship among the compositions of the +enriched components is necessary to establish linearly de- +pendent coexistence equations. Yet surprisingly, such be- +havior is not limited to phase diagrams with K > N. +We also find certain phase diagrams with K ≤ N that +are similarly sensitive to random perturbations in ϵ and +{⃗φ(α)} (Fig. 2b,c). +Regardless, when our inverse ap- +proach indicates that coexistence among the prescribed +phases is possible, exhaustive sampling of the grand po- +tential landscape confirms that these phases are globally +stable with high probability (red curve in Fig. 2c). +Taken together, these observations suggest that spe- +cial coexistence points, which are sensitive to small per- +turbations in ϵ, lie on manifolds of lower dimension than +the full ϵ-space. Coexistence is not limited to K ≤ N +condensed phases on these manifolds, although some of +these phases must become metastable if we move off the +manifold by perturbing the interaction matrix. +These +manifolds represent “interfaces” between volumes of ϵ- +space corresponding to condensed phases with different +sets of enriched components. +In other words, crossing +one of these interfaces by changing ϵ entails a discon- +tinuous transition from one set of condensed phases to +another, with phases from both sets stable on the inter- +face itself. We emphasize that our calculations are per- +formed far from critical points, since the Euclidean dis- +tance between all pairs of target phases {⃗φ(α)} is large. +Furthermore, these special coexistence points need not +have equimolar condensed-phase compositions; however, +the enriched-component volume fractions are not inde- +pendent on these manifolds (see Supplementary Infor- +mation for further discussion). +How are distinct multicomponent phase diagrams re- +lated in ϵ-space? To address this question, we can use the +Frobenius norm to measure of the distance between two +interaction matrices ϵr and ϵs, corresponding to differ- +ent phase diagrams with globally stable condensed phases +A +B +C +D +-space +FIG. 3. +Relationships among phase diagrams in ϵ- +space. (a) Low-dimensional representation of the interaction +matrices corresponding to the phase diagrams considered in +Fig. 1b. Circles with black outlines indicate phase diagrams +that are sensitive to random perturbations in ϵ (see text). +Multidimensional scaling [29] has been used to preserve dis- +tances in ϵ-space, taken here as the Frobenius norm. (b) Dis- +tances between pairs of matrices in panel a (black), and the +minimum distance required to switch from phase diagram r to +phase diagram s (red). Box plots indicate the quartiles of the +distance distributions as a function of the phase count differ- +ence, Ks−Kr. (c) The minimum number of entries of the sym- +metric ϵ matrix that must be changed to switch from phase +diagram r to phase diagram s, Drs; Dmax ≡ N(N + 1)/2. +(d) Asymmetry in the minimum number of elements changed +when switching between phase diagrams. +{⃗φ(α)}r and {⃗φ(α)}s, respectively (Fig. 3a and black dis- +tributions in Fig. 3b). Yet because the interaction ma- +trix that stabilizes a particular phase diagram is typi- +cally not unique, it is more useful to quantify the extent +to which an interaction matrix must be changed in or- +der to switch from one phase diagram to another. We +can accomplish this within our inverse design framework +by modifying the regularization heuristic in one of two +ways (see Supplementary Information for details). Start- +ing from the interaction matrix ϵr that solves the origi- +nal regularized SDP for phase diagram r, we identify the +“closest” matrix ϵs that solves phase diagram s by mini- +mizing the Frobenius norm ||ϵs−ϵr||fro (red distributions +in Fig. 3b). This distance can be infinitesimal if ϵr resides +on a low-dimensional manifold and the phase-diagram +change r → s reduces the phase count. Similarly, the +minimal distance between interaction matrices is typi- +cally larger when we add phases, such that Ks > Kr. +Alternatively, we can determine the smallest number of + +5 +1 +2 +3 +4 +1 +2 +3 +4 +5 +R=0.89 +R=1.00 +R=1.00 +R=1.00 +R=1.00 +R=1.00 +R=1.00 +R=1.00 +R=1.00 +1 +2 +3 +4 +5 +6 +R=1.00 +R=1.00 +R=1.00 +R=1.00 +R=1.00 +R=1.00 +A +B +C +Grand potential density +FIG. 4. Transferable predictions validate the pairwise +approximation. Simulated free-energy landscapes at phase +coexistence (|∆Ω| ≤ 0.007kBT) in a molecular model with +pair potentials derived from designed interaction matrices. +Examples are shown for mixtures with N = 5 species and +(a) K = 4, (b) 5, and (c) 6 condensed phases. +(N + 1)- +dimensional landscapes are projected onto two principal- +component coordinates, X1 and X2, for visualization. The +Pearson correlation coefficient, R, between the target and +equilibrium composition is shown for each condensed phase. +distinct matrix elements that must be changed to switch +phase diagrams. +This minimal number of elementwise +changes, Drs, is always greater than zero (Fig. 3c). Our +calculations reveal that Drs is also asymmetric with re- +spect to phase diagram changes r ↔ s and tends to in- +crease with the net number of added phases (Fig. 3d). +Finally, we assess whether the predictions of our in- +verse design approach apply beyond mean-field models. +Specifically, we consider fluids in which the potential en- +ergy can be written as a sum of short-ranged pair poten- +tials [23]. Note that Eq. (1) is only accurate at low con- +centrations for such fluids, since the higher-order virial +coefficients generically depend on the species-specific pair +potentials [23]. To this end, we perform free-energy cal- +culations using simulations of a multicomponent lattice +gas. We first design an interaction matrix, ϵ, for a target +phase diagram using the L = 1 Flory–Huggins SDP. We +then use this matrix to define the well-depths of the pair +potentials, uij(1 ≤ r/a < 2) ∝ ϵij, where r is the distance +between particles of types i and j and a is the lattice con- +stant (see Supplementary Information for details). We +first identify the free-energy basins in this model by run- +ning grand-canonical Monte Carlo simulations [30]. We +then sample reversible transitions between the dilute free- +energy basin and each of the condensed-phase basins [13]. +Finally, we reconstruct the free-energy landscapes in the +N-dimensional ⃗φ-space and adjust the chemical poten- +tials to bring all phases into coexistence [31], at which +point the grand potentials of all basins are all equal. +Our mean-field design approach results in non-mean- +field free-energy landscapes that are consistent with the +target phase diagrams. We carry out simulations with +five species and condensed-phase counts that are less +than, equal to, and greater than the number of compo- +nents (Fig. 4). In line with the predictions of the mean- +field model, we find that coexistence can be achieved +within sampling accuracy, even in the K > N case. Mi- +nor quantitative differences in the phase compositions oc- +cur in the simulation model due to inaccuracies in the +mean-field approximations; however, the enriched com- +ponents in all simulated phases match the designs. +In summary, we have introduced a method to de- +sign the phase behavior of multicomponent fluids with +pairwise—or approximately pairwise—interactions. Our +approach provides insight into the structure of the +interaction-matrix solution space, +revealing ways in +which the behavior of multicomponent fluids can differ +qualitatively from that of simple fluids, while also scal- +ing well to mixtures with tens or hundreds of compo- +nents. +In practice, it may not be possible to engineer +molecular interactions with the independence and pre- +cision necessary to construct all theoretically possible +phase diagrams. In this regard, our results indicate that +the physically relevant constraints on the phase behav- +ior of multicomponent fluids arise from the properties of +the intermolecular interactions, since thermodynamically +allowed phase diagrams can be surprisingly complex. 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Lett. 68, 9 +(1992). + +7 +SUPPLEMENTARY INFORMATION FOR “PROGRAMMABLE PHASE BEHAVIOR +IN FLUIDS WITH DESIGNABLE INTERACTIONS” +I. +APPLICATION TO THE MULTICOMPONENT FLORY–HUGGINS MODEL +In the numerical examples presented in the main text, we consider the special case of the multicomponent Flory– +Huggins polymer model [22], for which ⃗µv(⃗φ;⃗v) = − log(1−φT)−(1−1/Li). We introduce the degree of polymerization, +Li, for each component i in place of the molecular volume vi. In this section, we first write the thermodynamic +quantities for this model explicitly. We then show how to formulate the SDP for this specific model. +A. +Model definition +The Helmholtz free-energy density, F; chemical potential, ⃗µ; osmotic pressure, P; and Hessian matrix, ∂⃗µ(⃗φ)/∂⃗φ, +in the multicomponent Flory–Huggins model are +F = +N +� +i=1 +φi +Li +log φi + (1 − φT) log(1 − φT) + 1 +2 +N +� +i=1 +N +� +j=1 +ϵijφiφj +(S1) +µi = 1 +Li +log φi − log(1 − φT) − +� +1 − 1 +Li +� ++ +N +� +j=1 +ϵijφj +(S2) +P = − log(1 − φT) + +N +� +i=1 +φi +Li +− φT + 1 +2 +N +� +i=1 +N +� +j=1 +ϵijφiφj +(S3) +∂µi +∂φj += δij +Liφi ++ +1 +1 − φT ++ ϵij, +(S4) +respectively, where Li is the degree of polymerization of polymeric species i. +Before writing down the SDP constraints for a particular set of condensed-phase volume fractions, {⃗φ(α)}, we +consider a mixture with a fixed composition ⃗x. The resulting expressions will be utilized in subsequent sections. The +mixture composition is normalized such that �N +i=1 xi = 1. In order to calculate the total volume fraction at the +critical point, φ∗ +T(⃗x), we set the projection of the Hessian matrix along ⃗x to zero, +N +� +i=1 +N +� +j=1 +xi +∂µi +∂φj +xj = 1 +φT +N +� +i=1 +xi +Li ++ +1 +1 − φT ++ +N +� +i=1 +N +� +j=1 +xiϵijxj = 0, +(S5) +and differentiate with respect to φT, +− ∂⟨ϵ⟩x +∂φT += − +1 +(φT)2 +N +� +i=1 +xi +Li ++ +1 +(1 − φT)2 = 0, +(S6) +where ⟨ϵ⟩x ≡ �N +i=1 +�N +j=1 xiϵijxj. The critical volume fraction at fixed composition ⃗x is thus +φ∗ +T(⃗x) = +1 +1 + ⟨1/L⟩−1/2 +x +, +(S7) +where ⟨1/L⟩x ≡ �N +i=1 xi/Li. +Assuming that the chemical potential vector is known, the total volume fraction of a condensed phase with com- +position ⃗x can be approximated by setting the osmotic pressure equal to zero and projecting ⃗µ along ⃗x, +N +� +i=1 +xiµi ≡ ⟨µ⟩x = +N +� +i=1 +xi +Li +log φi + ⟨1/L⟩x − 1 − log(1 − φT) + φT⟨ϵ⟩x, +(S8) +to yield a non-linear equation for φT, +2 +φT +log(1 − φT) + ⟨1/L⟩ log φT − log(1 − φT) + 1 − ⟨1/L⟩x + +N +� +i=1 +xi +Li +log xi − ⟨µ⟩x = 0. +(S9) + +8 +We can also solve for the mean interaction, ⟨ϵ⟩x, in a condensed phase with composition ⃗x, +⟨ϵ⟩x = +2 +(φT)2 [log(1 − φT) + φT (1 − ⟨1/L⟩x)] . +(S10) +We can similarly obtain an expression for dilute-phase volume fractions in terms of ⃗µ by assuming that the osmotic +pressure is nearly zero, such that φ(0) +T +is very small, +φi ≃ exp [Liµi + (Li − 1)] . +(S11) +B. +SDP formulation +In this section, we use the notation {⃗φ(α)} to refer to the target volume fractions in the K condensed phases, +α = 1, . . . , K. We further assume that M (α) species are enriched in the α phase and that the target volume fractions +of the depleted components are set to zero. We can therefore denote the set of enriched components in phase α by +S(α) ≡ {i|δ(φ(α) +i +) = 0}, +(S12) +where δ(·) is the Dirac delta function. Thus, M (α) = N − �N +i=1 δ(φ(α) +i +) is the cardinality of the vector ⃗φ(α), and the +target total volume fraction in phase α is φ(α) +T += � +i∈S(α) φ(α) +i +. +The equal chemical potential constraints for enriched and depleted components, respectively, take the form +N +� +j=1 +Liφ(α) +j +ϵij − Liµi + log φ(α) +i +− Li log(1 − φ(α) +T ) − (Li − 1) = 0, +δ +� +φ(α) +i +� += 0 +(S13a) +N +� +j=1 +Liφ(α) +j +ϵij − Liµi + log +� +φ(α) +T ζ +M (α)(N − M (α)) +� +− Li log(1 − φ(α) +T ) − (Li − 1) ≥ 0, +δ +� +φ(α) +i +� += 1 +(S13b) +for each species index i = 1, . . . , N and each condensed phase α. We set the adjustable parameter ζ = 10−2 in this +work. The zero-osmotic-pressure constraint for each condensed phase α is +1 +2 +N +� +i=1 +N +� +j=1 +φ(α) +i +φ(α) +j +ϵij − log(1 − φ(α) +T ) − φ(α) +T ++ +N +� +i=1 +φ(α) +i +Li += 0. +(S14) +In order to place constraints on the Hessian matrices in the condensed phases, we define the augmented target volume +fractions +˜φ(α) +i += φ(α) +i ++ +φ(α) +T ζ +M (α)(N − M (α))δ +� +φ(α) +i +� +, +(S15) +so that we have ˜φ(α) +i +> 0 for all i and all α. Each condensed-phase scaled Hessian matrix must then satisfy +δij +˜φ(α) +i ++ +� +LiLj +1 − φ(α) +T ++ +� +LiLjϵij ⪰ λminδij, +(S16) +where λmin is the smallest allowed eigenvalue. We choose λmin = 1 in this work. Finally, we constrain the chemical +potentials such that the approximate total volume fraction of a roughly equimolar dilute phase is below the critical +volume fraction by utilizing Eq. (S11) and Eq. (S7), +log +N +� +i=1 +exp(Liµi + Li − 1) ≤ log +� +0.9 φ∗ +T +� +⃗x = +� 1 +N , . . . , 1 +N +��� +. +(S17) +Within the approximations of this convex relaxation, these constraints define the joint space of interaction matrices, +ϵ, and chemical potential vectors, ⃗µ, for which bulk phase coexistence can be established among the target condensed +phases and a dilute phase. + +9 +-space +convex relaxation +regularized solution +solution space for +coexistence of +target phases +solution space for +which the target phases +are globally stable +FIG. S1. Schematic of regularization in ϵ-space. Regularized convex optimization identifies a unique ϵ matrix, indicated by +the star in the diagram, and a corresponding ⃗µ vector. The heuristic regularization method described in SI Sec. I C is likely to +find a solution within the globally stable solution space (cyan region), if it exists. +C. +Regularization for global stability of target phases +Next, we regularize our convex optimization problem in order to identify an interaction matrix, ϵ, and chemical +potential vector, ⃗µ, that are least likely to result in stable off-target phases (Fig. S1). Off-target condensed phases +correspond to local minima of the grand-potential, Ω(⃗φ; ⃗µ, ϵ,⃗v), that lie below the grand potential of the dilute phase, +Ω(0). We therefore aim to maximize the grand potential everywhere in the domain ⃗φ, except at the target phases, +{⃗φ(0), ⃗φ(1), . . . , ⃗φ(K)}, where Ω(α) = Ω(0). Since the ⃗µ and ϵ-dependent contributions to the grand potential have the +form ω(⃗φ) ≡ +� +⃗φ⊤ϵ − ⃗µ +� +· ⃗φ, we define the objective function +L0 = +����� +�����(1 + δij)1/2 +¯ωij +⟨φT⟩{α} +����� +����� +fro ++ +����� +�����Li +� +µi − 1 +N +N +� +k=1 +µk +������ +����� +2 +. +(S18) +The first term is a scaled and shifted Euclidean norm of the unique ¯ωij elements, +¯ωij +⟨φT⟩{α} += ϵij − +1 +⟨φT⟩{α} +� +µi + µj +2 +− 1 +N +N +� +k=1 +µk +� +, +(S19) +where ⟨φT⟩{α} ≡ (1/K) �K +α=1 φ(α) +T , while the second term is a scaled standard deviation of the ⃗µ elements. The +notations || · ||fro and || · ||2 indicate the matrix Frobenius norm and vector Euclidean norm, respectively. +In the calculations presented in the main text, we solve an SDP in which we minimize L0 while obeying the +constraints described in SI Sec. I B. In order to suppress off-target condensed phases that are enriched in a single +component, we also introduce an additional constraint on the on-diagonal elements of ϵ, +min +i∈S(α) ϵii ≥ ⟨ϵ⟩x(α), +(S20) +where ⟨ϵ⟩x(α) is defined in Eq. (S10). As shown in Fig. 2c in the main text, this regularization heuristic has the +intended effect of biasing the SDP solution towards coexistence points for which no off-target phases are stable. +D. +Regularization for minimum interaction-matrix dissimilarity +In Fig. 3b–d of the main text, we illustrate how interaction matrices must be changed in order to switch from one +phase diagram to another while making minimal modifications to the interaction matrix. Let us assume that ϵr is the +interaction matrix that solves the regularized SDP for phase coexistence among a set of target phases {⃗φ(α)}r. To +identify the interaction matrix ϵs that is “closest” in some sense to this given initial matrix ϵr while satisfying phase +coexistence among a different set of target phases {⃗φ(α)}s, we define a new objective function +Ld(w) = w +����� +�����(1 + δij)1/2 +¯ωij +⟨φT⟩{α} +����� +����� +fro ++ +����� +�����Li +� +µi − 1 +N +N +� +k=1 +µk +������ +����� +2 ++ ||ϵs − ϵr||, +(S21) +where w ≥ 0 is an adjustable parameter. When minimizing the distance between ϵs and ϵr (main text Fig. 3b), we +use the Frobenius norm for the final term in Ld. When attempting to minimize the number of distinct elements of the + +10 +symmetric matrix ϵr that must be changed in order to establish coexistence among the target phases {⃗φ(α)}s (main +text Fig. 3c–d), we make use of the convex relaxation of vector cardinality; consequently, we use the L1 norm of the +upper triangle of ϵs −ϵr as the final term in Ld in this case. In both cases, we find the smallest value of the adjustable +parameter, w∗, for which ϵs results in the target phase diagram s with no off-target phases. This calculation is carried +out to a precision of w∗ ± 10−3. The distributions shown in Fig. 3b–d of the main text are computed using Ld(w∗), +with w∗ calculated independently for each phase diagram change r → s. +II. +NUMERICAL VERIFICATION OF PHASE COEXISTENCE IN MEAN-FIELD MODELS +After solving the SDP, we identify the exact coexistence point for the target condensed phases {⃗φ(α)}, if it exists, +by solving the non-linear coexistence equations to numerical precision. In this section, we describe the numerical +procedures that we follow to establish bulk phase coexistence and to check for stable off-target phases. +A. +Non-linear phase coexistence solver +The grand potential, Ω(⃗φ), of a mean-field model with an excess chemical potential in the form of Eq. (1) can be +written as +Ω(⃗φ; ⃗µ, ϵ,⃗v) ≡ +N +� +i=1 +� +dφi +�log φi +vi ++ µex,i(⃗φ) − µi +� += +N +� +i=1 +� +�φi log φi − 1 +vi ++ +� +dφi µv(⃗φ) + 1 +2 +N +� +j=1 +ϵijφiφj − µiφi +� +� . (S22) +In order to find a hyperplane that is co-tangent to the local minima of the grand potential corresponding to the dilute +and target condensed phases, we define the grand potential difference for each condensed phase α = 1, . . . , K, +∆Ω(α)(⃗µ; ϵ,⃗v) ≡ Ω(φ(α) +sp ; ⃗µ, ϵ,⃗v) − Ω(φ(0) +sp ; ⃗µ, ϵ,⃗v), +(S23) +where Ω(φ(α) +sp ) and Ω(φ(0) +sp ) indicate the grand potential evaluated at the stationary point (i.e., the local minimum) +of Ω(⃗φ; ⃗µ, ϵ,⃗v) nearest to phase α or the dilute phase, respectively. In practice, we identify these stationary points +by minimizing Ω(⃗φ; ⃗µ, ϵ,⃗v), starting from either a target condensed-phase volume fraction or from the approximate +dilute-phase volume fraction, Eq. (S11), using the Newton conjugate gradient trust-region algorithm [40]. We then +minimize the Euclidean norm of the K-dimensional ⃗∆Ω(⃗µ) vector by iteratively updating the chemical potential +vector ⃗µ and locating the stationary points to calculate ∆Ω(α)(⃗µ; ϵ,⃗v) for each condensed phase. Minimization of this +Euclidean norm is carried out using the Levenberg–Marquardt nonlinear least squares (NLLS) algorithm [27]. The +conditions for bulk-phase coexistence are satisfied when this norm reaches machine precision (≲ 10−14). +B. +Identification of stable off-target condensed phases +We perform a brute-force search for off-target stable phases by minimizing the grand potential at coexistence, +starting from randomly generated initial points in ⃗φ-space. +The grand potential at coexistence, Ω(⃗φ; ⃗µ), is first +determined via the nonlinear phase coexistence solver described above, which fixes ⃗µ. To perform one trial of the +search, we generate an initial point ⃗φtrial by sampling uniformly from the N-dimensional unit simplex, such that +φtrial,i > 0 for all components i = 1, . . . , N and �N +i=1 φtrial,i < 1. We then use the Newton conjugate gradient trust- +region algorithm [40] to minimize Ω(⃗φ) starting from this initial point. This algorithm terminates upon reaching a local +minimum, ⃗φsp, on the grand-potential surface. If ⃗φsp differs from the dilute, ⃗φ(0) +sp , and target-phase, ⃗φ(α) +sp , local minima, +then we compare the grand potential evaluated at this new local minimum, Ωsp = Ω(⃗φsp; ⃗µ), to the coexistence grand +potential, Ω(⃗φ(0) +sp ; ⃗µ). A new local minimum is deemed to be a stable off-target phase if Ωsp ≤ Ω(⃗φ(0) +sp ). +In practice, we carry out 104 trials in order to determine whether any off-target stable phases exist at a proposed +coexistence point (ϵ, ⃗µ). We find that this number of trials is sufficient to yield consistent, reproducible results for +mixtures with N ≤ 6. However, we emphasize that this approach is computationally intensive, since each trial involves +an N-dimensional minimization, and the number of trials must scale exponentially with N in order to carry out a +sufficiently exhaustive search. Thus, while we can use this algorithm to validate the results of our inverse design +regularization heuristic for small N, any direct usage of this brute-force approach (or any brute-force approach, for +that matter) would not be scalable to mixtures with a much larger number of components. + +11 +III. +GENERATION OF TARGET PHASE COMPOSITIONS +A. +Enumeration of target sets +In an effort to explore all possible phase diagrams, we enumerate “target sets” with a fixed number of components. +As in Eq. (S12), target sets label each of the N components as being either enriched or depleted in each of the K +condensed phases. We shall therefore refer to target sets as specifying the “topology” of the phase diagram. We +enumerate all phase-diagram topologies by generating (K, N) target sets that satisfy the following conditions: +1. Each of the N components is enriched in at least one of the K targets, and +2. No two components are enriched in precisely the same targets. +The second condition prevents consideration of phase diagrams that have fewer than N independent components. In +other words, if two components were to be enriched in precisely the same target phases, then the rows and columns of +ϵ associated with these components would also be directly related; therefore, a phase diagram with smaller N but an +equivalent topology could be constructed by grouping these components together. Note that these conditions result +in a finite lower bound on K. For example, these conditions cannot be satisfied using N = 6 components if K < 3. +Two (K, N) targets sets are isomorphic if they can be made identical by permuting the ordering of the components +and/or target phases. To account for this, we sort all target sets into isomorphic groups and consider one member of +each group in all calculations. Specifically, we choose a target set within an isomorphic group by sorting the target +phases in decreasing order of the enriched component cardinality, M (α), and the components in decreasing order of +the number of target phases in which each component appears. Sorting in this way allows us to compute the minimum +ϵ-space distance and minimum number of changed ϵ elements between target phase diagrams (main text Fig. 3b–d). +In the results presented in the main text, we generate “equimolar” target phases by choosing the target volume +fractions in the condensed phases to be φ(α) +i += φ(cond) +T +/M (α) if component i is enriched in phase α and φ(α) +i += 0 +otherwise. We then generate “non-equimolar” target phases by randomly scaling the volume fractions of the enriched +components in each phase of an equimolar target set. To this end, we define a scale factor, s > 0, and scale each +volume fraction by φ(α) +i +← φ(α) +i +[1 + exp(sη)], where η is a random number between 0 and 1. Finally, we adjust each +target phase such that the total volume fraction is equal to φ(cond) +T +using the transformation φ(α) +i +← (φ(cond) +T +/φ(α) +T )φ(α) +i +. +B. +Scaling of the maximum phase count based on graph theory +In Ref. [20], we showed that the feasibility of a related convex optimization problem can be predicted on the basis of +graph-theoretical arguments. Specifically, we showed that the problem of designing a mean-field free-energy landscape +with prescribed local minima reduces to a quadratic program (QP) if all the condensed phases are enriched in exactly +the same number of components, M, and the composition of each target phase is equimolar. Under these special +conditions, the feasibility of the QP can be predicted by considering the maximal cliques [41] within a graph, G, as +follows. The vertices of G correspond to the N species, and the adjacency matrix is defined according to +Gij = +� +1 +if components i and j are both enriched in any phase α +0 +otherwise. +(S24) +The components enriched in each target phase define a subset of the vertices of G, as noted in Eq. (S12). If any of +these subsets are not maximal cliques in G, then the QP is infeasible. +Extending this argument to the present work, we propose that the feasibility of phase coexistence among K +equimolar condensed phases that satisfy the equal-M condition described above can be predicted using the same +graph-theoretical approach. Thus, for these special cases, determining the phase count reduces to the problem of +finding maximal cliques in G. For example, to construct phase diagrams in which the phase count scales quadratically +with the number of components, N, we can enrich every condensed phase with precisely two components. This scaling +follows from Tur´an’s theorem [42], which states that the maximum number of edges of a graph free of 3-cliques is +N 2/4, in which case every edge is a maximal clique corresponding to a target phase. +We can also apply this argument to estimate the largest possible condensed-phase count in an N-component mixture. +Graphs with extremal clique counts can be realized by partitioning the N components into subsets of size 3 (assuming +N is divisible by 3), and creating edges between all pairs of components that are not in the same subset [43]. This +construction results in target phases enriched in precisely M = N/3 components, while the combinatorial nature +of this construction gives rise to a phase-count scaling that is exponential with respect to N, K ∼ 3N/3. We have + +12 +verified that equimolar target phases generated via this construction lead to phase coexistence with the prescribed +phase count. For example, applying our SDP approach to equimolar target phases with N = 15 and M = 5 results +in numerically precise phase coexistence (∆Ω < 4 × 10−13) among K = 243 condensed phases. +IV. +FREE-ENERGY CALCULATIONS IN A MOLECULAR MODEL WITH PAIR POTENTIALS +A. +Model definition +We consider a multicomponent three-dimensional square lattice-gas model in which particles interact via short- +ranged pair potentials. Specifically, if two lattice sites separated by a distance r are occupied by particles of types i +and j, then the additive contribution to the potential energy is +uij(r) = +� +� +� +� +� +r < a +∞ +a ≤ r < 2a +� 10 +z +� +ϵij +r ≥ 2a +0, +(S25) +where a is the lattice constant, z = 26 is the number of neighboring lattice sites within a distance 1 ≤ r/a < 2, and +ϵ is the designed interaction matrix obtained from the regularized L = 1 Flory–Huggins SDP. All particles of type i +are assigned a chemical potential µi. Vacant lattice sites are non-interacting and have chemical potential µ0 = 0. +B. +Free-energy calculations at phase coexistence +We use grand-canonical Monte Carlo simulations [30] to calculate the grand-potential free-energy landscape at +coexistence in this lattice model. +Following the method described in Ref. [13] and Ref. [20], we define an order +parameter ∆φ0α between the dilute phase and the α condensed phase, +∆φ0α(⃗φ) ≡ (⃗φ − ⃗φ(0)) · ˆν0α, +(S26) +where ˆναβ ≡ (⃗φ(α) − ⃗φ(0))/|⃗φ(α) − ⃗φ(0)| and ⃗φ(0) and ⃗φ(α) are the volume fractions at the grand-potential minima in +the dilute and α-phase free-energy basins, respectively. To sample trajectories that reversibly transit between these +two free-energy basins, we add a constraining potential in directions of concentration space orthogonal to ˆν0α, +U0α(⃗φ) ≡ k⊥ +��(⃗φ − ⃗φ(0)) − [(⃗φ − ⃗φ(0)) · ˆν0α]ˆν0α +��6. +(S27) +The efficiency of the simulation is improved by proposing particle exchanges from a lattice site occupied by a particle +(or a vacancy) of type i to a particle (or a vacancy) of type j with probability +pgen(i → j) = +� +� +� +� +� +0.5 +if j is a vacancy +(0.5 − 0.01)/M α +if j is enriched in phase α +0.01/(N − M α) +if j is depleted in phase α +(S28) +and then accounting for pgen in the Metropolis acceptance criteria [30]. +We first perform Wang–Landau simulations [20, 44] to compute the projected free-energy landscape, F0α(∆φ0α), +under the combined potential, HLG + U0α, +F0α(∆φ′) = − log +� +x +δ +� +∆φ0α[⃗φ(x)] − ∆φ′� +exp +� +−HLG(x) − U0α[⃗φ(x)] +� ++ const., +(S29) +where x represents a lattice configuration and HLG(x; ⃗µ, ϵ) is the multicomponent lattice-gas Hamiltonian for the +model described in SI Sec. IV A. We use an L × L × L periodic lattice with L = 6 and k⊥ = 406. Next, we perform +multicanonical simulations [45], using −F0α[⃗φ(x)] as a biasing potential to “flatten” the free-energy barrier separating +the dilute and α phases. We then use MBAR [31] to combine samples from the K multicanonical simulations, one for +each condensed phase. Reweighting the combined samples to the unbiased distribution, in which the probability of a +lattice configuration x is proportional to exp[−HLG(x)], yields the grand potential landscape, Ω(⃗φ; ⃗µ), from which we +can calculate the grand-potential differences between pairs of free-energy basins. Finally, we determine the coexistence +point by tuning ⃗µ and reweighting Ω(⃗φ; ⃗µ) such that the grand-potential differences among all pairs of phases vanish +to within statistical uncertainty. +This final step is accomplished using the algorithm described in Sec. II A. The +landscapes shown in Fig. 4 in the main text are constructed by projecting the grand potential at coexistence into a +two-dimensional space defined by the first two principal components of the reweighted simulation samples. + +13 +V. +FEATURES OF MULTIPHASE COEXISTENCE: EXTENDED RESULTS +A. +Illustrative example with N = 3 species +feasible +region +feasible +region +feas. +reg. +feas. +reg. +FIG. S2. Slices of the SDP solution space for the N = 3, K = 2 design problem described in this section. Inequality constraints +(blue line) and minimum eigenvalue constraints (contours and color mesh) delineate the feasible region (dashed white line). +As an illustrative example, we visualize the SDP constraints for a target phase diagram with three components +and two equimolar condensed phases: ⃗φ(1)/φ(cond) +T += (1/2, 1/2, 0) and ⃗φ(2)/φ(cond) +T += (1/2, 0, 1/2). In the zero-osmotic- +pressure approximation, Eq. (S14), terms involving ϵij only appear when components i and j are both enriched in +the α phase (φ(α) +i +> 0 and φ(α) +j +> 0), +ϵ12φ(1) +1 φ(1) +2 ++ ϵ11φ(1) +1 +2 + ϵ22φ(1) +2 +2 = b1 = log(1 − φ(1) +T ) + φ(1) +T +− �N +i=1 φ(α) +i +/Li, +(S30a) +ϵ13φ(2) +1 φ(2) +3 ++ ϵ11φ(2) +1 +2 + ϵ33φ(2) +3 +2 = b2 = log(1 − φ(2) +T ) + φ(2) +T +− �N +i=1 φ(α) +i +/Li, +(S30b) +leading to the constraints ϵ12 = ϵ13 and ϵ22 = ϵ33 when φ(2) +3 += φ(1) +2 +̸= φ(1) +1 += φ(2) +1 . The equality and inequality +constraints for the equal-chemical-potential conditions, Eq. (S13), can be written as +φ(1) +1 (ϵ11 + ϵ12 + ϵ13) − c1 = µ1 = φ(2) +1 (ϵ11 + ϵ12 + ϵ13) − c2 +(S31a) +φ(1) +2 (ϵ12 + ϵ22 + ϵ23) − c3 = µ2 ≤ φ(2) +2 (ϵ12 + ϵ22 + ϵ23) − c4 +(S31b) +φ(2) +3 (ϵ13 + ϵ23 + ϵ33) − c5 = µ3 ≤ φ(1) +3 (ϵ13 + ϵ23 + ϵ33) − c6 +(S31c) +where c1 = c2, c3 = c5, and c4 = c6 are constants depending only on Li and {⃗φ(α)}. Including the ⃗µ vector, there +are 9 variables with 5 distinct equality constraints. Therefore, we have 4 independent degrees of freedom, which we +choose to express in terms of ϵ11, ϵ12, ϵ22, and ϵ23. +In Fig. S2, we examine cross sections of this 4-dimensional design space in the ϵ12–ϵ23 plane. +After satisfying +the equality constraints described above, the inequality constraints restrict the solution to the domain bounded by +the hyperplane ϵ12 + ϵ22 + ϵ23 ≤ (c3 − c4)/[φ(1) +2 +− φ(2) +2 ]. The feasible solution space lies between this hyperplane +and the semidefinite cone [21] corresponding to the stability constraints, Eq. (S16). To visualize the cone, we plot +contour lines (along with an equivalent color mesh) showing the smallest eigenvalue of the Hessian matrix, Eq. (S16). +Here we use the parameters Li = 10 ∀i and φ(cond) +T += 0.99, for which the ϵ matrix and ⃗µ vector that solve the +regularized SDP are ϵ11 = ϵ22 = ϵ33 = −6.06, ϵ12 = ϵ13 = −9.09, ϵ23 = 0 and µ1 = µ2 = µ3 = −3.87, respectively. +After performing the common tangent plane construction, the condensed-phase volume fractions at coexistence are +⃗φ(1) = (0.495, 0.495, 0.000) and ⃗φ(2) = (0.495, 0.000, 0.495), which are consistent with the target compositions. + +14 +B. +Sensitivity analysis of designed interaction matrices +As described in the main text, we analyze the sensitivity of designed equimolar-target coexistence points to random +perturbations in both the interactions and the condensed-phase volume fractions. Extended results are shown in +Fig. S3, in which we systematically vary the strength of the zero-mean Gaussian noise added to the designed interaction +matrices, and Fig. S4, in which we systematically vary the scale factor used to alter the initially equimolar enriched- +component compositions in the target condensed phases (see Sec. III A). In both cases, increasing the magnitude of +the perturbation, either by increasing the standard deviation of the Gaussian noise (Fig. S3) or by increasing the +composition scale factor (Fig. S4), tends to reduce the probability that coexistence can be re-established among the +same number of target phases, with slightly perturbed volume fractions. For sufficiently large perturbations in ϵ- +space or ⃗φ-space, it is not possible to re-establish coexistence for any set of initial target phases. This behavior can be +understood by noting that large perturbations may cross a critical manifold, at which point the topology of the phase +diagram changes and it becomes impossible to re-establish coexistence among the original K target phases. For smaller +perturbations, random perturbations tend to destabilize one or more target phases in all cases if K > N, as well as +for some phase-diagram topologies with K ≤ N. However, in the following section, we show that in all cases where +small random perturbations tend to prevent phase coexistence among the target phases, coexisting non-equimolar +condensed phases can still be designed by perturbing the interaction matrices in a non-random manner. +〈p(coex)〉equimolar +noise std. dev. = 0.01 +0.05 +0.1 +0.5 +1 +FIG. S3. The probability that coexistence of target phases with N = 6 species can be re-established after zero-mean Gaussian +noise is added to the interaction matrix, as a function of the condensed-phase count, K, and the noise standard deviation +(cf. Fig. 2b in the main text). +〈p(coex)〉equimolar +composition +scale factor = 1 +2 +3 +4 +5 +〈p(global)〉equimolar +FIG. S4. The probability that coexistence (black curve) and global stability (red curve) of target phases with N = 6 species can +be established for condensed phases with non-equimolar compositions, assuming that the equivalent equimolar phase diagram +(i.e., the phase diagram with equimolar condensed phases having the same sets of enriched components) is feasible. Results +are shown as a function of the condensed-phase count, K, and the scale factor s (see SI Sec. III A) used to randomize the +compositions of the enriched components in the condensed phases (cf. Fig. 2c in the main text). + +15 +C. +Analysis of compositional constraints via iterative perturbation +phase-diagram +topology +phase-diagram +topology +-space +-space +A +D +phase-diagram +topology +Method I +B +random +perturbation +Method II +C +guided +perturbation +Method II: Illustrative example +perturbation step +FIG. S5. (a) Schematic of subspaces, each corresponding to a different phase-diagram topology, within the full ϵ-space. In +this illustration, the solution space corresponding to phase-diagram topology s has a lower dimension than the full ϵ-space. +(b,c) Schematics of two methods for iteratively perturbing a coexistence point while maintaining a given phase-diagram topology. +The equimolar coexistence point is indicated by a red star. In Method I, we randomly perturb ϵ (empty star), and then use +NLLS minimization to re-establish phase coexistence consistent with the target phase-diagram topology (green star). In Method +II, we perturb ϵ and ⃗µ in a manner that is consistent with linearized coexistence equations (see text). (d) Repeated application +of Method II results in diffusive behavior of the root-mean-squared distance between the volume fractions and the initial +coexistence point. Projecting the interaction matrices via multidimensional scaling [29] shows the path taken through ϵ-space. +As shown in Fig. S3, random perturbations to an interaction matrix can destabilize one or more of the target phases +in certain cases. This behavior can be understood by noting that the solution space corresponding to a particular +phase-diagram topology may have a lower dimension than the full ϵ-space (Fig. S5a). Thus, a random perturbation +to an equimolar coexistence point that moves off this lower-dimensional manifold must change the phase count. In +this section, we show that it is still possible to find non-equimolar coexistence points on low-dimensional manifolds. +One method for finding non-equimolar coexistence points makes direct use of the nonlinear coexistence solver +introduced in SI Sec. II A. Specifically, we use NLLS to minimize ⃗∆Ω(⃗µ, ϵ), except here we allow both ⃗µ and ϵ to +change. We also modify the least-squares objective function to force the NLLS solver to find a coexistence point +involving all K condensed phases that are present in target phase-diagram topology. +This brute-force approach +(Method I, Fig. S5b) generically leads to a new coexistence point among non-equimolar condensed phases. We note +that for phase-diagram topologies lying on low-dimensional manifolds in ϵ-space (e.g., topology s in Fig. S5b), the +changes in the volume fractions between the equimolar and non-equimolar coexistence points tend to be correlated +across target phases and components. +Alternatively, we can obtain non-equimolar phase diagrams by systematically perturbing ⃗µ (Method II, Fig. S5c). +Starting from a set of feasible equimolar target phases {⃗φ(α)}0 at a designed coexistence point (ϵ0, ⃗µ0), we can expand +Ω and ⃗µ to linear order in ∆ϵ and ∆⃗φ, +Ω(α) = Ω(α) +0 ++ +� +∂F +∂⃗φ +���� +(α) +0 +�⊤ +·∆⃗φ(α)+ +� +∂F +∂⃗ϵ +���� +(α) +0 +�⊤ +·∆⃗ϵ−⃗µ⊤ +0 ·∆⃗φ(α)−∆⃗µ⊤·⃗φ(α) +0 += Ω(α) +0 ++ +� +∂F +∂⃗ϵ +���� +(α) +0 +�⊤ +·∆⃗ϵ−∆⃗µ⊤·⃗φ(α) +0 , (S32) +where ⃗ϵ denotes the vector containing the independent elements of ϵ. Here we have assumed that (ϵ0, ⃗µ0) is located +far from a critical manifold. For phase coexistence to be maintained (to linear order) for some perturbation ∆ϵ, +Ω(α) = Ω(0) must hold for all condensed phases α = 0, . . . , K. Thus, from Eq. (S32), we obtain a system of linear +equations in the form A∆⃗ϵ = B∆⃗µ = ⃗b, where A ∈ RK×N(N+1)/2, B ∈ RK×N, ∆⃗ϵ ∈ RN(N+1)/2, and ∆⃗µ ∈ RN: +� +∂F +∂⃗ϵ +���� +(α) +0 +− ∂F +∂⃗ϵ +���� +(0) +0 +�⊤ +· ∆⃗ϵ = +� +⃗φ(α) +0 +− ⃗φ(0) +0 +�⊤ +· ∆µ +∀α = 1, . . . , K. +(S33) +The matrices A and B may be rank deficient when the volume fractions of the equimolar target phases are linearly +dependent. When col(A) = col(B), solutions are guaranteed for arbitrary perturbations, corresponding to cases in +which rank(A) = rank(B) = K ≤ N. If K > N, then rank(B) < rank(A) ≤ N, and random perturbations will in +general fail. However, it is still possible to find solutions for some perturbation ∆⃗ϵ if col(A) ∩ col(B) ̸= ∅. +To perturb an equimolar coexistence point via Method II, we rotate ⃗µ ∈ RN by a small angle θ in the plane specified +by two orthonormal vectors ˆn1 and ˆn2. To this end, we define the rotation +Rˆn1ˆn2(θ) = I + (ˆn2ˆn⊤ +1 − ˆn1ˆn⊤ +2 ) sin θ + (ˆn1ˆn⊤ +1 + ˆn2ˆn⊤ +2 )(cos θ + 1) +(S34) + +16 +such that the perturbed chemical-potential vector is +⃗µ = ⃗µ0 + ∆⃗µ = Rˆn1ˆn2(θ)⃗µ0. +(S35) +For example, if we choose ˆn1 = (1, 0, . . . , 0) and ˆn2 = (0, 1, . . . , 0), then we only perturb µ1 and µ2, leaving the +chemical potentials of the other components unchanged. In practice, we apply a sequence of rotations with uniformly +distributed random angles in the range [0, 0.005π) for all pairs of axes, and then solve for the perturbed interaction +matrix via A∆⃗ϵ = ⃗b. Finally, since this approach is only accurate to linear order, we fine-tune the coexistence point +using the nonlinear phase coexistence solver described in SI Sec. II A. Applying this method repeatedly produces +a random walk in ϵ-space, in which every interaction matrix corresponds to a coexistence point with the target +phase-diagram topology but, in general, non-equimolar condensed-phase volume fractions (Fig. S5d). As noted above, +the changes in the condensed-phase volume fractions relative to the initial equimolar coexistence point tend to be +correlated across phases and components when the target phase-diagram manifold in ϵ-space is low dimensional. + diff --git a/VNFOT4oBgHgl3EQf6jQA/content/tmp_files/load_file.txt b/VNFOT4oBgHgl3EQf6jQA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..14529947143a2bed69d47170fa4f8647440cb03d --- /dev/null +++ b/VNFOT4oBgHgl3EQf6jQA/content/tmp_files/load_file.txt @@ -0,0 +1,910 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf,len=909 +page_content='Programmable phase behavior in fluids with designable interactions Fan Chen and William M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Jacobs∗ Department of Chemistry, Princeton University, Princeton, NJ 08544, USA (Dated: January 31, 2023) Intracellular fluids can self-organize into phase-separated condensates as a result of evolved biomolecular interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' However, the extent to which the equilibrium phase behavior of a com- plex fluid can be controlled by tuning such interactions remains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Here we apply convex optimization to design fluids that demix into phases with prescribed compositions at thermody- namic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We then show how multicomponent phase diagrams can differ qualitatively from those of simple fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Our approach enables the rational design of multicomponent fluids with phase diagrams of arbitrary complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Intracellular mixtures of biopolymers can demix to form “biomolecular condensates” via the mechanism of liquid–liquid phase separation (LLPS) [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Since phase separation occurs spontaneously in vitro using naturally occurring or engineered biopolymers, biomolecular LLPS is widely believed to be governed primarily by equilib- rium thermodynamics [4–7], even though nonequilibrium processes may affect the phase behavior in vivo [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Al- though it is not surprising that heteropolymers can phase separate at high concentrations, it is remarkable that LLPS can establish coexisting condensates with distinct molecular compositions required for carrying out specific biological functions [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Despite recent progress in modeling multicomponent phase separation [12–19], the relationship between the “design space” of such specific interactions and the capacity for biological fluids to self- organize into chemically diverse droplets via LLPS re- mains poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Exploration of the design space for tunable LLPS is challenging due to the combinatorial complexity associ- ated with multicomponent phase coexistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In order to identify coexisting phases in a model multicomponent mixture, it is first necessary to locate all the candidate phases in a high-dimensional concentration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This constitutes a search problem whose complexity scales exponentially with the dimension of the concentration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' As a result, any algorithm for predicting or de- signing multicomponent phase diagrams that relies on computing phase coexistence from a prescribed set of in- teractions must either be limited to mixtures with a small number of components or employ additional assumptions in the search problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Such assumptions can qualita- tively affect the predictions of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This combinatorial complexity can be avoided by solv- ing the inverse problem of designing interactions to achieve a target phase diagram [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' By instead speci- fying the compositions of the coexisting phases as design criteria, this approach eliminates the need to search for candidate phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Furthermore, if the thermodynamic constraints on bulk phase coexistence can be cast as a convex optimization problem, then suitable interactions ∗ wjacobs@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='edu can be found using efficient algorithms [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Solving the inverse problem therefore allows us to associate an equi- librium phase diagram with a set of designed interactions, leading to a more complete picture of multicomponent phase behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Here we show that an inverse-problem approach can be applied to design equilibrium phase diagrams with ar- bitrary condensed-phase compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We first explain how this strategy can be applied to generic mean-field models with pairwise intermolecular interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We then show that this approach reveals unexpected fea- tures of multicomponent phase diagrams, which differ qualitatively from the intuitive phase behavior of sim- ple fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Finally, we perform molecular simulations and free-energy calculations to demonstrate that our insights apply beyond mean-field theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We begin by assuming that the intermolecular inter- actions in a multicomponent solution are pairwise addi- tive and that the elements of the symmetric interaction matrix, ϵ, are independently tunable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' With these as- sumptions, the vector of excess chemical potentials for all molecular species can be written in the form ⃗µex(⃗φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ϵ,⃗v) = ⃗µv(⃗φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='⃗v) + ϵ⃗φ, (1) where ⃗φ and ⃗v represent the volume fractions and molec- ular volumes of the species, respectively, and ⃗µv is inde- pendent of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' For simplicity, we assume that the ther- mal energy kBT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (1) describes the mean-field Flory–Huggins [22] and van der Waals [23] models, as well as approximate field-theoretic treatments of sequence- dependent heteropolymer mixtures [24], with appropriate choices of ⃗µv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The osmotic pressure, P(⃗φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ϵ,⃗v), which can be determined from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (1) via the Gibbs–Duhem rela- tion, is also linear with respect to ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Our objective is to find an N × N interaction matrix, ϵ, and an N-dimensional chemical potential vector, ⃗µ, that lead to equilibrium phase coexistence among a di- lute phase and K condensed phases in a solution with N solute species (“components”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The inverse design prob- lem is defined by the target volume fractions of each of the condensed phases, ⃗φ(α), indexed by α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' , K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In general, each target condensed phase con- sists of M (α) “enriched” components, which comprise the majority of the volume fraction of phase α, and N −M (α) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='12958v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='soft] 30 Jan 2023 2 A B C D convex relaxation feasible region coexistence coexistence regularization off-target phases FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Inverse design approach to multicomponent phase coexistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (a) Schematic of the design problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Each condensed-phase droplet (gray circle) has a distinct composition of the five molecular components (colors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The enriched-species compositions are indicated by pie charts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (b) Depiction of the solution space for a 3-component, 2-phase problem (see SI for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Inequality constraints (blue line) and minimum eigenvalue constraints (contours) delineate the feasible region (dashed white line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (c) Illustration of the common-hyperplane construction for two phases, α and β, of the convex relaxation solution shown in panel b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (d) Schematic of the regularization heuristic for eliminating stable (∆Ω ≤ 0) off-target phases γ and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' “depleted” components, which are found at negligible concentrations in phase α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Bulk phase coexistence oc- curs when all K + 1 phases have equal osmotic pressures and each molecular species has the same chemical poten- tial in each of the K + 1 phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Furthermore, all K + 1 phases must be stable with respect to concentration fluc- tuations, such that ∂⃗µ(⃗φ)/∂⃗φ is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' To find an ϵ and ⃗µ that satisfy these thermodynamic constraints, we perform a convex relaxation by making three minor approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' First, since the depleted components in each condensed phase have an insignificant effect on the phase diagram, we write the volume-fraction constraint for each depleted component j in every phase as an in- equality, such that φ(α) j < φ(α) depl ≡ φ(α) T /M (α)(N −M (α)), where φ(α) T ≡ �N i=1 φ(α) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Second, we assume that the contributions of the depleted components to ⃗µex in each condensed phase are negligible and can thus be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Finally, we assume that the total volume fraction in the dilute phase, φ(0) T , is very small, so that the osmotic pres- sure at coexistence is near zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' this approximation is valid far from a critical manifold when every component is enriched in at least one target condensed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Taken together, these conditions define a semidefinite program (SDP) that is convex with respect to ϵ and ⃗µ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 1b): µid,i(⃗φ(α);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='⃗v) + µex,i(⃗φ(α);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ϵ,⃗v) ≥ µi ∀i, α (2a) P(⃗φ(α);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ϵ,⃗v) = 0 ∀α (2b) ∂[⃗µid(⃗φ(α);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='⃗v) + ⃗µex(⃗φ(α);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ϵ,⃗v)]/∂⃗φ ≻ λminI ∀α (2c) φ(0) T (⃗µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='⃗v) < φ∗ T(⃗v), (2d) where µid,i = v−1 i log φ(α) i for any component i that is en- riched in the α phase, µid,i = v−1 i log φ(α) depl for any com- ponent i that is depleted in the α phase, and the equal- ity(inequality) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (2a) applies to enriched(depleted) components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (2c), the parameter λmin ≥ 0 places a lower bound on the smallest eigenvalue of the second- derivative matrix to guarantee thermodynamic stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The final constraint, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (2d), ensures that the volume fraction in the dilute phase, φ(0) T , is less than the criti- cal volume fraction, φ∗ T(⃗v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' this condition is independent of ϵ due to the zero-osmotic-pressure assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This program is straightforward to solve using modern convex optimization tools [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Moreover, it is possible to prove whether this convex relaxation is infeasible, mean- ing that no solution (ϵ, ⃗µ) exists for the target condensed- phase volume fractions {⃗φ(α)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Because the approxima- tions required to establish this convex relaxation are well controlled, we expect that there is a close correspondence between the feasible domain of {⃗φ(α)} and the domain on which thermodynamic coexistence can be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' To confirm that the precise thermodynamic conditions for bulk-phase coexistence can be satisfied, we next per- form a multicomponent generalization of the common- tangent construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Starting from the SDP solution (ϵ, ⃗µ), we adjust ⃗µ in order to fit a common tangent plane to the local minima of the grand potential, Ω(⃗φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ⃗µ, ϵ,⃗v) ≡ �N i=1 � dφi[v−1 i log φi + µex,i(⃗φ) − µi] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The con- ditions specified in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (2) imply that the grand potential evaluated at the SDP solution has local minima close to the prescribed target-phase and dilute-phase volume frac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Therefore, we can fit a common tangent plane by minimizing the norm of ⃗∆Ω(⃗µ), where ∆Ω(α)(⃗µ) is the difference between Ω(⃗φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ⃗µ) evaluated at the local mini- mum near the dilute phase and at the local minimum near the α condensed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This procedure converges rapidly using standard numerical methods [27], since the convex relaxation is a good approximation of this nonlin- ear hyperplane-fitting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In the extensive numeri- cal tests described below, we indeed find that a solution to the convex relaxation typically implies that the condi- tions for coexistence can be satisfied for the target phases to numerical precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This algorithm provides a scalable and highly accurate means for predicting whether prescribed target phases can be in simultaneous thermodynamic coexistence and, if so, for determining a coexistence point (ϵ, ⃗µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In gen- 3 eral, the convex relaxation specified by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (2) defines a continuous space of interaction matrices that solve the inverse design problem, with a unique ⃗µ corresponding to each point in this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' However, we have not yet considered the possibility that other “off-target” con- densed phases may be equally or even more stable than the target phases at the calculated coexistence point, meaning that the target phases are only in marginal or metastable coexistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This possibility can be addressed by introducing a regularization heuristic that attempts to maximize Ω(φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ⃗µ, ϵ,⃗v) away from the target phases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Specifically, based on the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (1), we seek to minimize both the norm of ϵ − ¯µ/⟨φ(α) T ⟩, where ¯µij = (µi +µj)/2 and ⟨φ(α) T ⟩ is the mean target-phase to- tal volume fraction, and the variance of the elements of ⃗µ (see Supplementary Information for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Regular- izing the SDP in this way tends to destabilize off-target phases while guaranteeing that the solution to our convex relaxation is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Our approach reveals unexpected features of multicom- ponent phase behavior, which we demonstrate in the case of a Flory–Huggins polymer model [22] with degree of polymerization ranging from L = 1 to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (See the Sup- plementary Information for the model and SDP defini- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=') Here we describe results for mixtures with N = 6 species, a sufficient number to uncover qualitative differ- ences with simple fluids while still permitting exhaustive searches for off-target phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' For simplicity, we choose the same total volume fraction, φα T = φ(cond) T , for each condensed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We begin by designing phase diagrams with “equimolar” target phases, in which every enriched component within a target phase has the same volume fraction ≃ φ(cond) T /M (α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Interestingly, we find that the feasibility of the SDP for any particular target phase di- agram is independent of L and φ(cond) T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' However, the probability that a solution to the regularized SDP re- sults in phase coexistence tends to increase with L and φ(cond) T ≫ φ∗ T (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 2a), since the convex relaxation be- comes a more accurate approximation as the coexistence pressure decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Intuition based on the phase behavior of simple mix- tures suggests that small changes in {⃗φ(α)} should result in small changes in ϵ, and vice versa, unless the mixture is near a critical point where two or more of the ⃗φ(α) merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' For example, small changes in the dimensionless interaction parameter in an incompressible binary mix- ture energy perturb the binodal but do not change the coexistence region qualitatively, as long as φT ≫ φ∗ T [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Furthermore, the Gibbs Phase Rule (GPR) [28], which relates the number of coexisting phases to the number of thermodynamic degrees of freedom, should be ex- pected to limit the number of condensed phases, K, for which our method can find a coexistence point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' For the N-component fluids that we study here, this expected bound is K ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Our inverse approach reveals multiple ways in which such intuition can mislead in multicomponent fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Sur- 〈p(coex)〉equimolar 〈p(coex)〉equimolar 〈p(global)〉equimolar A C B condensed phase count p(coex|feas) zero noise noise std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' = 10−3 space convex relaxation regularized solution solution space for phase coexistence coexisting phases are globally stable target phases : FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Features of multicomponent phase coexis- tence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (a) Schematic of ϵ-space and validation that SDP solutions (feas) result in coexistence (coex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This probabil- ity, p(coex|feas), approaches one as the degree of polymer- ization, L, increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Here we consider phase diagrams with N = 6 species and equimolar condensed-phase compositions (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Each class of isomorphic phase diagrams, which are equivalent under permutation of component and target-phase indices, is considered once in these calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (b) Sensi- tivity of phase coexistence to perturbations in the interaction matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We add zero-mean Gaussian noise to the interac- tion matrices that produce phase coexistence among target phases with equimolar compositions, and then attempt to re- establish coexistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The probability of success, averaged over many trials, is ⟨p(coex)⟩equimolar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (c) The probability that coexistence can be achieved for condensed phases with arbitrary compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Starting from feasible phase diagrams with equimolar condensed phases, we construct target phases by randomly scaling the enriched component compositions in each condensed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We also show whether the SDP so- lution leads to global phase coexistence, meaning no stable off-target phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In panels b–c, φ(cond) T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='95 and L = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' prisingly, we can design coexistence points where the condensed-phase count, K, is greater than the number of distinct species, N (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' At first glance, these exam- ples might appear to conflict with the GPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Furthermore, in terms of bulk phase coexistence, these examples imply that the lever rule, which relates the total concentrations of the various species in a mixture to the mole fractions of the coexisting phases, does not have a unique solu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' However, in mixtures with designed interactions, we can end up with coexistence equations that are lin- early dependent, allowing us to perform a common tan- gent plane construction when K > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This behavior can be understood by realizing that the design problem, with N(N+1)/2 tunable interaction-matrix parameters, is not overdetermined, and that convex optimization identifies interaction matrices that result in linearly dependent co- 4 existence equations at the prescribed {⃗φ(α)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' These un- usual coexistence conditions do not occur in mixtures with fewer than N = 5 species, but become increasingly common as the number of components increases (see Sup- plementary Information for further discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Consistent with this explanation, we find that small, random perturbations to the designed interaction matri- ces almost always preclude phase coexistence of the tar- get phases when K > N (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Specifically, we add zero-mean Gaussian noise to the designed matrix ϵ, and then attempt to perform a common tangent plane con- struction for phases close to the original target phases by tuning ⃗µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' After perturbation, only a subset of the original K condensed phases can be brought into coexis- tence with the dilute phase, while the remaining phases become metastable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Similarly, we can make random per- turbations to the initially equimolar compositions of the enriched components in each target phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' If we then at- tempt to solve the convex relaxation for these perturbed target phases, we almost always find that the SDP is infeasible when K > N (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This indicates that a particular relationship among the compositions of the enriched components is necessary to establish linearly de- pendent coexistence equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Yet surprisingly, such be- havior is not limited to phase diagrams with K > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We also find certain phase diagrams with K ≤ N that are similarly sensitive to random perturbations in ϵ and {⃗φ(α)} (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 2b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Regardless, when our inverse ap- proach indicates that coexistence among the prescribed phases is possible, exhaustive sampling of the grand po- tential landscape confirms that these phases are globally stable with high probability (red curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Taken together, these observations suggest that spe- cial coexistence points, which are sensitive to small per- turbations in ϵ, lie on manifolds of lower dimension than the full ϵ-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Coexistence is not limited to K ≤ N condensed phases on these manifolds, although some of these phases must become metastable if we move off the manifold by perturbing the interaction matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' These manifolds represent “interfaces” between volumes of ϵ- space corresponding to condensed phases with different sets of enriched components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In other words, crossing one of these interfaces by changing ϵ entails a discon- tinuous transition from one set of condensed phases to another, with phases from both sets stable on the inter- face itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We emphasize that our calculations are per- formed far from critical points, since the Euclidean dis- tance between all pairs of target phases {⃗φ(α)} is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Furthermore, these special coexistence points need not have equimolar condensed-phase compositions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' however, the enriched-component volume fractions are not inde- pendent on these manifolds (see Supplementary Infor- mation for further discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' How are distinct multicomponent phase diagrams re- lated in ϵ-space?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' To address this question, we can use the Frobenius norm to measure of the distance between two interaction matrices ϵr and ϵs, corresponding to differ- ent phase diagrams with globally stable condensed phases A B C D space FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Relationships among phase diagrams in ϵ- space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (a) Low-dimensional representation of the interaction matrices corresponding to the phase diagrams considered in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Circles with black outlines indicate phase diagrams that are sensitive to random perturbations in ϵ (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Multidimensional scaling [29] has been used to preserve dis- tances in ϵ-space, taken here as the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (b) Dis- tances between pairs of matrices in panel a (black), and the minimum distance required to switch from phase diagram r to phase diagram s (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Box plots indicate the quartiles of the distance distributions as a function of the phase count differ- ence, Ks−Kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (c) The minimum number of entries of the sym- metric ϵ matrix that must be changed to switch from phase diagram r to phase diagram s, Drs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Dmax ≡ N(N + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (d) Asymmetry in the minimum number of elements changed when switching between phase diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' {⃗φ(α)}r and {⃗φ(α)}s, respectively (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 3a and black dis- tributions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Yet because the interaction ma- trix that stabilizes a particular phase diagram is typi- cally not unique, it is more useful to quantify the extent to which an interaction matrix must be changed in or- der to switch from one phase diagram to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We can accomplish this within our inverse design framework by modifying the regularization heuristic in one of two ways (see Supplementary Information for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Start- ing from the interaction matrix ϵr that solves the origi- nal regularized SDP for phase diagram r, we identify the “closest” matrix ϵs that solves phase diagram s by mini- mizing the Frobenius norm ||ϵs−ϵr||fro (red distributions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This distance can be infinitesimal if ϵr resides on a low-dimensional manifold and the phase-diagram change r → s reduces the phase count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Similarly, the minimal distance between interaction matrices is typi- cally larger when we add phases, such that Ks > Kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Alternatively, we can determine the smallest number of 5 1 2 3 4 1 2 3 4 5 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='89 R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='00 R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='00 R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='00 R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='00 R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='00 R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='00 R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='00 R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='00 1 2 3 4 5 6 R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='00 R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='00 R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='00 R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='00 R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='00 R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='00 A B C Grand potential density FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Transferable predictions validate the pairwise approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Simulated free-energy landscapes at phase coexistence (|∆Ω| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='007kBT) in a molecular model with pair potentials derived from designed interaction matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Examples are shown for mixtures with N = 5 species and (a) K = 4, (b) 5, and (c) 6 condensed phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (N + 1)- dimensional landscapes are projected onto two principal- component coordinates, X1 and X2, for visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The Pearson correlation coefficient, R, between the target and equilibrium composition is shown for each condensed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' distinct matrix elements that must be changed to switch phase diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This minimal number of elementwise changes, Drs, is always greater than zero (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Our calculations reveal that Drs is also asymmetric with re- spect to phase diagram changes r ↔ s and tends to in- crease with the net number of added phases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Finally, we assess whether the predictions of our in- verse design approach apply beyond mean-field models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Specifically, we consider fluids in which the potential en- ergy can be written as a sum of short-ranged pair poten- tials [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (1) is only accurate at low con- centrations for such fluids, since the higher-order virial coefficients generically depend on the species-specific pair potentials [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' To this end, we perform free-energy cal- culations using simulations of a multicomponent lattice gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We first design an interaction matrix, ϵ, for a target phase diagram using the L = 1 Flory–Huggins SDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We then use this matrix to define the well-depths of the pair potentials, uij(1 ≤ r/a < 2) ∝ ϵij, where r is the distance between particles of types i and j and a is the lattice con- stant (see Supplementary Information for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We first identify the free-energy basins in this model by run- ning grand-canonical Monte Carlo simulations [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We then sample reversible transitions between the dilute free- energy basin and each of the condensed-phase basins [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Finally, we reconstruct the free-energy landscapes in the N-dimensional ⃗φ-space and adjust the chemical poten- tials to bring all phases into coexistence [31], at which point the grand potentials of all basins are all equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Our mean-field design approach results in non-mean- field free-energy landscapes that are consistent with the target phase diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We carry out simulations with five species and condensed-phase counts that are less than, equal to, and greater than the number of compo- nents (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In line with the predictions of the mean- field model, we find that coexistence can be achieved within sampling accuracy, even in the K > N case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Mi- nor quantitative differences in the phase compositions oc- cur in the simulation model due to inaccuracies in the mean-field approximations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' however, the enriched com- ponents in all simulated phases match the designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In summary, we have introduced a method to de- sign the phase behavior of multicomponent fluids with pairwise—or approximately pairwise—interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Our approach provides insight into the structure of the interaction-matrix solution space, revealing ways in which the behavior of multicomponent fluids can differ qualitatively from that of simple fluids, while also scal- ing well to mixtures with tens or hundreds of compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In practice, it may not be possible to engineer molecular interactions with the independence and pre- cision necessary to construct all theoretically possible phase diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In this regard, our results indicate that the physically relevant constraints on the phase behav- ior of multicomponent fluids arise from the properties of the intermolecular interactions, since thermodynamically allowed phase diagrams can be surprisingly complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Ad- ditional constraints on the physicochemical properties of the molecular components should therefore be added as an additional layer of our design framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Overall, our results highlight the need to understand the extent to which molecular interactions can be tuned indepen- dently in phase-separating (bio)chemical fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We an- ticipate that our theoretical approach will play an im- portant role in ongoing efforts to unravel the connections between molecular design and multicomponent phase be- havior [32–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This research was partially supported by the NSF through the Princeton University’s Materials Research Science and Engineering Center DMR-2011750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 6 [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Hyman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Weber, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' J¨ulicher, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Cell Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Biol 30, 39 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' [2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Shin and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Brangwynne, Science 357, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Landau, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 86, 2050 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' [45] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Berg and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Neuhaus, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 68, 9 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 7 SUPPLEMENTARY INFORMATION FOR “PROGRAMMABLE PHASE BEHAVIOR IN FLUIDS WITH DESIGNABLE INTERACTIONS” I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' APPLICATION TO THE MULTICOMPONENT FLORY–HUGGINS MODEL In the numerical examples presented in the main text, we consider the special case of the multicomponent Flory– Huggins polymer model [22], for which ⃗µv(⃗φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='⃗v) = − log(1−φT)−(1−1/Li).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We introduce the degree of polymerization, Li, for each component i in place of the molecular volume vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In this section, we first write the thermodynamic quantities for this model explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We then show how to formulate the SDP for this specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Model definition The Helmholtz free-energy density, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' chemical potential, ⃗µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' osmotic pressure, P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' and Hessian matrix, ∂⃗µ(⃗φ)/∂⃗φ, in the multicomponent Flory–Huggins model are F = N � i=1 φi Li log φi + (1 − φT) log(1 − φT) + 1 2 N � i=1 N � j=1 ϵijφiφj (S1) µi = 1 Li log φi − log(1 − φT) − � 1 − 1 Li � + N � j=1 ϵijφj (S2) P = − log(1 − φT) + N � i=1 φi Li − φT + 1 2 N � i=1 N � j=1 ϵijφiφj (S3) ∂µi ∂φj = δij Liφi + 1 1 − φT + ϵij, (S4) respectively, where Li is the degree of polymerization of polymeric species i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Before writing down the SDP constraints for a particular set of condensed-phase volume fractions, {⃗φ(α)}, we consider a mixture with a fixed composition ⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The resulting expressions will be utilized in subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The mixture composition is normalized such that �N i=1 xi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In order to calculate the total volume fraction at the critical point, φ∗ T(⃗x), we set the projection of the Hessian matrix along ⃗x to zero, N � i=1 N � j=1 xi ∂µi ∂φj xj = 1 φT N � i=1 xi Li + 1 1 − φT + N � i=1 N � j=1 xiϵijxj = 0, (S5) and differentiate with respect to φT, − ∂⟨ϵ⟩x ∂φT = − 1 (φT)2 N � i=1 xi Li + 1 (1 − φT)2 = 0, (S6) where ⟨ϵ⟩x ≡ �N i=1 �N j=1 xiϵijxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The critical volume fraction at fixed composition ⃗x is thus φ∗ T(⃗x) = 1 1 + ⟨1/L⟩−1/2 x , (S7) where ⟨1/L⟩x ≡ �N i=1 xi/Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Assuming that the chemical potential vector is known, the total volume fraction of a condensed phase with com- position ⃗x can be approximated by setting the osmotic pressure equal to zero and projecting ⃗µ along ⃗x, N � i=1 xiµi ≡ ⟨µ⟩x = N � i=1 xi Li log φi + ⟨1/L⟩x − 1 − log(1 − φT) + φT⟨ϵ⟩x, (S8) to yield a non-linear equation for φT, 2 φT log(1 − φT) + ⟨1/L⟩ log φT − log(1 − φT) + 1 − ⟨1/L⟩x + N � i=1 xi Li log xi − ⟨µ⟩x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S9) 8 We can also solve for the mean interaction, ⟨ϵ⟩x, in a condensed phase with composition ⃗x, ⟨ϵ⟩x = 2 (φT)2 [log(1 − φT) + φT (1 − ⟨1/L⟩x)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S10) We can similarly obtain an expression for dilute-phase volume fractions in terms of ⃗µ by assuming that the osmotic pressure is nearly zero, such that φ(0) T is very small, φi ≃ exp [Liµi + (Li − 1)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S11) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' SDP formulation In this section, we use the notation {⃗φ(α)} to refer to the target volume fractions in the K condensed phases, α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We further assume that M (α) species are enriched in the α phase and that the target volume fractions of the depleted components are set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We can therefore denote the set of enriched components in phase α by S(α) ≡ {i|δ(φ(α) i ) = 0}, (S12) where δ(·) is the Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Thus, M (α) = N − �N i=1 δ(φ(α) i ) is the cardinality of the vector ⃗φ(α), and the target total volume fraction in phase α is φ(α) T = � i∈S(α) φ(α) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The equal chemical potential constraints for enriched and depleted components, respectively, take the form N � j=1 Liφ(α) j ϵij − Liµi + log φ(α) i − Li log(1 − φ(α) T ) − (Li − 1) = 0, δ � φ(α) i � = 0 (S13a) N � j=1 Liφ(α) j ϵij − Liµi + log � φ(α) T ζ M (α)(N − M (α)) � − Li log(1 − φ(α) T ) − (Li − 1) ≥ 0, δ � φ(α) i � = 1 (S13b) for each species index i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' , N and each condensed phase α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We set the adjustable parameter ζ = 10−2 in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The zero-osmotic-pressure constraint for each condensed phase α is 1 2 N � i=1 N � j=1 φ(α) i φ(α) j ϵij − log(1 − φ(α) T ) − φ(α) T + N � i=1 φ(α) i Li = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S14) In order to place constraints on the Hessian matrices in the condensed phases, we define the augmented target volume fractions ˜φ(α) i = φ(α) i + φ(α) T ζ M (α)(N − M (α))δ � φ(α) i � , (S15) so that we have ˜φ(α) i > 0 for all i and all α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Each condensed-phase scaled Hessian matrix must then satisfy δij ˜φ(α) i + � LiLj 1 − φ(α) T + � LiLjϵij ⪰ λminδij, (S16) where λmin is the smallest allowed eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We choose λmin = 1 in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Finally, we constrain the chemical potentials such that the approximate total volume fraction of a roughly equimolar dilute phase is below the critical volume fraction by utilizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S11) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S7), log N � i=1 exp(Liµi + Li − 1) ≤ log � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='9 φ∗ T � ⃗x = � 1 N , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' , 1 N ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S17) Within the approximations of this convex relaxation, these constraints define the joint space of interaction matrices, ϵ, and chemical potential vectors, ⃗µ, for which bulk phase coexistence can be established among the target condensed phases and a dilute phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 9 space convex relaxation regularized solution solution space for coexistence of target phases solution space for which the target phases are globally stable FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Schematic of regularization in ϵ-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Regularized convex optimization identifies a unique ϵ matrix, indicated by the star in the diagram, and a corresponding ⃗µ vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The heuristic regularization method described in SI Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' I C is likely to find a solution within the globally stable solution space (cyan region), if it exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Regularization for global stability of target phases Next, we regularize our convex optimization problem in order to identify an interaction matrix, ϵ, and chemical potential vector, ⃗µ, that are least likely to result in stable off-target phases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Off-target condensed phases correspond to local minima of the grand-potential, Ω(⃗φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ⃗µ, ϵ,⃗v), that lie below the grand potential of the dilute phase, Ω(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We therefore aim to maximize the grand potential everywhere in the domain ⃗φ, except at the target phases, {⃗φ(0), ⃗φ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' , ⃗φ(K)}, where Ω(α) = Ω(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Since the ⃗µ and ϵ-dependent contributions to the grand potential have the form ω(⃗φ) ≡ � ⃗φ⊤ϵ − ⃗µ � ⃗φ, we define the objective function L0 = ����� �����(1 + δij)1/2 ¯ωij ⟨φT⟩{α} ����� ����� fro + ����� �����Li � µi − 1 N N � k=1 µk ������ ����� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S18) The first term is a scaled and shifted Euclidean norm of the unique ¯ωij elements, ¯ωij ⟨φT⟩{α} = ϵij − 1 ⟨φT⟩{α} � µi + µj 2 − 1 N N � k=1 µk � , (S19) where ⟨φT⟩{α} ≡ (1/K) �K α=1 φ(α) T , while the second term is a scaled standard deviation of the ⃗µ elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The notations || · ||fro and || · ||2 indicate the matrix Frobenius norm and vector Euclidean norm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In the calculations presented in the main text, we solve an SDP in which we minimize L0 while obeying the constraints described in SI Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' I B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In order to suppress off-target condensed phases that are enriched in a single component, we also introduce an additional constraint on the on-diagonal elements of ϵ, min i∈S(α) ϵii ≥ ⟨ϵ⟩x(α), (S20) where ⟨ϵ⟩x(α) is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 2c in the main text, this regularization heuristic has the intended effect of biasing the SDP solution towards coexistence points for which no off-target phases are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Regularization for minimum interaction-matrix dissimilarity In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 3b–d of the main text, we illustrate how interaction matrices must be changed in order to switch from one phase diagram to another while making minimal modifications to the interaction matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Let us assume that ϵr is the interaction matrix that solves the regularized SDP for phase coexistence among a set of target phases {⃗φ(α)}r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' To identify the interaction matrix ϵs that is “closest” in some sense to this given initial matrix ϵr while satisfying phase coexistence among a different set of target phases {⃗φ(α)}s, we define a new objective function Ld(w) = w ����� �����(1 + δij)1/2 ¯ωij ⟨φT⟩{α} ����� ����� fro + ����� �����Li � µi − 1 N N � k=1 µk ������ ����� 2 + ||ϵs − ϵr||, (S21) where w ≥ 0 is an adjustable parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' When minimizing the distance between ϵs and ϵr (main text Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 3b), we use the Frobenius norm for the final term in Ld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' When attempting to minimize the number of distinct elements of the 10 symmetric matrix ϵr that must be changed in order to establish coexistence among the target phases {⃗φ(α)}s (main text Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 3c–d), we make use of the convex relaxation of vector cardinality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' consequently, we use the L1 norm of the upper triangle of ϵs −ϵr as the final term in Ld in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In both cases, we find the smallest value of the adjustable parameter, w∗, for which ϵs results in the target phase diagram s with no off-target phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This calculation is carried out to a precision of w∗ ± 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The distributions shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 3b–d of the main text are computed using Ld(w∗), with w∗ calculated independently for each phase diagram change r → s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' NUMERICAL VERIFICATION OF PHASE COEXISTENCE IN MEAN-FIELD MODELS After solving the SDP, we identify the exact coexistence point for the target condensed phases {⃗φ(α)}, if it exists, by solving the non-linear coexistence equations to numerical precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In this section, we describe the numerical procedures that we follow to establish bulk phase coexistence and to check for stable off-target phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Non-linear phase coexistence solver The grand potential, Ω(⃗φ), of a mean-field model with an excess chemical potential in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (1) can be written as Ω(⃗φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ⃗µ, ϵ,⃗v) ≡ N � i=1 � dφi �log φi vi + µex,i(⃗φ) − µi � = N � i=1 � �φi log φi − 1 vi + � dφi µv(⃗φ) + 1 2 N � j=1 ϵijφiφj − µiφi � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S22) In order to find a hyperplane that is co-tangent to the local minima of the grand potential corresponding to the dilute and target condensed phases, we define the grand potential difference for each condensed phase α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' , K, ∆Ω(α)(⃗µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ϵ,⃗v) ≡ Ω(φ(α) sp ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ⃗µ, ϵ,⃗v) − Ω(φ(0) sp ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ⃗µ, ϵ,⃗v), (S23) where Ω(φ(α) sp ) and Ω(φ(0) sp ) indicate the grand potential evaluated at the stationary point (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=', the local minimum) of Ω(⃗φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ⃗µ, ϵ,⃗v) nearest to phase α or the dilute phase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In practice, we identify these stationary points by minimizing Ω(⃗φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ⃗µ, ϵ,⃗v), starting from either a target condensed-phase volume fraction or from the approximate dilute-phase volume fraction, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S11), using the Newton conjugate gradient trust-region algorithm [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We then minimize the Euclidean norm of the K-dimensional ⃗∆Ω(⃗µ) vector by iteratively updating the chemical potential vector ⃗µ and locating the stationary points to calculate ∆Ω(α)(⃗µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ϵ,⃗v) for each condensed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Minimization of this Euclidean norm is carried out using the Levenberg–Marquardt nonlinear least squares (NLLS) algorithm [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The conditions for bulk-phase coexistence are satisfied when this norm reaches machine precision (≲ 10−14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Identification of stable off-target condensed phases We perform a brute-force search for off-target stable phases by minimizing the grand potential at coexistence, starting from randomly generated initial points in ⃗φ-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The grand potential at coexistence, Ω(⃗φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ⃗µ), is first determined via the nonlinear phase coexistence solver described above, which fixes ⃗µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' To perform one trial of the search, we generate an initial point ⃗φtrial by sampling uniformly from the N-dimensional unit simplex, such that φtrial,i > 0 for all components i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' , N and �N i=1 φtrial,i < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We then use the Newton conjugate gradient trust- region algorithm [40] to minimize Ω(⃗φ) starting from this initial point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This algorithm terminates upon reaching a local minimum, ⃗φsp, on the grand-potential surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' If ⃗φsp differs from the dilute, ⃗φ(0) sp , and target-phase, ⃗φ(α) sp , local minima, then we compare the grand potential evaluated at this new local minimum, Ωsp = Ω(⃗φsp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ⃗µ), to the coexistence grand potential, Ω(⃗φ(0) sp ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ⃗µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' A new local minimum is deemed to be a stable off-target phase if Ωsp ≤ Ω(⃗φ(0) sp ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In practice, we carry out 104 trials in order to determine whether any off-target stable phases exist at a proposed coexistence point (ϵ, ⃗µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We find that this number of trials is sufficient to yield consistent, reproducible results for mixtures with N ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' However, we emphasize that this approach is computationally intensive, since each trial involves an N-dimensional minimization, and the number of trials must scale exponentially with N in order to carry out a sufficiently exhaustive search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Thus, while we can use this algorithm to validate the results of our inverse design regularization heuristic for small N, any direct usage of this brute-force approach (or any brute-force approach, for that matter) would not be scalable to mixtures with a much larger number of components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 11 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' GENERATION OF TARGET PHASE COMPOSITIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Enumeration of target sets In an effort to explore all possible phase diagrams, we enumerate “target sets” with a fixed number of components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' As in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S12), target sets label each of the N components as being either enriched or depleted in each of the K condensed phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We shall therefore refer to target sets as specifying the “topology” of the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We enumerate all phase-diagram topologies by generating (K, N) target sets that satisfy the following conditions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Each of the N components is enriched in at least one of the K targets, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' No two components are enriched in precisely the same targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The second condition prevents consideration of phase diagrams that have fewer than N independent components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In other words, if two components were to be enriched in precisely the same target phases, then the rows and columns of ϵ associated with these components would also be directly related;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' therefore, a phase diagram with smaller N but an equivalent topology could be constructed by grouping these components together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Note that these conditions result in a finite lower bound on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' For example, these conditions cannot be satisfied using N = 6 components if K < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Two (K, N) targets sets are isomorphic if they can be made identical by permuting the ordering of the components and/or target phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' To account for this, we sort all target sets into isomorphic groups and consider one member of each group in all calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Specifically, we choose a target set within an isomorphic group by sorting the target phases in decreasing order of the enriched component cardinality, M (α), and the components in decreasing order of the number of target phases in which each component appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Sorting in this way allows us to compute the minimum ϵ-space distance and minimum number of changed ϵ elements between target phase diagrams (main text Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 3b–d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In the results presented in the main text, we generate “equimolar” target phases by choosing the target volume fractions in the condensed phases to be φ(α) i = φ(cond) T /M (α) if component i is enriched in phase α and φ(α) i = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We then generate “non-equimolar” target phases by randomly scaling the volume fractions of the enriched components in each phase of an equimolar target set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' To this end, we define a scale factor, s > 0, and scale each volume fraction by φ(α) i ← φ(α) i [1 + exp(sη)], where η is a random number between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Finally, we adjust each target phase such that the total volume fraction is equal to φ(cond) T using the transformation φ(α) i ← (φ(cond) T /φ(α) T )φ(α) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Scaling of the maximum phase count based on graph theory In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' [20], we showed that the feasibility of a related convex optimization problem can be predicted on the basis of graph-theoretical arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Specifically, we showed that the problem of designing a mean-field free-energy landscape with prescribed local minima reduces to a quadratic program (QP) if all the condensed phases are enriched in exactly the same number of components, M, and the composition of each target phase is equimolar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Under these special conditions, the feasibility of the QP can be predicted by considering the maximal cliques [41] within a graph, G, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The vertices of G correspond to the N species, and the adjacency matrix is defined according to Gij = � 1 if components i and j are both enriched in any phase α 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S24) The components enriched in each target phase define a subset of the vertices of G, as noted in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' If any of these subsets are not maximal cliques in G, then the QP is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Extending this argument to the present work, we propose that the feasibility of phase coexistence among K equimolar condensed phases that satisfy the equal-M condition described above can be predicted using the same graph-theoretical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Thus, for these special cases, determining the phase count reduces to the problem of finding maximal cliques in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' For example, to construct phase diagrams in which the phase count scales quadratically with the number of components, N, we can enrich every condensed phase with precisely two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This scaling follows from Tur´an’s theorem [42], which states that the maximum number of edges of a graph free of 3-cliques is N 2/4, in which case every edge is a maximal clique corresponding to a target phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We can also apply this argument to estimate the largest possible condensed-phase count in an N-component mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Graphs with extremal clique counts can be realized by partitioning the N components into subsets of size 3 (assuming N is divisible by 3), and creating edges between all pairs of components that are not in the same subset [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This construction results in target phases enriched in precisely M = N/3 components, while the combinatorial nature of this construction gives rise to a phase-count scaling that is exponential with respect to N, K ∼ 3N/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We have 12 verified that equimolar target phases generated via this construction lead to phase coexistence with the prescribed phase count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' For example, applying our SDP approach to equimolar target phases with N = 15 and M = 5 results in numerically precise phase coexistence (∆Ω < 4 × 10−13) among K = 243 condensed phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' FREE-ENERGY CALCULATIONS IN A MOLECULAR MODEL WITH PAIR POTENTIALS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Model definition We consider a multicomponent three-dimensional square lattice-gas model in which particles interact via short- ranged pair potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Specifically, if two lattice sites separated by a distance r are occupied by particles of types i and j, then the additive contribution to the potential energy is uij(r) = � � � � � r < a ∞ a ≤ r < 2a � 10 z � ϵij r ≥ 2a 0, (S25) where a is the lattice constant, z = 26 is the number of neighboring lattice sites within a distance 1 ≤ r/a < 2, and ϵ is the designed interaction matrix obtained from the regularized L = 1 Flory–Huggins SDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' All particles of type i are assigned a chemical potential µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Vacant lattice sites are non-interacting and have chemical potential µ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Free-energy calculations at phase coexistence We use grand-canonical Monte Carlo simulations [30] to calculate the grand-potential free-energy landscape at coexistence in this lattice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Following the method described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' [13] and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' [20], we define an order parameter ∆φ0α between the dilute phase and the α condensed phase, ∆φ0α(⃗φ) ≡ (⃗φ − ⃗φ(0)) · ˆν0α, (S26) where ˆναβ ≡ (⃗φ(α) − ⃗φ(0))/|⃗φ(α) − ⃗φ(0)| and ⃗φ(0) and ⃗φ(α) are the volume fractions at the grand-potential minima in the dilute and α-phase free-energy basins, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' To sample trajectories that reversibly transit between these two free-energy basins, we add a constraining potential in directions of concentration space orthogonal to ˆν0α, U0α(⃗φ) ≡ k⊥ ��(⃗φ − ⃗φ(0)) − [(⃗φ − ⃗φ(0)) · ˆν0α]ˆν0α ��6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S27) The efficiency of the simulation is improved by proposing particle exchanges from a lattice site occupied by a particle (or a vacancy) of type i to a particle (or a vacancy) of type j with probability pgen(i → j) = � � � � � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='5 if j is a vacancy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='5 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='01)/M α if j is enriched in phase α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='01/(N − M α) if j is depleted in phase α (S28) and then accounting for pgen in the Metropolis acceptance criteria [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We first perform Wang–Landau simulations [20, 44] to compute the projected free-energy landscape, F0α(∆φ0α), under the combined potential, HLG + U0α, F0α(∆φ′) = − log � x δ � ∆φ0α[⃗φ(x)] − ∆φ′� exp � −HLG(x) − U0α[⃗φ(x)] � + const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=', (S29) where x represents a lattice configuration and HLG(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ⃗µ, ϵ) is the multicomponent lattice-gas Hamiltonian for the model described in SI Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We use an L × L × L periodic lattice with L = 6 and k⊥ = 406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Next, we perform multicanonical simulations [45], using −F0α[⃗φ(x)] as a biasing potential to “flatten” the free-energy barrier separating the dilute and α phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We then use MBAR [31] to combine samples from the K multicanonical simulations, one for each condensed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Reweighting the combined samples to the unbiased distribution, in which the probability of a lattice configuration x is proportional to exp[−HLG(x)], yields the grand potential landscape, Ω(⃗φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ⃗µ), from which we can calculate the grand-potential differences between pairs of free-energy basins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Finally, we determine the coexistence point by tuning ⃗µ and reweighting Ω(⃗φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' ⃗µ) such that the grand-potential differences among all pairs of phases vanish to within statistical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This final step is accomplished using the algorithm described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The landscapes shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 4 in the main text are constructed by projecting the grand potential at coexistence into a two-dimensional space defined by the first two principal components of the reweighted simulation samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 13 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' FEATURES OF MULTIPHASE COEXISTENCE: EXTENDED RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Illustrative example with N = 3 species feasible region feasible region feas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' feas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Slices of the SDP solution space for the N = 3, K = 2 design problem described in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Inequality constraints (blue line) and minimum eigenvalue constraints (contours and color mesh) delineate the feasible region (dashed white line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' As an illustrative example, we visualize the SDP constraints for a target phase diagram with three components and two equimolar condensed phases: ⃗φ(1)/φ(cond) T = (1/2, 1/2, 0) and ⃗φ(2)/φ(cond) T = (1/2, 0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In the zero-osmotic- pressure approximation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S14), terms involving ϵij only appear when components i and j are both enriched in the α phase (φ(α) i > 0 and φ(α) j > 0), ϵ12φ(1) 1 φ(1) 2 + ϵ11φ(1) 1 2 + ϵ22φ(1) 2 2 = b1 = log(1 − φ(1) T ) + φ(1) T − �N i=1 φ(α) i /Li, (S30a) ϵ13φ(2) 1 φ(2) 3 + ϵ11φ(2) 1 2 + ϵ33φ(2) 3 2 = b2 = log(1 − φ(2) T ) + φ(2) T − �N i=1 φ(α) i /Li, (S30b) leading to the constraints ϵ12 = ϵ13 and ϵ22 = ϵ33 when φ(2) 3 = φ(1) 2 ̸= φ(1) 1 = φ(2) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The equality and inequality constraints for the equal-chemical-potential conditions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S13), can be written as φ(1) 1 (ϵ11 + ϵ12 + ϵ13) − c1 = µ1 = φ(2) 1 (ϵ11 + ϵ12 + ϵ13) − c2 (S31a) φ(1) 2 (ϵ12 + ϵ22 + ϵ23) − c3 = µ2 ≤ φ(2) 2 (ϵ12 + ϵ22 + ϵ23) − c4 (S31b) φ(2) 3 (ϵ13 + ϵ23 + ϵ33) − c5 = µ3 ≤ φ(1) 3 (ϵ13 + ϵ23 + ϵ33) − c6 (S31c) where c1 = c2, c3 = c5, and c4 = c6 are constants depending only on Li and {⃗φ(α)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Including the ⃗µ vector, there are 9 variables with 5 distinct equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Therefore, we have 4 independent degrees of freedom, which we choose to express in terms of ϵ11, ϵ12, ϵ22, and ϵ23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' S2, we examine cross sections of this 4-dimensional design space in the ϵ12–ϵ23 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' After satisfying the equality constraints described above, the inequality constraints restrict the solution to the domain bounded by the hyperplane ϵ12 + ϵ22 + ϵ23 ≤ (c3 − c4)/[φ(1) 2 − φ(2) 2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The feasible solution space lies between this hyperplane and the semidefinite cone [21] corresponding to the stability constraints, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' To visualize the cone, we plot contour lines (along with an equivalent color mesh) showing the smallest eigenvalue of the Hessian matrix, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Here we use the parameters Li = 10 ∀i and φ(cond) T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='99, for which the ϵ matrix and ⃗µ vector that solve the regularized SDP are ϵ11 = ϵ22 = ϵ33 = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='06, ϵ12 = ϵ13 = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='09, ϵ23 = 0 and µ1 = µ2 = µ3 = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='87, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' After performing the common tangent plane construction, the condensed-phase volume fractions at coexistence are ⃗φ(1) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='495, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='495, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='000) and ⃗φ(2) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='495, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='000, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='495), which are consistent with the target compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 14 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Sensitivity analysis of designed interaction matrices As described in the main text, we analyze the sensitivity of designed equimolar-target coexistence points to random perturbations in both the interactions and the condensed-phase volume fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Extended results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' S3, in which we systematically vary the strength of the zero-mean Gaussian noise added to the designed interaction matrices, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' S4, in which we systematically vary the scale factor used to alter the initially equimolar enriched- component compositions in the target condensed phases (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' III A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In both cases, increasing the magnitude of the perturbation, either by increasing the standard deviation of the Gaussian noise (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' S3) or by increasing the composition scale factor (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' S4), tends to reduce the probability that coexistence can be re-established among the same number of target phases, with slightly perturbed volume fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' For sufficiently large perturbations in ϵ- space or ⃗φ-space, it is not possible to re-establish coexistence for any set of initial target phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This behavior can be understood by noting that large perturbations may cross a critical manifold, at which point the topology of the phase diagram changes and it becomes impossible to re-establish coexistence among the original K target phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' For smaller perturbations, random perturbations tend to destabilize one or more target phases in all cases if K > N, as well as for some phase-diagram topologies with K ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' However, in the following section, we show that in all cases where small random perturbations tend to prevent phase coexistence among the target phases, coexisting non-equimolar condensed phases can still be designed by perturbing the interaction matrices in a non-random manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 〈p(coex)〉equimolar noise std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='5 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The probability that coexistence of target phases with N = 6 species can be re-established after zero-mean Gaussian noise is added to the interaction matrix, as a function of the condensed-phase count, K, and the noise standard deviation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 2b in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 〈p(coex)〉equimolar composition scale factor = 1 2 3 4 5 〈p(global)〉equimolar FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The probability that coexistence (black curve) and global stability (red curve) of target phases with N = 6 species can be established for condensed phases with non-equimolar compositions, assuming that the equivalent equimolar phase diagram (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=', the phase diagram with equimolar condensed phases having the same sets of enriched components) is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Results are shown as a function of the condensed-phase count, K, and the scale factor s (see SI Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' III A) used to randomize the compositions of the enriched components in the condensed phases (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 2c in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' 15 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Analysis of compositional constraints via iterative perturbation phase-diagram topology phase-diagram topology space space A D phase-diagram topology Method I B random perturbation Method II C guided perturbation Method II: Illustrative example perturbation step FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (a) Schematic of subspaces, each corresponding to a different phase-diagram topology, within the full ϵ-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In this illustration, the solution space corresponding to phase-diagram topology s has a lower dimension than the full ϵ-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (b,c) Schematics of two methods for iteratively perturbing a coexistence point while maintaining a given phase-diagram topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' The equimolar coexistence point is indicated by a red star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In Method I, we randomly perturb ϵ (empty star), and then use NLLS minimization to re-establish phase coexistence consistent with the target phase-diagram topology (green star).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In Method II, we perturb ϵ and ⃗µ in a manner that is consistent with linearized coexistence equations (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (d) Repeated application of Method II results in diffusive behavior of the root-mean-squared distance between the volume fractions and the initial coexistence point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Projecting the interaction matrices via multidimensional scaling [29] shows the path taken through ϵ-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' S3, random perturbations to an interaction matrix can destabilize one or more of the target phases in certain cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This behavior can be understood by noting that the solution space corresponding to a particular phase-diagram topology may have a lower dimension than the full ϵ-space (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' S5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Thus, a random perturbation to an equimolar coexistence point that moves off this lower-dimensional manifold must change the phase count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In this section, we show that it is still possible to find non-equimolar coexistence points on low-dimensional manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' One method for finding non-equimolar coexistence points makes direct use of the nonlinear coexistence solver introduced in SI Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Specifically, we use NLLS to minimize ⃗∆Ω(⃗µ, ϵ), except here we allow both ⃗µ and ϵ to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We also modify the least-squares objective function to force the NLLS solver to find a coexistence point involving all K condensed phases that are present in target phase-diagram topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' This brute-force approach (Method I, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' S5b) generically leads to a new coexistence point among non-equimolar condensed phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' We note that for phase-diagram topologies lying on low-dimensional manifolds in ϵ-space (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=', topology s in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' S5b), the changes in the volume fractions between the equimolar and non-equimolar coexistence points tend to be correlated across target phases and components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Alternatively, we can obtain non-equimolar phase diagrams by systematically perturbing ⃗µ (Method II, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' S5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Starting from a set of feasible equimolar target phases {⃗φ(α)}0 at a designed coexistence point (ϵ0, ⃗µ0), we can expand Ω and ⃗µ to linear order in ∆ϵ and ∆⃗φ, Ω(α) = Ω(α) 0 + � ∂F ∂⃗φ ���� (α) 0 �⊤ ∆⃗φ(α)+ � ∂F ∂⃗ϵ ���� (α) 0 �⊤ ∆⃗ϵ−⃗µ⊤ 0 ·∆⃗φ(α)−∆⃗µ⊤·⃗φ(α) 0 = Ω(α) 0 + � ∂F ∂⃗ϵ ���� (α) 0 �⊤ ∆⃗ϵ−∆⃗µ⊤·⃗φ(α) 0 , (S32) where ⃗ϵ denotes the vector containing the independent elements of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Here we have assumed that (ϵ0, ⃗µ0) is located far from a critical manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' For phase coexistence to be maintained (to linear order) for some perturbation ∆ϵ, Ω(α) = Ω(0) must hold for all condensed phases α = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Thus, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S32), we obtain a system of linear equations in the form A∆⃗ϵ = B∆⃗µ = ⃗b, where A ∈ RK×N(N+1)/2, B ∈ RK×N, ∆⃗ϵ ∈ RN(N+1)/2, and ∆⃗µ ∈ RN: � ∂F ∂⃗ϵ ���� (α) 0 − ∂F ∂⃗ϵ ���� (0) 0 �⊤ ∆⃗ϵ = � ⃗φ(α) 0 − ⃗φ(0) 0 �⊤ ∆µ ∀α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S33) The matrices A and B may be rank deficient when the volume fractions of the equimolar target phases are linearly dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' When col(A) = col(B), solutions are guaranteed for arbitrary perturbations, corresponding to cases in which rank(A) = rank(B) = K ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' If K > N, then rank(B) < rank(A) ≤ N, and random perturbations will in general fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' However, it is still possible to find solutions for some perturbation ∆⃗ϵ if col(A) ∩ col(B) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' To perturb an equimolar coexistence point via Method II, we rotate ⃗µ ∈ RN by a small angle θ in the plane specified by two orthonormal vectors ˆn1 and ˆn2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' To this end, we define the rotation Rˆn1ˆn2(θ) = I + (ˆn2ˆn⊤ 1 − ˆn1ˆn⊤ 2 ) sin θ + (ˆn1ˆn⊤ 1 + ˆn2ˆn⊤ 2 )(cos θ + 1) (S34) 16 such that the perturbed chemical-potential vector is ⃗µ = ⃗µ0 + ∆⃗µ = Rˆn1ˆn2(θ)⃗µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' (S35) For example, if we choose ˆn1 = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' , 0) and ˆn2 = (0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' , 0), then we only perturb µ1 and µ2, leaving the chemical potentials of the other components unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' In practice, we apply a sequence of rotations with uniformly distributed random angles in the range [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content='005π) for all pairs of axes, and then solve for the perturbed interaction matrix via A∆⃗ϵ = ⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Finally, since this approach is only accurate to linear order, we fine-tune the coexistence point using the nonlinear phase coexistence solver described in SI Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' Applying this method repeatedly produces a random walk in ϵ-space, in which every interaction matrix corresponds to a coexistence point with the target phase-diagram topology but, in general, non-equimolar condensed-phase volume fractions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' S5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} +page_content=' As noted above, the changes in the condensed-phase volume fractions relative to the initial equimolar coexistence point tend to be correlated across phases and components when the target phase-diagram manifold in ϵ-space is low dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFOT4oBgHgl3EQf6jQA/content/2301.12958v1.pdf'} diff --git a/WNE4T4oBgHgl3EQfMwy5/content/tmp_files/2301.04951v1.pdf.txt b/WNE4T4oBgHgl3EQfMwy5/content/tmp_files/2301.04951v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..187b73d146057154ed6a822a9c6aaa1c0e13257f --- /dev/null +++ b/WNE4T4oBgHgl3EQfMwy5/content/tmp_files/2301.04951v1.pdf.txt @@ -0,0 +1,1500 @@ +Synchronization transitions on connectome graphs with external force +Géza Ódor (1), István Papp(1), Shengfeng Deng(1) and Jeffrey Kelling (2,3) +(1) Centre for Energy Research, +Institute of Technical Physics and Materials Science, +P. O. Box 49, H-1525 Budapest, Hungary +(2) Faculty of Natural Sciences, +Chemnitz University of Technology, +09111 Chemnitz, Germany +(3) Department of Information Services and Computing, +Helmholtz-Zentrum Dresden-Rossendorf, +P.O.Box 51 01 19, 01314 Dresden, Germany +We investigate the synchronization transition of the Shinomoto-Kuramoto model on networks of +the fruit-fly and two large human connectomes. This model contains a force term, thus is capable of +describing critical behavior in the presence of external excitation. By numerical solution we deter- +mine the crackling noise durations with and without thermal noise and show extended non-universal +scaling tails characterized by 2 < τt < 2.8, in contrast with the Hopf transition of the Kuramoto +model, without the force τt = 3.1(1). Comparing the phase and frequency order parameters we +find different transition points and fluctuations peaks as in case of the Kuramoto model. Using the +local order parameter values we also determine the Hurst (phase) and β (frequency) exponents and +compare them with recent experimental results obtained by fMRI. We show that these exponents, +characterizing the auto-correlations are smaller in the excited system than in the resting state and +exhibit module dependence. +I. +INTRODUCTION +The critical brain hypothesis has been confirmed ex- +perimentally many times since the pioneering electrode +experiments in [1]. +Power law (PL) distributed neu- +ronal avalanches were shown in neuronal recordings (spik- +ing activity and local field potentials, LFPs) of neural +cultures in vitro [2–4], LFP signals in vivo [5], field +potentials and functional magnetic resonance imaging +(fMRI) blood-oxygen-level-dependent (BOLD) signals in +vivo [6, 7], voltage imaging in vivo [8], 10–100 and single- +unit or multi-unit spiking and calcium-imaging activ- +ity in vivo [9–12]. +Furthermore, source reconstructed +magneto- and electroencephalographic recordings (MEG +and EEG), characterizing the dynamics of ongoing cor- +tical activity, have also shown nonuniveral PL scaling in +neuronal long-range temporal correlations [13, 14]. Op- +tical methods, like light-sheet microscopy with GCaMP +zebrafish larvae [15] or calcium imaging recordings of dis- +sociated neuronal cultures [16] also show PL scaling. +From a theoretical point of view the hypothesis is also +very attractive as critical systems possess optimal com- +putational capabilities as well as provide efficient long +range communications, memory and sensitivity [17–28]. +Homogeneous critical systems exhibit universal scaling +behavior and many experiments claim indeed a mean- +field class behavior of the branching process [29, 30] gen- +erated by self-organized criticality [31]. However, neu- +ral systems are very non-homogeneous, thus it is natu- +ral to expect non-universal behavior, known in statisti- +cal physics within the field of quenched disordered mod- +els [32, 33]. Indeed some experiments [13, 14, 16] show +that the measured exponents are not universal, signif- +icantly different from the mean-field class ones of the +branching process. +Furthermore, external sources can move the system +away from criticality [34] or tune it to other classes like +the isotropic percolation [16, 35] or tricritical points [36]. +More complex models than the two-state branching +process, can also exhibit hybrid type of phase transi- +tions, like threshold models [37], models with inhibitory +nodes [38] or models with oscillatory units [39]. Subsys- +tems can also show different scaling behavior and may be +within different distances from criticality [40]. +For quenched disordered models it has recently been +shown [41, 42], that even for weak time dependence the +semi-critical, dynamical scaling, which occurs in an ex- +tended control parameter region of criticality, in the so +called Griffihts Phases [43] (GP), remains stable. Fur- +thermore, even when the network dimension is high, one +does not find the usual mean-field behavior, but in the +presence of modules a GP [35, 37, 44–47] or Griffihts ef- +fects [48] and a different, sometimes logarithmically slow +scaling at the critical point [32]. +The big advantage of critical universality is that more +realistic models for the brain, like the integrate and fire +models [49], can also show the same criticality as simpler +ones like in a recent work +[50], which derives Hopf bi- +furcation criticality or in a more experimental study [51] +of neural cultures agreement with isotropic percolation +avalanche size distributions is obtained. But of course +the directed percolation criticality [52, 53], which occurs +in branching processes [1] is the main example for the +universality principle [54]. Therefore, the study of sim- +pler models, for which numerical analysis can be done +are very useful for the brain science [28, 33]. +Recently threshold models and Kuramoto type of mod- +els have been analyzed on different, available connectome +arXiv:2301.04951v1 [cond-mat.dis-nn] 12 Jan 2023 + +2 +networks and GP behavior was reported [33, 42, 55–58]. +This behavior is also called as frustrated synchroniza- +tion [59–61] and has been analyzed within the framework +of a Kuramoto like models, albeit lacking quenched self- +frequencies. +From the experimental directions the different behav- +ior in modules of brains of the mouse [40], by phenomeno- +logical renormalization-group analysis of the spectrum of +electrode spikes, and humans [62], via Hurst and β expo- +nents analysis of fMRI; quasi-critical (off-critical) scaling +like behavior has been shown. Here we attempt to model +this using the Shinomoto–Kuramoto (SK) model on con- +nectomes of the fruit-fly (FF) and humans. This is an +extension of the Kuramoto model [63], which itself does +not have an external source, that can describe the resting +state critical behavior at the Hopf transition towards a +model with a periodic external driving force, thus may +be appropriate to characterize criticality with an excita- +tion [39]. +II. +MODELS AND METHODS +In this Section we introduce the synchronization +model, followed by an overview of different connectome +graphs, on which we run the numerical analysis. Finally +we discuss the method of local synchronization to dig into +the details of the spatio-temporal simulations of these +brain systems. +A. +The Shinomoto–Kuramoto (SK) model +We consider an extension of the Kuramoto model [63] +of interacting oscillators sitting at the nodes of a network, +whose phases θj(t), j = 1, 2 . . . , N evolve according to the +following set of dynamical equations +˙θj(t) = ω0 +j + K +� +k +Wjk sin[θk(t) − θj(t)] +(1) ++ F sin(θj(t)) + ϵηj(t) . +Here, ω0 +j is the so-called self-frequency of the jth oscil- +lator, which is drawn from a Gaussian distribution with +zero mean and unit variance. +The summation is per- +formed over adjacent nodes, coupled by the Wjk ma- +trix. Up to this point we have the classical Kuramoto +model [63]. In the Shinomoto extension [64], we have a +Gaussian annealed noise term ηj(t), with an amplitude +ϵ, and to describe excitation, a site dependent periodic +force term, proportional to a coupling F. +Sakaguchi [65] was the first to study the periodically +forced Kuramoto model. In numerical simulations, how- +ever, he found that the state of forced entrainment was +not always attained: macroscopic fractions of the system +self-synchronized at a different frequency from that of +the drive, indicating that this sub-population had broken +away and established its own collective rhythm. Analyt- +ically improvements were provided in [66–68] and found +a rich phase space of the SK model. +Recently, in [39] the avalanche behavior of the full +equation was investigated, albeit with site independent +self-frequencies ω0 +j = ω. The authors explored the phase +diagram, besides the forceless Hopf transition a so-called +saddle node invariant cycle (SNIC) and a hybrid type +of bifurcation was compered. In a very recent publica- +tion [47] this numerical analysis has been continued on +Erdős–Rényi (ER) and hierarchical modular networks, +motivated by brain research. Considering quenched ω0 +j -s +with bi-modal frequency distributions the authors claim +the emergence of Griffiths effects by the broadening of +the synchronization transition region. +Here we study the SK model using quenched ω0 +j -s with +and without annealed noise ηj(t) on real connectomes. +In particular we test if the chaoticity, generated by the +quenched ω0 +j -s generates the same phase transition be- +havior and avalanches as with the presence of the stochas- +tic noise. We measured the Kuramoto phase order pa- +rameter: +z(t) = r(t) exp iθ(t) = 1/N +� +j +exp [iθj(t)] , +(2) +by increasing the sampling time δt = 0.01. +Here 0 ≤ +r(t) ≤ 1 gauges the overall coherence and θ(t) is the av- +erage phase. The set of equations (2) was solved by the +steppers Runge–Kutta-4 (RK4), for the noisy, or by the +Bulrisch–Stoer [69, 70] (BS) for the noiseless cases, be- +cause in the presence of noise the adaptive BS fails to +work. Here and in earlier studies [57] we found that the +stronger stochastic noise makes the results non-reliable, +while application of other steppers slow down the nu- +merical solution. For the noisy cases we also tried the +Euler–Maruyama solver [71], which has a stronger math- +ematical foundation for stochastic differential equations. +This had to be restricted to testing purposes only, as this +first-order solver is orders of magnitude slower than the +RK4 for the same precision. +We integrated the set of equations numerically for +103−104 independent initial conditions, by different ω0 +j -s +and sample averages of the phases +R(t) = ⟨r(t)⟩ +(3) +and of the variance of the frequencies +Ω(t) = 1 +N +N +� +j=1 +(ω(t) − ω2 +j (t)) +(4) +were determined, where N denotes the number of nodes. +In the steady state, which we determined by visual in- +spection of R(t) and Ω(t), we measured their half values +and the standard deviations: σ(R(t)), σ(Ω(t)) in order +to locate the transition points. Note, that σ(R(t)) is just +a version of the SK order parameter employed by [72] +for discrete version of oscillatory models. In case of the + +3 +Kuramoto equation the fluctuations of both order pa- +rameters show a peak, albeit at different K′ +c (for phases) +and Kc (for frequency) values in the case of the KKI-18 +connectome. For graph dimensions 3 < d < 4 a crossover +transition is expected for R and phase transition for Ω. +In the case of the FF, having d > 5 we found Kc ≃ K′ +c, +which is expected for real phase transitions at large sizes, +where both order parameters converge to a finite value +in the infinite size limit [58]. +B. +Connectome graphs +The connectome is defined as the structural network +of neural connections in the brain [73]. +For the fruit- +fly connectome, we used the hemibrain data-set (v1.0.1) +from [74], which has NF F = 21 662 nodes and LF F = +3 413 160 edges, out of which the largest single connected +component contains N = 21 615 and L = 3 410 247 di- +rected and weighted edges. +The number of incoming +edges varies between 1 and 2708. The weights are inte- +ger numbers, varying between 1 and 4299. The average +node degree is ⟨k⟩ = 315.129 (for the in-degrees it is: +157.6), while the average weighted degree is ⟨w⟩ = 628. +The adjacency matrix, visualized in [58] where one can +see a rather homogeneous, almost structureless network, +however it is not random. For example, the degree dis- +tribution is much wider than that of a random ER graph +and exhibits a fat tail. The analysis in [58] found a weight +distribution p(w) with a heavy tail, assuming a PL form, +an exponent −2.9(2) could be fitted for the w > 100 +region. +The human brain has ≈ 1011 neurons, which cur- +rent imaging techniques cannot comprehensively resolve +at the scale of single neurons. We used graphs on the +coarse-grained, level with ≈ 106 nodes obtained by dif- +fusion tensor imaging [75]. This method has generally +been found to be in good agreement with ground-truth +data from histological tract tracing [76]. +Inferred net- +works of structural connections were made available by +the Open Connectome Project and previously analyzed +by [77]. These graphs are symmetric, weighted networks, +where the weights measure the number of fiber tracts be- +tween nodes. The network topology study found a certain +level of universality in the topological features of the ten +large human connectomes investigated: degree distribu- +tions, graph dimensions, clustering and small world coef- +ficients. These can be observed in Tables 3 and 4 of [77]. +Therefore, two networks, called KKI-18, and KKI-113 +were selected to be the representatives in further studies. +The graphs, downloaded in 2015 from the Open Con- +nectome project repository [78], were generated via the +MIGRAINE pipeline [79], publicly available from [80]. +KKI-18 comprises a large component with N = 804 092 +nodes connected via 41 523 908 undirected edges and sev- +eral small disconnected sub-components, which were ig- +nored in the modeling. Similarly, the extracted largest +connected component of KKI-113 contains 799 133 nodes +connected by 48 096 500 undirected and weighted edges. +The large number of nodes is because of other parcel- +lations closer to voxel resolution being used. +For in- +stance, there are approximately 1.8 million voxels in the +brain mask of a 1 mm resolution standard-aligned MRI. +The graphs exhibit a hierarchical modular structure, be- +cause they are constructed from cerebral regions of the +Desikan–Killany–Tourville parcellations, which is stan- +dard in neuroimaging [81, 82] providing (at least) two +different scales. +The modularity quotient of a network is defined by [83] +Q = +1 +N⟨k⟩ +� +ij +� +Aij − kikj +N⟨k⟩ +� +δ(gi, gj), +(5) +the maximum of this value characterizes how modular +a network is, where Aij is the adjacency matrix, ki, kj +are the node degrees of i and j and δ(gi, gj) is 1 when +nodes i and j were found to be in the same commu- +nity, or 0 otherwise. However, this value is not indepen- +dent of the community detection method. If our detec- +tion method produces lower modularity than the max- +imum achieved, it means our algorithm has fallen be- +hind others. Community detection algorithms based on +modularity optimization will get the closest to the actual +modular properties of the network. We calculated the +modularity using community structures detected by the +Louvain method [84], the results for each network were: +QF F ≈ 0.631, QKKI−18 ≈ 0.913, QKKI−113 ≈ 0.915. +The FF is a small-world network, according to the defi- +nition of the coefficient [85]: +σW = CW /Cr +L/Lr +, +(6) +because σF F = 9.5 is much larger than unity. Here CW +denotes the Watts clustering coefficient, and L the av- +erage path length. Cr and Lr are the reference values +of random networks with the same sizes and average de- +grees. The same is true for the human connectomes, as +their σW is in the range between 400 and 1000 [77]. +The effective graph (topological) dimension, obtained +by the breadth-first search algorithm is d = 5.4(5). This +is defined by N(r) ∼ rd, where the number of nodes +N(r) with chemical distance r or less from the seeds are +counted and averages are calculated over the many trials. +For the Open Connectome data, power-law fits in the +range 1 ≤ r ≤ 5 suggest topological dimensions between +d = 3 and d = 4 [77]. +As these structural connectome graphs exhibit heavy- +tailed weight distributions, probably as a result of learn- +ing, there exist hubs, which could fully determine the be- +havior of neighboring nodes and suppress the occurrence +of critical behavior in the models [55]. In reality, on top of +the structural weights, there exist inhibition/excitation +mechanisms, which control the dynamics of the neural +system and provide a local homeostasis. As we do not +know the details of these mechanisms, in earlier stud- + +4 +ies [33, 42, 55–58], the weight normalization scheme +W ′ +jk = Wjk/ +� +k +Wjk +(7) +was applied to achieve this artificially. +This way we +equalize the sensitivity of nodes to the incoming excita- +tion. We do the same in the simulations presented here. +C. +Analysis of the local synchronization +As the connectomes are very heterogeneous, built up +from modules we also measured the local Kuramoto order +parameter Ri(t), defined as the partial sum of phases for +the neighbors of node i +Ri(t) = +1 +Ni.neigh +������ +Ni.neigh +� +j +Aijeiθj(t) +������ +, +(8) +and the local Ωi(t) defined as +Ωi(t) = +1 +Ni.neigh +������ +Ni.neigh +� +j +(ω(t) − ωj(t))2 +������ +. +(9) +The local Kuramoto order parameter was initially sug- +gested by [86, 87] to quantify the local synchronization of +nodes, which allows us to follow the synchronization pro- +cess by mapping the solutions on the connectome graphs. +The necessity of storing the states of the system at each +time step requires large amount of hard drive storage. +Thus we analyzed the local order parameters in a time +period of 50 time-steps as stop time with time increment +of dt′ = 0.1 , in the steady state. To study it in more de- +tail we also separated the networks into communities. Al- +though, these communities should be separated according +to anatomical and/or functional properties [88], we chose +as a first approximation a community detection method +based on global optimization of the modularity [84]. This +method yielded 9 modules in FF network, 130 communi- +ties in KKI-113 and 134 modules in the giant component +of KKI-18. For detecting community structure that is +closer to the real anatomical functional communities just +by using the network topology, one might require other +algorithms, which analyze the network with more depth, +or even using fuzzy clustering methods [89, 90]. +We studied the long-term persistence of the local order +parameters with the Hurst and β exponents. The Hurst +exponent measures the degree of self-similarity of a time +series, based on the assumption of an Ornstein–Ulenbeck +process, that the measured values will go back to its av- +erage in just a few time-steps. The Hurst exponent is +defined as follows: +E +�Z(n) +S(n) +� += CnH, +(10) +where E is the expectation value of the rescaled range +Z/S and Z(n) is the cumulative deviate of the series until +the first number of n data points (n = (tmax − t0)/dt′), +while S(n) is the sum of the standard deviations until +that point. We averaged the first local parameter val- +ues within the communities and calculated the Hurst ex- +ponent over the n points in the time period t, where +Sj(n) = �Mj,comm +i +Ri(t) are community averages and +Mj,comm is the number of nodes in the community. We +calculated the Hurst exponents for all communities. +Similarly the power spectral scaling exponent, β, is +used for quantifying long range correlations in time se- +ries. The power spectral density is the modulus of the +Fourier transform, if the spectrum of the process satisfies +a power-law scaling relation: +S(f) = +������ +N +� +j=0 +Ωj(t)e−2πifj/N +������ +2 +≈ 1/f β, +(11) +where fj = �Mi,comm +j +Ωi(t) and β must be obtained by +using a linear fit to the logarithmic axes of the Fourier +transform periodigram [62]. +III. +FORCE DRIVEN SYNCHRONIZATION +TRANSITION +First we determined the synchronization transition be- +havior of the Shinomoto–Kuramoto model on different +connectomes by calculating the global order parameters +R and Ω as well as their fluctuations as the function of +the force control parameter, which mimics the external +excitation of the system. +After that we measured the +crackling noise distributions within the neighborhood of +these transitions +A. +Global order parameters +We started the numerical analysis of SK on the fruit-fly +connectome at the global coupling value K = 1.3, which +was found to be asynchronous without a force in [58]. +For each F value we determined the steady state by fol- +lowing the evolution of the control parameters starting +from random initial θ-s via visual inspection. Averaging +was done over many independent samples, correspond- +ing to different initial ωj self-frequencies. The transient +regimes were short, in the range of 10-100 time steps and +we could not see PL growth as in case of the Hopf tran- +sition of the Kuramoto model. But the Kuramoto order +parameter curves exhibit R(t) ∝ ln(t)x(K) type of growth +(see upper inset of Fig. 1), as in case of activated scaling +in disordered systems [32]. +To locate the transition we plotted the steady state val- +ues of R and Ω and their fluctuations on Fig. 1. The half +values provide estimates: F ′ +c ≃ 0.22 and Fc ≃ 0.35. One +can see smooth fluctuation peaks of σ(R) at F ′ ≃ 0.04 +and of σ(Ω) at F ≃ 0.2. +Thus, the two different or- +der parameters seem to exhibit different synchronization + +5 +points. The frequency fluctuation peak agrees roughly +with F ′ +c ≃ 0.22, but the phase fluctuation peak occurs +at a much lower value. This, in contrast with the Hopf +transition of FF and the random network, where fluctu- +ation peaks were roughly the same position, where we +knew that the dimension is d > 4. +As σ(R) is also called SK order parameter, which char- +acterizes the transition in excitable systems, its approach +to zero as F increases agrees with the SNIC transition re- +sult of [39], albeit that was obtained in the synchronous +phase. We have also run SK in the synchronous phase of +FF, using K = 2, and we found similar results as in the +asynchronous phase. +0 +1 +2 +3 +F +0 +0.2 +0.4 +0.6 +0.8 +1 +R, Ω +R ε=0.01 +R ε=0 +Ω ε=0.01 +0 +0.1 +0.2 +0.3 +0.4 +F +0.001 +0.006 +0.011 +0.016 +0.021 +0.026 +σ(R) +σ(Ω) +10 +0 +10 +1 +10 +2 +t +0.0 +0.2 +0.4 +0.6 +0.8 +R +FIG. 1: Order parameter dependence on F for the fruit-fly +connectome for the noisy (black bullet) and the noiseless (red +boxes) cases at K = 1.3. The blue diamonds show the steady- +state Ω values with noise. Lower inset: Variances of R and +Ω for the noisy case. Upper inset: Time dependence of the +noisy R(t), for F = 0, 0.02, 0.03, 0.04, 0.07, 0.1, 0.2, 0.3, 0.4 +(bottom to top curves). +Results with and without a small noise with ampli- +tude ϵ = 0.01 did not show observable differences, so the +chaotic noise from the quenched disorder is capable to +compete with the ordering effect of the force. +As the next step we performed the same analysis of +the human connectomes at K = 1, which is in the asyn- +chronous phase without a force [56]. Figure 2 shows the +steady state values both for R and Ω in case of K = 1 +for KKI-113. Again the annealed noise does not mod- +ify the results and seems to be unnecessary to produce a +synchronization transition. We estimated: F ′ +c ≃ 0.4 and +Fc ≃ 0.55 by the half values or R and Ω respectively. The +fluctuation peaks of the two order parameters are again +far away from each other: F ′ ≃ 0.05 versus F ≃ 0.4. +Again the fluctuation peak of Ω is close to F ′ +c ≃ 0.4. +For the connectome KKI-18 we enlarged the fluctu- +ation peak results on Figure 3. The smeared synchro- +nization ’peaks’ happen at similar values as for KKI-113: +0 +0.5 +1 +1.5 +2 +F +0 +0.2 +0.4 +0.6 +0.8 +1 +R, Ω +R, ε=0 +R, ε=0.01 +Ω ε=0.01 +0 +0.2 +0.4 +0.6 +0.8 +1 +F +0 +0.0005 +0.001 +0.0015 +0.002 +0.0025 +σ(R), σ(Ω) +FIG. 2: Order parameter dependence on F for the KKI-113 +for the noisy and the noiseless cases at K = 1. Inset: Vari- +ances of R and Ω for the noisy case. +F ′ ≃ 0.05 and F ≃ 0.5 within numerical precision. The +transition points, estimated by the half values of R is +F ′c ≃ 0.4 and of Ω is Fc ≃ 0.55. Again, the σ(R) peaks +are much lower than the other transition point estimates. +0 +0.2 +0.4 +0.6 +0.8 +1 +F +0 +0.001 +0.002 +σ(R), σ(Ω) +R ε=0 +R ε=0.01 +Ω ε=0.01 +0 +0.2 0.4 0.6 0.8 +F +0 +0.2 +0.4 +0.6 +0.8 +R, Ω +FIG. 3: Fluctuations of R and Ω as the function of F for the +KKI-18, for the noisy and the noiseless cases at K = 1. Inset: +Order parameters for the noisy and noiseless cases.. +B. +Avalanche durations +We investigated avalanches similarly to the local field +potential experiments and as it was done in simulations +of spike-like events [39]. However, we did not threshold +individual variables θ(t)i, but the global order param- + +6 +eter R(t), which is a sum of them. +This has the ad- +vantage of a much faster algorithm, allowing to consider +larger statistics and the lack of ambiguity in avalanche +definitions +[91–93]. The disadvantage is that spatially +independent avalanches overlapping in time accidentally +may be unified, thus the duration times can be larger +and we do not have information on the spatio-temporal +sizes, thus on the exponent τ. Still, we think that in- +vestigating this coarse-grained description of avalanches, +which has also been measured in experiments, as a kind +of crackling noise [94] in the case of zebrafish larvae [15], +describes a possible critical behavior. +Results of local +characterization of the synchronization will be shown in +Sects. 8, III D, IV B. +As in [39] here we also found that the choice of thresh- +old T(F) value did not change the scaling behavior of +the duration distributions if it was chosen within the +fluctuation range Rmin < T(F) < Rmax corresponding +to F, that was determined numerically after several runs +on different initial conditions. +For thresholds we used +the mean value of R(t), obtained in the steady state +by sample and time averaging up to tmax = 104. +By +the integration we used uniform random distributions +θi(0) ∈ (0, 2π) and the initial frequencies were set to be +˙θi(0) = ω0 +i . Following measurements of the avalanche du- +ration ∆(t) = ti − ti′, defined between subsequent cross- +ing of an up event: R(ti) > T and a down one: R(t′ +i) < T, +we applied a histogramming to determine the probability +distributions p(∆(t)). +Fig. 4 shows the pdf-s p(∆(t)) results for the fruit-fly +in case of K = 1.3, ϵ = 0.01 and different forces. We can +see F dependent extended PL tails, with continuously +changing exponents: 2.1 < α < 2.8, which are somewhat +smaller, but close to the experimental values for the ze- +brafish: α = 3.0(1) [15]. +10 +0 +10 +1 +10 +2 +10 +3 +∆t +10 +−9 +10 +−7 +10 +−5 +10 +−3 +10 +−1 +p(∆t) +0.03 +0.06 +0.2 +2.0 +t +−2.12(6) +t +−2.64(6) +FIG. 4: Avalanche duration distributions on the fruit-fly con- +nectome for different forces, shown by the legends and at +K = 1.3, ϵ = 0.01. Dashed lines are PL fits for ∆t > 100. +Similar results are obtained in case of the two human +connectomes as shown on Figs.4,5. Furthermore, the re- +sults do not change without the additive noise, or in case +of a force in the synchronized phase (see graphs in the +Appendix). +10 +0 +10 +1 +10 +2 +10 +3 +∆t +10 +−8 +10 +−6 +10 +−4 +10 +−2 +10 +0 +p(∆t) +0.1 +0.35 +0.4 +0.5 +∆t +−2.40(9) +∆t +−2.17 +FIG. 5: Avalanche duration distributions on the KKI-113 con- +nectome for different forces, shown by the legends and at +K = 1, ϵ = 0.01. +Dashed lines are PL fits for ∆t > 20. +10 +0 +10 +1 +10 +2 +10 +3 +∆t +10 +−8 +10 +−6 +10 +−4 +10 +−2 +10 +0 +p(∆t) +0.25 +0.4 +0.45 +∆t +−2.21(4) +∆t +−1.94(6) +∆t +−2.7(3) +FIG. 6: Avalanche duration distributions on the KKI-18 con- +nectome for different forces, shown by the legends and at +K = 1, ϵ = 0.01. +Dashed lines are PL fits for ∆t > 20. +C. +Local order parameters snapshots +We have plotted with Wolfram Mathematica [95] snap- +shots of the local order parameters of the FF at different + +7 +force values in increasing order for the average local pa- +rameters (see Fig. 7). The giant component of the graph +was plotted with 21 615 nodes, however with a very few +75 657 edges for better visualization, where we sorted the +links of each node by their weights in a decreasing manner +and then randomly chose the first nr links, where nr is a +random integer between 1 and nm = 6. Since the graph +is a modular small-world graph, this kind of represen- +tation can be a close visual representation of the actual +network. The color coding on the figure is a logarith- +mic (log3) binned scale between 0 and 1 (0.01, 0.03, 0.09, +0.27, 0.81, 1.) representing the Ri values of each node at +time step, indicated on the top left of each figure. +Top row plots are results without force, second row at +F = 0.04, third row at F = 0.1 and last row is at F = 1.0. +Similarly to the β exponent’s case, we notice that the +average local parameter R is not increasing linearly with +the force at the same time-step. There is a maximum +around 0.1, thus it does not have a linear correlation +with the force. Without force the steady state has more +fluctuations and the communities are more observable +through visualization. By increasing the force every node +comes into the same local state. +D. +Hurst and β exponent results +The H and β exponents measure the self-similarity of +a time series, when power-law behavior (10), (11) can be +observed. H and β values lower than 0.5 describe anti- +correlated signals. On the other hand, values between 0.5 +and 1 mean signals with long range correlations in time. +First, we separated the communities in all FF, KK- +18, KKI-113 connectomes with the Louvain modularity +optimizing algorithm. Then, we calculated the H and +β exponents for each community for the local parame- +ters. In case of the FF the results (see Fig. 8 with force +could similarly be differentiated from the results without +force as in the [62] experiments with rest and task driven +measures. Simulation results without force seem to have +longer correlations in time, resembling to the fMRI mea- +surements at the rest phase. +The same conclusion however cannot be found in the +case of the human connectomes (see Fig. 9). It appears +that even with a relatively high force the exponents re- +main close to each other and close to those of the “rest" +phase. In the case of FF higher force led to less “rest" in +the system resembling more like task driven behaviour. +IV. +HOPF SYNCHRONIZATION TRANSITION +WITHOUT FORCE +We have rerun this analysis for the fruit-fly connec- +tome using the standard Kuramoto equation for different +couplings, i.e. near the Hopf synchronization transition +discussed in [58]. +36.6 +12.6 +36.6 +12.6 +36.6 +12.6 +no force +F=0.04 +F=0.1 +F=1.0 +36.6 +12.6 +36.6 +12.6 +FIG. 7: Here we see the evolution of the local order param- +eters of a sub-graph of the fruit-fly connectome at different +time steps: t = 12.6, 36.6. +The upper row shows Ri map +without a force, the lowest one with F = 1.0. +A. +Crackling noise analysis +Earlier mean-field type of phase transition was found +at Kc ≃ 1.7(2). As we can see on Fig. 10 the crackling +noise duration analysis results in faster than PL decays +of p(∆t) for K < 1.4 and an inflection point with up +veering decays for K ≥ 1.65 couplings. At K = 1.5 we +can observe a PDF, with PL decay for 30 < t < 300, +which can be fitted by the exponent α = 3.03(3). +As in [58] we do not find an extended scaling region +with non-universal exponents suggesting a GP. So, the +crackling noise exponent, presumably the mean-field class +exponent of the Hopf transition, describing the resting + +R +0.81 +0.27 +0.09 +0.03 +0.01R +0.81 +0.27 +0.09 +0.03 +0.01R +0.81 +0.27 +0.09 +0.03 +0.01R +0.81 +0.27 +0.09 +0.03 +0.01R +0.81 +0.27 +0.09 +0.03 +0.01R +0.81 +0.27 +0.09 +0.03 +0.01R +0.81 +0.27 +0.09 +0.03 +0.010.81 +0.27 +0.09 +0.03 +0.018 +0.0 +0.5 +1.0 +⟨β⟩ +1 +⟩ +3 +4 +5 +6 +7 +8 +9 +communities +0.6 +0.8 +1.0 +⟨H⟩ +F⟨=⟨1.0 +no⟨force +F⟨=⟨0.04 +F⟨=⟨0.1 +FIG. 8: Hurst and beta exponents of all fruit-fly connectome +communities. In the forceless case at the critical Hopf tran- +sition coupling, the H exponent is the largest for every com- +munity. With forces these values drop for each community. +This shows a resemblance with the rest and non-rest studies +of different brain areas in [62], showing ⟨H⟩ ≈ 1.0 at resting +state and ⟨H⟩ ≈ 0.7 at task driven states. +0.8 +1.0 +⟨β⟩ +0 +20 +40 +60 +80 +100 +120 +140 +communities +0.90 +0.95 +1.00 +⟨H⟩ +KKI-113⟩⟨no⟨force +KK-18⟩⟨no⟨force +KKI-113⟩⟨F⟨=⟨0.4 +FIG. 9: Hurst and β exponents of all human connectomes’ +communities. KKI-113 is presented with and without force +terms and KK-18 without the force terms. +state, should be this value. This is a rather large expo- +nent and is difficult to reproduce by simulations, because +large systems are needed to see the scaling region before +an exponential cutoff. We assume that this was not seen +in [39], where N = 500 nodes were used. Another rea- +son might be that in [39] an annealed Kuramoto model +was simulated, lacking the quenched self-frequencies. Or +perhaps because [39] used thresholds of the θi(t) vari- +ables and identified avalanches by estimating the spatio- +temporal size of the activity avalanches. +But indeed the scaling region we observe is rathernar- +row, even though we know that the Kuramoto model +exhibits a critical synchronization transition here. +10 +0 +10 +1 +10 +2 +10 +3 +∆t +10 +−6 +10 +−5 +10 +−4 +10 +−3 +10 +−2 +10 +−1 +p(∆t) +1.3 +1.4 +1.5 +1.65 +∆t +−3.1(1) +FIG. 10: Avalanche duration distributions on the fruit-fly con- +nectome without force for K different couplings. +B. +Hurst and β analysis of local variables +We cannot exclude the possibility that doing the +avalanche analysis on the local angles θi(t), would lead +to a lack of PL-s as it was claimed in [39]. Since iden- +tification of avalanches of local variables is a rather dif- +ficult and ambiguous task, requiring careful binning, to +check the scaling of local phase and frequency data we +performed auto-correlation measurements and estimated +the Hurst and β exponents as before. +The "no-force" result on Figs. 8 and +9 show strong +auto-correlations, indications of criticality as in the brain +experiments [62]. In fact the exponents are larger (close +to 1), than in case of the Shinomoto–Kuramoto model +calculations. This suggests that the external excitation +results in a less correlated scaling behavior of the neural +systems than in the resting state. These results are in +agreement with the experimental findings of [62]. +V. +CONCLUSIONS +In conclusion our numerical analysis of synchroniza- +tion models on different connectome graphs show that in +the case of excitation we can find PL scaling of duration +of the crackling noise of the activity, defined by thresh- +olds of R. By solving the Shinomoto-Kuramoto model +numerically we concluded that even without the additive +noise we can find similar synchronization transition as +with the full Langevin equation. +The observed PL tails exhibit some dependence on the +amplitude of the force, which may be related to GP het- +erogeneity effects, but can also arise as the consequence +of quasi-critical, scaling like behavior reported in the dis- +crete models of Ref. [34]. We estimated the extension of +the synchronization transition region by the fluctuations + +9 +of R and Ω and found an extended, smeared transition +region. +This makes it difficult to define the transition +points. We attempted it in two different ways: half val- +ues and fluctuation peaks of the order parameters in the +steady state. The F ′ +c from the R half values are some- +what smaller than the Fc of Ω-s and agree with the fre- +quency fluctuation peaks. While the phase fluctuations +peaks were found to be much smaller both for the FF +and the human connectomes. This is very different from +the Kuramoto Hopf transition results [58]. +However, σ(R) describes a transition of the SK order +parameter, introduced for excitable systems. In the case +of the synchronized transition in FF the addition of force +results in a quick decay of the SK order parameter like +in the SNIC transition. We have not reached a region, +showing hybrid phase transition reported in [39], possi- +bly by the lack of strong noise. +We avoided to apply +strong noise, because that makes the numerical solution +less precise or very slow. A systematic finite-size scaling +study of this transition would be necessary to settle this +issue. +In case of initial conditions with random phase vari- +ables the R(t) curves at the transition point do not show +PL growth as in case of the Kuramoto model, but a loga- +rithmic growth, similar to strong random fixed points of +models of statistical physics. +We also investigated the local order parameters and +found frustrated synchronization with Chimera like +states, coexistence of synchronized and asynchronous do- +mains. Performing auto-correlation analysis on the lo- +cal order parameters we found strong auto-correlation in +the resting (Kuramoto) state at criticality and somewhat +weaker ones in presence of an external force. In the latter +case the H and β exponents take their maximal values, +where the fluctuations of R(t) are maximal, i.e at the +transition. +We also investigated the module dependence of H and +β by decomposing the connectomes via community de- +tection algorithms. We observed variations amongst the +communities suggesting different levels of criticality, but +the identification of communities with real brain regions +is a further task to be completed. Our simulated H and +β exponents are in agreement with recent experimental +findings [62]. +Acknowledgments +We thank Róbert Juhász for the useful comments. +Support from the Hungarian National Research, Devel- +opment and Innovation Office NKFIH (K128989) is ac- +knowledged. Most of the numerical work was done on +KIFU supercomputers of Hungary. +Appendix +Here we show avalanche duration PDF-s without noise +in case of the KKI-113 connectome on Fig. 11. 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Research, Graph, https://reference.wolfram.com/ +language/ref/Graph.html (2022). + diff --git a/WNE4T4oBgHgl3EQfMwy5/content/tmp_files/load_file.txt b/WNE4T4oBgHgl3EQfMwy5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a2c9ee9ce9240e1c9b782cb66a286ca95903a8f7 --- /dev/null +++ b/WNE4T4oBgHgl3EQfMwy5/content/tmp_files/load_file.txt @@ -0,0 +1,1192 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf,len=1191 +page_content='Synchronization transitions on connectome graphs with external force Géza Ódor (1), István Papp(1), Shengfeng Deng(1) and Jeffrey Kelling (2,3) (1) Centre for Energy Research, Institute of Technical Physics and Materials Science, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Box 49, H-1525 Budapest, Hungary (2) Faculty of Natural Sciences, Chemnitz University of Technology, 09111 Chemnitz, Germany (3) Department of Information Services and Computing, Helmholtz-Zentrum Dresden-Rossendorf, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='Box 51 01 19, 01314 Dresden, Germany We investigate the synchronization transition of the Shinomoto-Kuramoto model on networks of the fruit-fly and two large human connectomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' This model contains a force term, thus is capable of describing critical behavior in the presence of external excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' By numerical solution we deter- mine the crackling noise durations with and without thermal noise and show extended non-universal scaling tails characterized by 2 < τt < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='8, in contrast with the Hopf transition of the Kuramoto model, without the force τt = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Comparing the phase and frequency order parameters we find different transition points and fluctuations peaks as in case of the Kuramoto model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Using the local order parameter values we also determine the Hurst (phase) and β (frequency) exponents and compare them with recent experimental results obtained by fMRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We show that these exponents, characterizing the auto-correlations are smaller in the excited system than in the resting state and exhibit module dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' INTRODUCTION The critical brain hypothesis has been confirmed ex- perimentally many times since the pioneering electrode experiments in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Power law (PL) distributed neu- ronal avalanches were shown in neuronal recordings (spik- ing activity and local field potentials, LFPs) of neural cultures in vitro [2–4], LFP signals in vivo [5], field potentials and functional magnetic resonance imaging (fMRI) blood-oxygen-level-dependent (BOLD) signals in vivo [6, 7], voltage imaging in vivo [8], 10–100 and single- unit or multi-unit spiking and calcium-imaging activ- ity in vivo [9–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Furthermore, source reconstructed magneto- and electroencephalographic recordings (MEG and EEG), characterizing the dynamics of ongoing cor- tical activity, have also shown nonuniveral PL scaling in neuronal long-range temporal correlations [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Op- tical methods, like light-sheet microscopy with GCaMP zebrafish larvae [15] or calcium imaging recordings of dis- sociated neuronal cultures [16] also show PL scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' From a theoretical point of view the hypothesis is also very attractive as critical systems possess optimal com- putational capabilities as well as provide efficient long range communications, memory and sensitivity [17–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Homogeneous critical systems exhibit universal scaling behavior and many experiments claim indeed a mean- field class behavior of the branching process [29, 30] gen- erated by self-organized criticality [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' However, neu- ral systems are very non-homogeneous, thus it is natu- ral to expect non-universal behavior, known in statisti- cal physics within the field of quenched disordered mod- els [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Indeed some experiments [13, 14, 16] show that the measured exponents are not universal, signif- icantly different from the mean-field class ones of the branching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Furthermore, external sources can move the system away from criticality [34] or tune it to other classes like the isotropic percolation [16, 35] or tricritical points [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' More complex models than the two-state branching process, can also exhibit hybrid type of phase transi- tions, like threshold models [37], models with inhibitory nodes [38] or models with oscillatory units [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Subsys- tems can also show different scaling behavior and may be within different distances from criticality [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' For quenched disordered models it has recently been shown [41, 42], that even for weak time dependence the semi-critical, dynamical scaling, which occurs in an ex- tended control parameter region of criticality, in the so called Griffihts Phases [43] (GP), remains stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Fur- thermore, even when the network dimension is high, one does not find the usual mean-field behavior, but in the presence of modules a GP [35, 37, 44–47] or Griffihts ef- fects [48] and a different, sometimes logarithmically slow scaling at the critical point [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The big advantage of critical universality is that more realistic models for the brain, like the integrate and fire models [49], can also show the same criticality as simpler ones like in a recent work [50], which derives Hopf bi- furcation criticality or in a more experimental study [51] of neural cultures agreement with isotropic percolation avalanche size distributions is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' But of course the directed percolation criticality [52, 53], which occurs in branching processes [1] is the main example for the universality principle [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Therefore, the study of sim- pler models, for which numerical analysis can be done are very useful for the brain science [28, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Recently threshold models and Kuramoto type of mod- els have been analyzed on different, available connectome arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='04951v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='dis-nn] 12 Jan 2023 2 networks and GP behavior was reported [33, 42, 55–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' This behavior is also called as frustrated synchroniza- tion [59–61] and has been analyzed within the framework of a Kuramoto like models, albeit lacking quenched self- frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' From the experimental directions the different behav- ior in modules of brains of the mouse [40], by phenomeno- logical renormalization-group analysis of the spectrum of electrode spikes, and humans [62], via Hurst and β expo- nents analysis of fMRI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' quasi-critical (off-critical) scaling like behavior has been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Here we attempt to model this using the Shinomoto–Kuramoto (SK) model on con- nectomes of the fruit-fly (FF) and humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' This is an extension of the Kuramoto model [63], which itself does not have an external source, that can describe the resting state critical behavior at the Hopf transition towards a model with a periodic external driving force, thus may be appropriate to characterize criticality with an excita- tion [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' MODELS AND METHODS In this Section we introduce the synchronization model, followed by an overview of different connectome graphs, on which we run the numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Finally we discuss the method of local synchronization to dig into the details of the spatio-temporal simulations of these brain systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The Shinomoto–Kuramoto (SK) model We consider an extension of the Kuramoto model [63] of interacting oscillators sitting at the nodes of a network, whose phases θj(t), j = 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' , N evolve according to the following set of dynamical equations ˙θj(t) = ω0 j + K � k Wjk sin[θk(t) − θj(t)] (1) + F sin(θj(t)) + ϵηj(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Here, ω0 j is the so-called self-frequency of the jth oscil- lator, which is drawn from a Gaussian distribution with zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The summation is per- formed over adjacent nodes, coupled by the Wjk ma- trix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Up to this point we have the classical Kuramoto model [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' In the Shinomoto extension [64], we have a Gaussian annealed noise term ηj(t), with an amplitude ϵ, and to describe excitation, a site dependent periodic force term, proportional to a coupling F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Sakaguchi [65] was the first to study the periodically forced Kuramoto model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' In numerical simulations, how- ever, he found that the state of forced entrainment was not always attained: macroscopic fractions of the system self-synchronized at a different frequency from that of the drive, indicating that this sub-population had broken away and established its own collective rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Analyt- ically improvements were provided in [66–68] and found a rich phase space of the SK model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Recently, in [39] the avalanche behavior of the full equation was investigated, albeit with site independent self-frequencies ω0 j = ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The authors explored the phase diagram, besides the forceless Hopf transition a so-called saddle node invariant cycle (SNIC) and a hybrid type of bifurcation was compered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' In a very recent publica- tion [47] this numerical analysis has been continued on Erdős–Rényi (ER) and hierarchical modular networks, motivated by brain research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Considering quenched ω0 j -s with bi-modal frequency distributions the authors claim the emergence of Griffiths effects by the broadening of the synchronization transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Here we study the SK model using quenched ω0 j -s with and without annealed noise ηj(t) on real connectomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' In particular we test if the chaoticity, generated by the quenched ω0 j -s generates the same phase transition be- havior and avalanches as with the presence of the stochas- tic noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We measured the Kuramoto phase order pa- rameter: z(t) = r(t) exp iθ(t) = 1/N � j exp [iθj(t)] , (2) by increasing the sampling time δt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Here 0 ≤ r(t) ≤ 1 gauges the overall coherence and θ(t) is the av- erage phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The set of equations (2) was solved by the steppers Runge–Kutta-4 (RK4), for the noisy, or by the Bulrisch–Stoer [69, 70] (BS) for the noiseless cases, be- cause in the presence of noise the adaptive BS fails to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Here and in earlier studies [57] we found that the stronger stochastic noise makes the results non-reliable, while application of other steppers slow down the nu- merical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' For the noisy cases we also tried the Euler–Maruyama solver [71], which has a stronger math- ematical foundation for stochastic differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' This had to be restricted to testing purposes only, as this first-order solver is orders of magnitude slower than the RK4 for the same precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We integrated the set of equations numerically for 103−104 independent initial conditions, by different ω0 j -s and sample averages of the phases R(t) = ⟨r(t)⟩ (3) and of the variance of the frequencies Ω(t) = 1 N N � j=1 (ω(t) − ω2 j (t)) (4) were determined, where N denotes the number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' In the steady state, which we determined by visual in- spection of R(t) and Ω(t), we measured their half values and the standard deviations: σ(R(t)), σ(Ω(t)) in order to locate the transition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Note, that σ(R(t)) is just a version of the SK order parameter employed by [72] for discrete version of oscillatory models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' In case of the 3 Kuramoto equation the fluctuations of both order pa- rameters show a peak, albeit at different K′ c (for phases) and Kc (for frequency) values in the case of the KKI-18 connectome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' For graph dimensions 3 < d < 4 a crossover transition is expected for R and phase transition for Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' In the case of the FF, having d > 5 we found Kc ≃ K′ c, which is expected for real phase transitions at large sizes, where both order parameters converge to a finite value in the infinite size limit [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Connectome graphs The connectome is defined as the structural network of neural connections in the brain [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' For the fruit- fly connectome, we used the hemibrain data-set (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='1) from [74], which has NF F = 21 662 nodes and LF F = 3 413 160 edges, out of which the largest single connected component contains N = 21 615 and L = 3 410 247 di- rected and weighted edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The number of incoming edges varies between 1 and 2708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The weights are inte- ger numbers, varying between 1 and 4299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The average node degree is ⟨k⟩ = 315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='129 (for the in-degrees it is: 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6), while the average weighted degree is ⟨w⟩ = 628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The adjacency matrix, visualized in [58] where one can see a rather homogeneous, almost structureless network, however it is not random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' For example, the degree dis- tribution is much wider than that of a random ER graph and exhibits a fat tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The analysis in [58] found a weight distribution p(w) with a heavy tail, assuming a PL form, an exponent −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='9(2) could be fitted for the w > 100 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The human brain has ≈ 1011 neurons, which cur- rent imaging techniques cannot comprehensively resolve at the scale of single neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We used graphs on the coarse-grained, level with ≈ 106 nodes obtained by dif- fusion tensor imaging [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' This method has generally been found to be in good agreement with ground-truth data from histological tract tracing [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Inferred net- works of structural connections were made available by the Open Connectome Project and previously analyzed by [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' These graphs are symmetric, weighted networks, where the weights measure the number of fiber tracts be- tween nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The network topology study found a certain level of universality in the topological features of the ten large human connectomes investigated: degree distribu- tions, graph dimensions, clustering and small world coef- ficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' These can be observed in Tables 3 and 4 of [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Therefore, two networks, called KKI-18, and KKI-113 were selected to be the representatives in further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The graphs, downloaded in 2015 from the Open Con- nectome project repository [78], were generated via the MIGRAINE pipeline [79], publicly available from [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' KKI-18 comprises a large component with N = 804 092 nodes connected via 41 523 908 undirected edges and sev- eral small disconnected sub-components, which were ig- nored in the modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Similarly, the extracted largest connected component of KKI-113 contains 799 133 nodes connected by 48 096 500 undirected and weighted edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The large number of nodes is because of other parcel- lations closer to voxel resolution being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' For in- stance, there are approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='8 million voxels in the brain mask of a 1 mm resolution standard-aligned MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The graphs exhibit a hierarchical modular structure, be- cause they are constructed from cerebral regions of the Desikan–Killany–Tourville parcellations, which is stan- dard in neuroimaging [81, 82] providing (at least) two different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The modularity quotient of a network is defined by [83] Q = 1 N⟨k⟩ � ij � Aij − kikj N⟨k⟩ � δ(gi, gj), (5) the maximum of this value characterizes how modular a network is, where Aij is the adjacency matrix, ki, kj are the node degrees of i and j and δ(gi, gj) is 1 when nodes i and j were found to be in the same commu- nity, or 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' However, this value is not indepen- dent of the community detection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' If our detec- tion method produces lower modularity than the max- imum achieved, it means our algorithm has fallen be- hind others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Community detection algorithms based on modularity optimization will get the closest to the actual modular properties of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We calculated the modularity using community structures detected by the Louvain method [84], the results for each network were: QF F ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='631, QKKI−18 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='913, QKKI−113 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The FF is a small-world network, according to the defi- nition of the coefficient [85]: σW = CW /Cr L/Lr , (6) because σF F = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='5 is much larger than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Here CW denotes the Watts clustering coefficient, and L the av- erage path length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Cr and Lr are the reference values of random networks with the same sizes and average de- grees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The same is true for the human connectomes, as their σW is in the range between 400 and 1000 [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The effective graph (topological) dimension, obtained by the breadth-first search algorithm is d = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' This is defined by N(r) ∼ rd, where the number of nodes N(r) with chemical distance r or less from the seeds are counted and averages are calculated over the many trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' For the Open Connectome data, power-law fits in the range 1 ≤ r ≤ 5 suggest topological dimensions between d = 3 and d = 4 [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' As these structural connectome graphs exhibit heavy- tailed weight distributions, probably as a result of learn- ing, there exist hubs, which could fully determine the be- havior of neighboring nodes and suppress the occurrence of critical behavior in the models [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' In reality, on top of the structural weights, there exist inhibition/excitation mechanisms, which control the dynamics of the neural system and provide a local homeostasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' As we do not know the details of these mechanisms, in earlier stud- 4 ies [33, 42, 55–58], the weight normalization scheme W ′ jk = Wjk/ � k Wjk (7) was applied to achieve this artificially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' This way we equalize the sensitivity of nodes to the incoming excita- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We do the same in the simulations presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Analysis of the local synchronization As the connectomes are very heterogeneous, built up from modules we also measured the local Kuramoto order parameter Ri(t), defined as the partial sum of phases for the neighbors of node i Ri(t) = 1 Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='neigh ������ Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='neigh � j Aijeiθj(t) ������ , (8) and the local Ωi(t) defined as Ωi(t) = 1 Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='neigh ������ Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='neigh � j (ω(t) − ωj(t))2 ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' (9) The local Kuramoto order parameter was initially sug- gested by [86, 87] to quantify the local synchronization of nodes, which allows us to follow the synchronization pro- cess by mapping the solutions on the connectome graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The necessity of storing the states of the system at each time step requires large amount of hard drive storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Thus we analyzed the local order parameters in a time period of 50 time-steps as stop time with time increment of dt′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='1 , in the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' To study it in more de- tail we also separated the networks into communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Al- though, these communities should be separated according to anatomical and/or functional properties [88], we chose as a first approximation a community detection method based on global optimization of the modularity [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' This method yielded 9 modules in FF network, 130 communi- ties in KKI-113 and 134 modules in the giant component of KKI-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' For detecting community structure that is closer to the real anatomical functional communities just by using the network topology, one might require other algorithms, which analyze the network with more depth, or even using fuzzy clustering methods [89, 90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We studied the long-term persistence of the local order parameters with the Hurst and β exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The Hurst exponent measures the degree of self-similarity of a time series, based on the assumption of an Ornstein–Ulenbeck process, that the measured values will go back to its av- erage in just a few time-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The Hurst exponent is defined as follows: E �Z(n) S(n) � = CnH, (10) where E is the expectation value of the rescaled range Z/S and Z(n) is the cumulative deviate of the series until the first number of n data points (n = (tmax − t0)/dt′), while S(n) is the sum of the standard deviations until that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We averaged the first local parameter val- ues within the communities and calculated the Hurst ex- ponent over the n points in the time period t, where Sj(n) = �Mj,comm i Ri(t) are community averages and Mj,comm is the number of nodes in the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We calculated the Hurst exponents for all communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Similarly the power spectral scaling exponent, β, is used for quantifying long range correlations in time se- ries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The power spectral density is the modulus of the Fourier transform, if the spectrum of the process satisfies a power-law scaling relation: S(f) = ������ N � j=0 Ωj(t)e−2πifj/N ������ 2 ≈ 1/f β, (11) where fj = �Mi,comm j Ωi(t) and β must be obtained by using a linear fit to the logarithmic axes of the Fourier transform periodigram [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' FORCE DRIVEN SYNCHRONIZATION TRANSITION First we determined the synchronization transition be- havior of the Shinomoto–Kuramoto model on different connectomes by calculating the global order parameters R and Ω as well as their fluctuations as the function of the force control parameter, which mimics the external excitation of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' After that we measured the crackling noise distributions within the neighborhood of these transitions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Global order parameters We started the numerical analysis of SK on the fruit-fly connectome at the global coupling value K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='3, which was found to be asynchronous without a force in [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' For each F value we determined the steady state by fol- lowing the evolution of the control parameters starting from random initial θ-s via visual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Averaging was done over many independent samples, correspond- ing to different initial ωj self-frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The transient regimes were short, in the range of 10-100 time steps and we could not see PL growth as in case of the Hopf tran- sition of the Kuramoto model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' But the Kuramoto order parameter curves exhibit R(t) ∝ ln(t)x(K) type of growth (see upper inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 1), as in case of activated scaling in disordered systems [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' To locate the transition we plotted the steady state val- ues of R and Ω and their fluctuations on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The half values provide estimates: F ′ c ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='22 and Fc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' One can see smooth fluctuation peaks of σ(R) at F ′ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='04 and of σ(Ω) at F ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Thus, the two different or- der parameters seem to exhibit different synchronization 5 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The frequency fluctuation peak agrees roughly with F ′ c ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='22, but the phase fluctuation peak occurs at a much lower value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' This, in contrast with the Hopf transition of FF and the random network, where fluctu- ation peaks were roughly the same position, where we knew that the dimension is d > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' As σ(R) is also called SK order parameter, which char- acterizes the transition in excitable systems, its approach to zero as F increases agrees with the SNIC transition re- sult of [39], albeit that was obtained in the synchronous phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We have also run SK in the synchronous phase of FF, using K = 2, and we found similar results as in the asynchronous phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 0 1 2 3 F 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='8 1 R, Ω R ε=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='01 R ε=0 Ω ε=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='026 σ(R) σ(Ω) 10 0 10 1 10 2 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='8 R FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 1: Order parameter dependence on F for the fruit-fly connectome for the noisy (black bullet) and the noiseless (red boxes) cases at K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The blue diamonds show the steady- state Ω values with noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Lower inset: Variances of R and Ω for the noisy case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Upper inset: Time dependence of the noisy R(t), for F = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='07, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4 (bottom to top curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Results with and without a small noise with ampli- tude ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='01 did not show observable differences, so the chaotic noise from the quenched disorder is capable to compete with the ordering effect of the force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' As the next step we performed the same analysis of the human connectomes at K = 1, which is in the asyn- chronous phase without a force [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Figure 2 shows the steady state values both for R and Ω in case of K = 1 for KKI-113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Again the annealed noise does not mod- ify the results and seems to be unnecessary to produce a synchronization transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We estimated: F ′ c ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4 and Fc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='55 by the half values or R and Ω respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The fluctuation peaks of the two order parameters are again far away from each other: F ′ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='05 versus F ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Again the fluctuation peak of Ω is close to F ′ c ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' For the connectome KKI-18 we enlarged the fluctu- ation peak results on Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The smeared synchro- nization ’peaks’ happen at similar values as for KKI-113: 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='5 2 F 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='8 1 R, Ω R, ε=0 R, ε=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='01 Ω ε=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='8 1 F 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='0025 σ(R), σ(Ω) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 2: Order parameter dependence on F for the KKI-113 for the noisy and the noiseless cases at K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Inset: Vari- ances of R and Ω for the noisy case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' F ′ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='05 and F ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='5 within numerical precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The transition points, estimated by the half values of R is F ′c ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4 and of Ω is Fc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Again, the σ(R) peaks are much lower than the other transition point estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='8 1 F 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='002 σ(R), σ(Ω) R ε=0 R ε=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='01 Ω ε=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='8 F 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='8 R, Ω FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 3: Fluctuations of R and Ω as the function of F for the KKI-18, for the noisy and the noiseless cases at K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Inset: Order parameters for the noisy and noiseless cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='. B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Avalanche durations We investigated avalanches similarly to the local field potential experiments and as it was done in simulations of spike-like events [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' However, we did not threshold individual variables θ(t)i, but the global order param- 6 eter R(t), which is a sum of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' This has the ad- vantage of a much faster algorithm, allowing to consider larger statistics and the lack of ambiguity in avalanche definitions [91–93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The disadvantage is that spatially independent avalanches overlapping in time accidentally may be unified, thus the duration times can be larger and we do not have information on the spatio-temporal sizes, thus on the exponent τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Still, we think that in- vestigating this coarse-grained description of avalanches, which has also been measured in experiments, as a kind of crackling noise [94] in the case of zebrafish larvae [15], describes a possible critical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Results of local characterization of the synchronization will be shown in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 8, III D, IV B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' As in [39] here we also found that the choice of thresh- old T(F) value did not change the scaling behavior of the duration distributions if it was chosen within the fluctuation range Rmin < T(F) < Rmax corresponding to F, that was determined numerically after several runs on different initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' For thresholds we used the mean value of R(t), obtained in the steady state by sample and time averaging up to tmax = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' By the integration we used uniform random distributions θi(0) ∈ (0, 2π) and the initial frequencies were set to be ˙θi(0) = ω0 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Following measurements of the avalanche du- ration ∆(t) = ti − ti′, defined between subsequent cross- ing of an up event: R(ti) > T and a down one: R(t′ i) < T, we applied a histogramming to determine the probability distributions p(∆(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 4 shows the pdf-s p(∆(t)) results for the fruit-fly in case of K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='3, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='01 and different forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We can see F dependent extended PL tails, with continuously changing exponents: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='1 < α < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='8, which are somewhat smaller, but close to the experimental values for the ze- brafish: α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='0(1) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 10 0 10 1 10 2 10 3 ∆t 10 −9 10 −7 10 −5 10 −3 10 −1 p(∆t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='0 t −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='12(6) t −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='64(6) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 4: Avalanche duration distributions on the fruit-fly con- nectome for different forces, shown by the legends and at K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='3, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Dashed lines are PL fits for ∆t > 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Similar results are obtained in case of the two human connectomes as shown on Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Furthermore, the re- sults do not change without the additive noise, or in case of a force in the synchronized phase (see graphs in the Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 10 0 10 1 10 2 10 3 ∆t 10 −8 10 −6 10 −4 10 −2 10 0 p(∆t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='5 ∆t −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='40(9) ∆t −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='17 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 5: Avalanche duration distributions on the KKI-113 con- nectome for different forces, shown by the legends and at K = 1, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Dashed lines are PL fits for ∆t > 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 10 0 10 1 10 2 10 3 ∆t 10 −8 10 −6 10 −4 10 −2 10 0 p(∆t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='45 ∆t −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='21(4) ∆t −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='94(6) ∆t −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='7(3) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 6: Avalanche duration distributions on the KKI-18 con- nectome for different forces, shown by the legends and at K = 1, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Dashed lines are PL fits for ∆t > 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Local order parameters snapshots We have plotted with Wolfram Mathematica [95] snap- shots of the local order parameters of the FF at different 7 force values in increasing order for the average local pa- rameters (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The giant component of the graph was plotted with 21 615 nodes, however with a very few 75 657 edges for better visualization, where we sorted the links of each node by their weights in a decreasing manner and then randomly chose the first nr links, where nr is a random integer between 1 and nm = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Since the graph is a modular small-world graph, this kind of represen- tation can be a close visual representation of the actual network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The color coding on the figure is a logarith- mic (log3) binned scale between 0 and 1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='09, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='27, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='81, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=') representing the Ri values of each node at time step, indicated on the top left of each figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Top row plots are results without force, second row at F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='04, third row at F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='1 and last row is at F = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Similarly to the β exponent’s case, we notice that the average local parameter R is not increasing linearly with the force at the same time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' There is a maximum around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='1, thus it does not have a linear correlation with the force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Without force the steady state has more fluctuations and the communities are more observable through visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' By increasing the force every node comes into the same local state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Hurst and β exponent results The H and β exponents measure the self-similarity of a time series, when power-law behavior (10), (11) can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' H and β values lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='5 describe anti- correlated signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' On the other hand, values between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='5 and 1 mean signals with long range correlations in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' First, we separated the communities in all FF, KK- 18, KKI-113 connectomes with the Louvain modularity optimizing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Then, we calculated the H and β exponents for each community for the local parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' In case of the FF the results (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 8 with force could similarly be differentiated from the results without force as in the [62] experiments with rest and task driven measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Simulation results without force seem to have longer correlations in time, resembling to the fMRI mea- surements at the rest phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The same conclusion however cannot be found in the case of the human connectomes (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' It appears that even with a relatively high force the exponents re- main close to each other and close to those of the “rest" phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' In the case of FF higher force led to less “rest" in the system resembling more like task driven behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' HOPF SYNCHRONIZATION TRANSITION WITHOUT FORCE We have rerun this analysis for the fruit-fly connec- tome using the standard Kuramoto equation for different couplings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' near the Hopf synchronization transition discussed in [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 no force F=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='04 F=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='1 F=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 7: Here we see the evolution of the local order param- eters of a sub-graph of the fruit-fly connectome at different time steps: t = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6, 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The upper row shows Ri map without a force, the lowest one with F = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Crackling noise analysis Earlier mean-field type of phase transition was found at Kc ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='7(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' As we can see on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 10 the crackling noise duration analysis results in faster than PL decays of p(∆t) for K < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4 and an inflection point with up veering decays for K ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='65 couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' At K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='5 we can observe a PDF, with PL decay for 30 < t < 300, which can be fitted by the exponent α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='03(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' As in [58] we do not find an extended scaling region with non-universal exponents suggesting a GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' So, the crackling noise exponent, presumably the mean-field class exponent of the Hopf transition, describing the resting R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='01R 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='0 ⟨β⟩ 1 ⟩ 3 4 5 6 7 8 9 communities 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='0 ⟨H⟩ F⟨=⟨1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='0 no⟨force F⟨=⟨0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='04 F⟨=⟨0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 8: Hurst and beta exponents of all fruit-fly connectome communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' In the forceless case at the critical Hopf tran- sition coupling, the H exponent is the largest for every com- munity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' With forces these values drop for each community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' This shows a resemblance with the rest and non-rest studies of different brain areas in [62], showing ⟨H⟩ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='0 at resting state and ⟨H⟩ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='7 at task driven states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='0 ⟨β⟩ 0 20 40 60 80 100 120 140 communities 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='00 ⟨H⟩ KKI-113⟩⟨no⟨force KK-18⟩⟨no⟨force KKI-113⟩⟨F⟨=⟨0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 9: Hurst and β exponents of all human connectomes’ communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' KKI-113 is presented with and without force terms and KK-18 without the force terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' state, should be this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' This is a rather large expo- nent and is difficult to reproduce by simulations, because large systems are needed to see the scaling region before an exponential cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We assume that this was not seen in [39], where N = 500 nodes were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Another rea- son might be that in [39] an annealed Kuramoto model was simulated, lacking the quenched self-frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Or perhaps because [39] used thresholds of the θi(t) vari- ables and identified avalanches by estimating the spatio- temporal size of the activity avalanches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' But indeed the scaling region we observe is rathernar- row, even though we know that the Kuramoto model exhibits a critical synchronization transition here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 10 0 10 1 10 2 10 3 ∆t 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 p(∆t) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='65 ∆t −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='1(1) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 10: Avalanche duration distributions on the fruit-fly con- nectome without force for K different couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Hurst and β analysis of local variables We cannot exclude the possibility that doing the avalanche analysis on the local angles θi(t), would lead to a lack of PL-s as it was claimed in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Since iden- tification of avalanches of local variables is a rather dif- ficult and ambiguous task, requiring careful binning, to check the scaling of local phase and frequency data we performed auto-correlation measurements and estimated the Hurst and β exponents as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The "no-force" result on Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 8 and 9 show strong auto-correlations, indications of criticality as in the brain experiments [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' In fact the exponents are larger (close to 1), than in case of the Shinomoto–Kuramoto model calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' This suggests that the external excitation results in a less correlated scaling behavior of the neural systems than in the resting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' These results are in agreement with the experimental findings of [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' CONCLUSIONS In conclusion our numerical analysis of synchroniza- tion models on different connectome graphs show that in the case of excitation we can find PL scaling of duration of the crackling noise of the activity, defined by thresh- olds of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' By solving the Shinomoto-Kuramoto model numerically we concluded that even without the additive noise we can find similar synchronization transition as with the full Langevin equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The observed PL tails exhibit some dependence on the amplitude of the force, which may be related to GP het- erogeneity effects, but can also arise as the consequence of quasi-critical, scaling like behavior reported in the dis- crete models of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We estimated the extension of the synchronization transition region by the fluctuations 9 of R and Ω and found an extended, smeared transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' This makes it difficult to define the transition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We attempted it in two different ways: half val- ues and fluctuation peaks of the order parameters in the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The F ′ c from the R half values are some- what smaller than the Fc of Ω-s and agree with the fre- quency fluctuation peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' While the phase fluctuations peaks were found to be much smaller both for the FF and the human connectomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' This is very different from the Kuramoto Hopf transition results [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' However, σ(R) describes a transition of the SK order parameter, introduced for excitable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' In the case of the synchronized transition in FF the addition of force results in a quick decay of the SK order parameter like in the SNIC transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We have not reached a region, showing hybrid phase transition reported in [39], possi- bly by the lack of strong noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We avoided to apply strong noise, because that makes the numerical solution less precise or very slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' A systematic finite-size scaling study of this transition would be necessary to settle this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' In case of initial conditions with random phase vari- ables the R(t) curves at the transition point do not show PL growth as in case of the Kuramoto model, but a loga- rithmic growth, similar to strong random fixed points of models of statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We also investigated the local order parameters and found frustrated synchronization with Chimera like states, coexistence of synchronized and asynchronous do- mains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Performing auto-correlation analysis on the lo- cal order parameters we found strong auto-correlation in the resting (Kuramoto) state at criticality and somewhat weaker ones in presence of an external force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' In the latter case the H and β exponents take their maximal values, where the fluctuations of R(t) are maximal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='e at the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We also investigated the module dependence of H and β by decomposing the connectomes via community de- tection algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' We observed variations amongst the communities suggesting different levels of criticality, but the identification of communities with real brain regions is a further task to be completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Our simulated H and β exponents are in agreement with recent experimental findings [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Acknowledgments We thank Róbert Juhász for the useful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Support from the Hungarian National Research, Devel- opment and Innovation Office NKFIH (K128989) is ac- knowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Most of the numerical work was done on KIFU supercomputers of Hungary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Appendix Here we show avalanche duration PDF-s without noise in case of the KKI-113 connectome on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' One see only a slight variation of the PL tail exponents around −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='2, but they are close to the noisy case results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 10 0 10 1 10 2 10 3 ∆t 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 p(∆t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='42 t −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='17(3) t −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='2(1) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 11: Avalanche duration distributions on the KKI-113 connectome for different forces, shown by the legends and at K = 1, without noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Dashed lines are PL fits for ∆t > 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Similarly, in case of the FF with the application of force in the synchronized phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' K = 2 the PL tails fitted for t > 20 do not differ to much, they can be characterized by an exponent −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content='21(1) as one can see on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' The inset shows the rapid drop of the SK order parameter as the function of the force and the maximum both of σ(R), σ(Ω) are at F ≃ 0 [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Beggs and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Plenz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Neuroscience 23, 11167 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' [2] A.' 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Neuroscience 33, 7079 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Tagliazucchi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Balenzuela, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Fraiman, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' Chialvo, Frontiers in Physiology 3, 15 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE4T4oBgHgl3EQfMwy5/content/2301.04951v1.pdf'} +page_content=' [8] G.' metadata={'source': 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b/WtE2T4oBgHgl3EQfYQdR/content/tmp_files/2301.03852v1.pdf.txt @@ -0,0 +1,1277 @@ +BLE Protocol in IoT Devices and Smart Wearable +Devices: Security and Privacy Threats +Tushar Nagrare*, Parul Sindhewad, Faruk Kazi +Electronics and Telecommunication Engineering Department, VJTI, Mumbai, India +Email: *tpnagrare m20@et.vjti.ac.in +Abstract—Bluetooth Low Energy (BLE) has become the pri- +mary transmission media due to its extremely low energy +consumption, good network scope, and data transfer speed for +the Internet of Things (IoT) and smart wearable devices. With +the exponential boom of the Internet of Things (IoT) and the +Bluetooth Low Energy (BLE) connection protocol, a requirement +to discover defensive techniques to protect it with practical +security analysis. Unfortunately, IoT-BLE is at risk of spoofing +assaults where an attacker can pose as a gadget and provide its +users a harmful information. Furthermore, due to the simplified +strategy of this protocol, there were many security and privacy +vulnerabilities. Justifying this quantitative security analysis with +STRIDE Methodology change to create a framework to deal with +protection issues for the IoT-BLE sensors. Therefore, providing +probable attack scenarios for various exposures in this analysis, +and offer mitigating strategies. In light of this authors performed +STRIDE threat modeling to understand the attack surface for +smart wearable devices supporting BLE. The study evaluates +different exploitation scenarios Denial of Service (DoS), Elevation +of privilege, Information disclosure, spoofing, Tampering, and re- +pudiation on MI Band, One plus Band, Boat Storm smartwatch, +and Fire Bolt Invincible. +Index Terms—Bluetooth Low Energy, STRIDE Model, Quan- +titative Experimental Analysis, Wearable Device. +I. INTRODUCTION +All IoT devices feature at least one sensor unit, enabling +more direct integration between the natural environment and +the computer system via communication protocols such as +BLE, Wi-Fi, and ZigBee [1], [2]. BLE is an IoT commu- +nication protocol that focused on low power needs, fewer +channel hopping, and improved security over prior versions. +BLE, often known as Bluetooth Smart (BS), is the most widely +used IoT communication technology [3]. It is a Wireless +Personal Area Network (WPAN) era BLE initially included in +the BCS (Bluetooth Core Specification) in June 2010 and has +several superior features over regular bluetooth. IoT has been +increasingly used in corporate systems, healthcare systems, +army packages, beacons, novel household items, and various +packages. There will be 15 billion connected IoT devices +by 2022 [4]. Nearly all modern operating systems support +Bluetooth and BLE, including Windows 10, Linux, Android, +and Mac OS [5]. +There are three steps to the BLE pairing process as shown +in fig 2. First, both devices notify each other which pairing +technique to employ and what the BLE device may do +and anticipate in the beginning. A short-term key (STK) is +generated and processed in stage two. In order to generate the +Fig. 1. BLE Protocol Stack [25] +STK, the two devices must agree on a temporary key (TK) that +is combined with random integers. The STK is not transferred +between devices. BLE is now utilized in billions of devices, +it is critical to look at security flaws. +Wearable devices like MI band 4, Boat Storm smartwatch, +One-plus Band and Fire bolt Invincible smart watch these all +personal and industrial gadgets make our lives easier but they +also posses high threat. BLE’s vulnerabilities might be fatal be- +cause of its broad use in healthcare applications. We found that +wearable devices like MI band 4, Boat Storm smartwatch, One +plus Band and Fire bolt Invincible smart watch has insecure +pairing, inappropriate authentication, and poor protocol imple- +mentation exposes them to eavesdropping, pin cracking, and +other attacks [6]. Security hazards include revealing personal +information [7]. They write Personal security risks, identifiable +information, and critical infrastructure attacks the Internet +of Things developing technology is worrisome for instance, +accounting and financial data storage [8]. He write Concerns +around IoT trust and sensor integration include the secure IoT +nodes are vulnerable because of threats to sensors[5]. Several +studies on BLE security and privacy risks have been published +independently, some researchers look at the protocol’s secu- +rity design and perform specific attacks that take advantage +of the protocol’s inadequate implementation [9], [10], [11]. +Additionally, security researchers from academia and industry +presented multiple attacks scenarios against IoT-BLE devices + +Enables user to Interact +App +Application +with Application +Interface with Profiles +Generic Access Profile +on Physical Layer +Service framework for +Generic Access Profile +Data Communication +Host +Attribute +Security +Device Pairing +Protocol +Manager +Host +Controller +Logica Link Control and +Data Encapsulation +Interface +Adaption Protocol +Link Layer +Advertising, Scaning and +Maintaining Connections +Controller +Basic Radio Frequency +Physical Layer +TransmissionFig. 2. BLE Connections Phases [3] +by performing attacks on IoT-BLE device via attack tools +such as btlejack, adafruit, ubertooth one at various security +conferences [12], [13]. This study aims to evaluate wearable +device threat surface. +Fig. 3. BLE Analysis proposed model +This paper’s contributions can be summarised as follows: +• With the help of STRIDE threat analysis of wearable +devices, several vulnerabilities identified in multiple well- +known (most recent) versions of the BLE protocol are +utilized to categorize BLE risks. +• Present various examples of attacks on BLE wearable +devices, along with the tools used to do them and discuss +the lessons learned. +• Suggest mitigation technique for observed attack threat +surface. +II. TOOLS USED FOR ATTACKING AND ANALYSING BLE +This section details open-source security researchers that +have created hardware and software tools to analyze various +BLE vulnerabilities. Security researchers have used MITM, +passive eavesdropping, and bluetooth encryption. These tools +demonstrate security vulnerabilities on standard IoT devices +as shown the analysis based in the mind map as shown. +This list of open source, free tools help security researchers, +BLE developers, and testers build up low-cost application +components, identify and assess vulnerabilities, and safeguard +BLE-enabled IoT devices. This section enlist various tools to +perform attack on BLE wearable device. +A. Hardware tools +1) Ubertooth: A free tool for exploring Bluetooth is called +Ubertooth. Ubertooth offers gear that enables passive inspec- +tion of Bluetooth and BLE device communication. The Jan- +uary 2011-released Ubertooth One can detect and demodulate +2.4GHz Frequency band transmissions with a bandwidth of +1MHz [15]. Ubertooth sniffs the data and displays a visual of +the local traffic sorted by frequencies and transmission inten- +sity in dBm. This study approximates the overall Bluetooth +traffic in the region, including distance diagnostics searching +for items outside the purview of this project [4],[14]. The +intensity signal strength on the channel frequency is used to +structure the Ubertooth spectrum analyzer as shown in fig 12 +the green lines show the strongest signal for that identified +frequency, while the white lines show activities now. Wireless +technology transmits data over the air, making it simple to +intercept or tamper with these data packets. +2) ADA-FRUIT BLE Sniffer: Adafruit is a Bluetooth Low +Energy (BLE) package capturing device similar to Ubertooth. +Adafruit was created by Adafruit Industry[16]. It has the +ability to intercept sent data packets. Analyze two BLE devices +with this tool. Data gathered via Wireshark. +B. Software tools +1) Btlejack: Btlejack is used to spy, jam, and hijack Blue- +tooth Low Energy devices. One or more of these devices +are now supported, and one or more are running the special +firmware. Based on the BBC Micro: Bit. This utility’s latest +version (2.0) supports BLE 4.0 and 5.0. However, only 1 +Mbps uncoded PHY is supported by BLE 5.0 [14]. BtleJack +controls the connection by actively disconnecting the master +and moving it about in the connection, taking and sniffing. + +Master +Slave +EstablishedLLConnection +SecurityRequest(optional) +Pairing Request +Phasel +Pairing Responses +DetermineTempraryKey +PairingOverSMP +Phasell +LegacyParingorSecureConnections +Pairing Confirmation +STK basedencrypted connection Eshtablished +Key Distribution +Phase Il +KeyDistribution +Key DistributionSystematic LiteratureReview +QualitativeContentAnalysis +Threats +Vulnerabilities +Analysisof ThreatsandVulnerabilities +ThreatModeling/ +STRIDEFramework +Mapping2) Wireshark: Wireshark is a packet analyzer that is avail- +able for free. It is an open-source network protocol analyzer +that frequently uses software for packet analysis. To help with +data gathering, it may collect information via Ethernet, Blue- +tooth, USB ports, and other communication mediums Using +a specific filter, Wireshark’s graphical user interface (GUI) +users can see captured packets in fig. 4 of some wearable +devices as where vendors can’t used encryption techniques +data is transferred directly in plain text as seen. +Fig. 4. Sniffed Packets in Wireshark +3) Btlejuice: Cauquil created BtleJuice, which he exhibited +at the 2016 DefCon24 conference as a MITM attack against +BLE devices. Intercept score, intercept proxy, specialized +web interface, python, and Node JS binding are the four +components of Btlejuice. The key components are interception +cores and proxies. These two components function indepen- +dently and communicate via the Web Socket protocol btlejuice +functions as a proxy between BLE accessories and mobile +applications. +III. SECURITY PREREQUISITES FROM STRIDE THREAT +ANALYSIS +Looking at the DFD of wearable devices, we can derive se- +curity requirements and evaluate the attack scenarios acquired +from the analysis of wearable devices like MI band 4, Boat +Storm smartwatch, One plus Band, and Firebolt Invincible +smartwatch since, Determining security needs, we consider +the attack scenarios derived from the DFD diagram since the +security requirement is meant to stop attackers from attaining +their objectives, below are some security requirements. +1) Adding timestamp in the packet: Incorporating a times- +tamp into the transmission for all data exchange and connec- +tion processes, a timestamp should be included in the packet +to avoid replay attacks. +2) Authentication: Even if an attacker edits a message, the +message authentication mechanism can identify the change +and prevent it from being inserted. Also, to avoid connecting +to an attacker’s device, stealing accounts, and circumventing +the login procedure, the user authentication process should be +employed. +3) Encryption: An attacker’s ability to read and change +the original communication is hampered by encryption. As a +result, message alteration and network packet sniffing should +be prevented by the packet encryption procedure. Additionally, +before saving any sensitive data in the database, it must be +encrypted. The attacker won’t be able to access the data with +even physical access to the storage. +4) Traffic analysis and intrusion detection: Attack detection +and traffic analysis It’s tough to stop excessive data transmis- +sion completely. As a result, we must mitigate it on the server +side using traffic analysis and penetration testing. +5) Secure account management: Account management that +is safe to avoid account theft, all users must maintain their +accounts securely. The service provider should inform the user +about the possibility of account theft and safeguard the data +store. +The Microsoft threat modeling tool provides a threat mod- +eling report based on our model as data flow diagram shown +in fig 5. It generates 111 threats on its own. Based on prior +research on smart band security,and identified 44 significant +threats that may achieve the attacker’s aim. also, given the +practical analysis below in attack analysis section A and B +with all attack scenario on IoT-BLE wearable devices. +IV. ATTACK ANALYSIS OF WEARABLE DEVICE +The security architecture of BLE is distinct from that of +classic bluetooth. Low power consumption, computationally +restricted sensors, and connection with IoT devices are all +supported by BLE. With BLE, we have to choose between +performance, security, privacy concerns, and low power usage. +Different ways for secure connections are included in the BLE +standard. Device binding, link-layer encryption, and device +safe-list Unfortunately, many IoT-BLE devices do not imple- +ment these security methods properly. As a result, numerous +security threats arise because many of these risks are caused +by vulnerabilities in a common underlying architecture or +protocol, the same mitigating procedure applies. Some attacks +are mutually beneficial. This section focuses on how to arrange +and perform assaults using the threat model sequence. +The user information is sent between the smartphone and the +smart band as demonstrated in the DFD in fig. 5 smartphone. +also shown each threats to the specific BLE wearable devices +(MI Band 4, Boat Storm Smart watch, Oneplus Band and +Fire bolt invincible) as shown in the table identified threat for +analysed wearable devices which derived from the STRIDE +threat analysis. Examining the system with an emphasis on +user data and the smart band service should be done to +overcome the security issues in BLE devices. +A. Passive Attack +The first step in any attack scenarios is passive listening. As- +sailants secretly monitor every communication between linked +devices, resulting in a wide range of destructive attacks. By +interfering with data transmission somehow, we can eavesdrop +on and intercept every conveyed data. Passive eavesdropping +attacks are particularly vulnerable since data is delivered +wirelessly, and an attacker only needs an interceptor such as +Micro bit V2 to intercept wireless communications of devices + +Wireshark.Packet2250.eth0and/tmp/onepipe +X +oooo ooo. = Reserved: +0X00 +Header length: 24 +DLT:147 +Reserved: +36750c0000620900cf2e185d16091400 +DLT: 147, Payload: btle (Bluetooth Low Energy +LinkLayer +BluetoothLowEnergyLinkLayer +Accesing Packets via Link Layer +Access Address: +ox8e89bed6 +PacketHeader: +0x2304 (PDU Type: SCAN_RSP,TxAdd: Public) +Advertising Address: eb:ac:08:56:79:28 (eb:ac:08:56:79:28) +scan +Response +Data:10094d6920536d6172742042616e6420340302e0fe0716e0fe82010000 +CRC: +0xf8d1b9 +0000 +000018 +0093000000 +36750c +00 +00 +62 +09 +00 +6u. +b +0010 +cf 2e 18 5d16 09 14 00 +d6 be 89 8e +04 23 28 +79 +井（ +0020 + 6 0 80 +V +MiSmart B +0030 +61 6e 64 20 34 +03 +02 e0 +fe 07 16 e0 fe 82 01 00 +and 4 +0040 +o01f8b9d +Adressing Packets + Here seen details in plain text makers should +use Encryption techniquesFig. 5. Data Flow Diagram + +PPIDIRECTCOMMUNICATION +1MasterDevice +Slave Device +-PP +Measured Data +Secure +Connection +L.PPI +Smart Phone +PPI +Service +PPr +PPi +Response +PPI +Exchange User +Smart Band +Alert +PPI +Information +PPI +一 +PPI +Request +Alert +PPl +Information +Authorized User +五 +PM +PPI +Authentication +ServiceRequestk +P +PPI +AppDataNotification +EPPI +(Request&Authentication) +Mi Band4 +BoatStorm +OneplusBand +Fire-BoltInvincible +Shows all the Data Communication Between Master and Slave Device +Shows Specific Data Communicationof BLEbased Specific Wearable DevicesTABLE I +IDENTIFIED THREATS FOR ANALYSED WEARABLE DEVICES +Denial of Services +Elevation of Privilege +Information Disclosure +Repudiation +Spoofing +Tampering +Mi Band 4 +Yes +No +Yes +No +Yes +No +Boat Storm Smart Watch +No +No +Yes +No +No +No +One Plus Band +Yes +No +No +No +Yes +No +Fire-Bolt Invincible +Yes +Yes +Yes +No +Yes +Yes +Fig. 6. Security Analysis Mind-map +Fig. 7. Attackers Testbed +(MI Band 4, One plus Band and Fire bolt invincible) this IoT- +BLE wearables are particularly vulnerable to this attack due +to its predictable and obvious channel switching. +Fig. 8. Passive Attack +Most BLE systems like MI Band 4 and fire bolt invincible +smart watch and Boat storm are vulnerable to this sort of attack +due to the insufficient protection of the BLE standard and +poor encryption algorithms, as well as several critical exchange +mechanisms attackers to decode data easily. + +BruteForce +Denial of +BD ADDR +Service +Attack +Ubertooth +Attack +one +Adafruit +Bluetooth +Spoofing +MITM +MI Band +Nrf +Blue +BTLEJack +Connect +Bluejacking +Printing +Attack +Sniffing +Bluejacking +Bluetooth +Attack +MITM +BTLEJack +Bluetooth +Wearable +BTLEJack +Boat Storm +OnePlus +Bluejacking +MITM +Devices Security +Ubertooth +Attack +Smartwatch +Band +One +BruteForce +Analysis +BD ADDR +Denial of +Adafruit +Attack +Service +Attack +Sniffing +Adafruit +Denial of +Bluetooth +Service +Blue +NrF +MITM +Attack +Printing +connect +Fire Bolt +BTLEJack +Invincible +Bluejacking +Spoofing +Attack +Ubertooth +Denial of +Adafruit +One +Service +BruteForce +Attack +BD_ADDR +Attack0x6e4d5787 +BtleJack version 2.0 +[i] Detected sniffers: +>Sniffer #0:fwversion 2.0 +withconnection0x6e4d5787 +Attackers +ox1fffffc01f +Window +hijacki +itisallyourslo/ +a678cf50df610a +Ubertooth +67a11cd7 +One +Adafruit +BLE Sniffer +Micro bit +V2 +MI band +One plus +Fire bolt +band +InvincibleDevices are being paired +Device A +Device B +Original Connection +(Master) +(Slave) +Attacker +sniffs Data +AttackerFig. 9. Sniffing packets using Micro bit V2 +B. Active Attacks +Listening actively, the attacker disrupts communication and +takes information in these assaults. MITM and Replay are +two active eavesdropping attack versions to compromise data +integrity, attackers actively participate in the communication +process in MITM [11]. In contrast to MITM, a replay attack +does not compromise the sender or recipient instead, the +attacker captures the packet and re-transmits it. +Fig. 10. Active Attacks +1) Brute-Force BD-ADDR Attack: As previously stated, the +bluetooth protocol employs Frequency Hopping Spread Spec- +trum (FHSS) to avoid interference by hopping between unique +channels of the 2.4GHz ISM radio band [5]. The pseudo- +random records utilised to clock this hopping behaviour are +obtained from the ”main device” for BD-ADDR to sniff data +from a device, you must account for hops, which necessitates +knowing the BD-ADDR format. Bluetooth frames do not carry +the entire BD-ADDR, but do include the Lower Address +Part(LAP), which is a reduction of 24 bits (three bytes). The +BD-ADDR is really a combination of the LAP and 1 byte that +makes up the upper address part component (UAP). Since the +LAPs are broadcast in every frame, the Ubertooth One can +capture them passively to calculate the UAP and LAP parts. +After a while, the discovery of MAC ID’s acquired from the +UAP and LAP sniffing of devices by using Ubertooth-Rx in +scan mode (using the -z flag) for a few minutes and letting +the gadgets communicate closely with each other. +Fig. 11. Sniffing LAP’s and UAP’s using ubertooth to capture MAC Id’s +Ubertooth allows to find gadgets in both concealed and +non-discoverable modes. Ubertooth isn’t a full-fledged BTLE +device here; it’s just a sniffer that gathers LAP and UAP to +form addresses and sends an inquiry to the appropriate BTLE +device. +2) Bluetooth Man-In-The-Middle attack: There has been +researching on MITM attack tactics in IoT systems for both +traditional Bluetooth and BLE, and MITM is a common type +of wireless communication threat [6]. By placing oneself in the +center of the BLE peripheral devices, an attacker may conduct +a MITM attack upon them. The attacker intercepts and alters +a packet supplied by several devices before being sent to the +other. Neither device knows its data is blocked or modified by +a suspicious device. +BLE device makers must rigorously adhere to BLE protocol +binding and encryption requirements to prevent MITM. It +is also best to bypass using the matching approach without +updating devices. Secure connections are also recommended +for developers since they offer far more robust cryptographic +protection than older connections. If the central device (Mobile +Phone) is aware that the matching device (MI Band 4) has I/O +capabilities, the MITM flag should be supplied during pairing. +Fig. 12. Man in the Middle Attack + +Original Connection +Device A +Device B +X +(Master) +(Slave) +Diconnected Original +Connection +New Connection +Attacker +Manipulation of Data+ +tushar@tushar: +systime=1643292844 +ch=76 +LAP=a6731f +cLk100n=107649464 +ctk1=17224 +-69 +55 +-14 +SySt +ime=1643292845 +ch=71 +LAP=a6731f +Lkioo +73978 +c1k1=19004 +55 +snr=55 +ch=26 +LAP=a6731f +120973200 +snres +ch=45 +LAP=a6731f +- +czk1 +19880 +15 +292845 +che +ch=17 +LAP=a6731f +74344 +1643292846 +ch=19 +LAP=a6731f +29835867 +207 +15 +ch=38 +AP +ch=38 +LAP=a6731f +80369 +300 +16 +ch=1o +LAP: +ch=29 +ch +LAP=a6731f +snr +=5 +ch=20 +LAP +SS= +ch=20 +LAP=a6731f +608 +snr=5 +292847 +ch=37 +LAP=a6731f +59562 +AP +0ec490 +ch=11 +LAP=0eC490 +snr +-16 +2848 +ch=32 +LAP=a6731 +55 +ch=49 +LAP: +a67311 +13292849 +ch=59 +LAP=a6731f +46850 +snr +17 +50 +ch=14 +LAP +311 +=55 +ch=33 +7891 +13 +43292850 +ch=33 +LAP=a6731f +56869 +55 +2850 +ch=50 +AP: +15 +ch=50 +LAP=a6731f +15689 +55 +-9 +ch=43 +LAP: +sn +14 +chE77 +AP +1643 +292851 +ch=17 +LAP: +a6731f +snr=-16 +2852 +ch=25 +LAP +1906 +snr=55 +5ystime=1643292853 +ch=37 +LAP=a6731f +211032307 +?55 +systime=1643292853 +ch=71 +LAP=a6731f +- +clk100nS=213271415 +cLk1=34123 +S=0 +55 +snr=55 +chE3e +c1k100nS=219931148 +ck1=35189 +n=-55 +snr=-16 +Lap and Uap Sniffing gives the result of surrounding +?:FA:A6:73:1F +00-00-fa-a6-73-1f +FH +map: +0x5dd73aef67a906088834 +MAC address of BLE based Devices(tusharkali)-[~] +-s btlejack -f 0x6e4d5787 -t +BtleJack version 2.0 +Allinterfacesshown +[i] Detected sniffers: +> Sniffer #o: fw version 2.0 +[i] Synchronizing with connection 0x6e4d5787 +v CRCInit = 0xe2e0de +V Channel Map = ox1fffffc01f + Hop interval e- 12 +v Hop increment =e80 +Svnchronizedhijacking in progress +1 +Connection successfuily hijacked, it is all yours o/ +Taking over the ongoing connection +btlejack> +discover +Discovering and Sniffing ongoing connection +Spoofing and Sending unexpected data to the +Command +not +found +wearable device which accepted by slave device +1b +c4 +e4 +a6 +500+ +61 +e +De +e2 +93 +8b +50 +09 +e6 +of +8a +02 +30 +5d +97 +fd d5 b7 +e3 +15 +e +7459b +66 +db +d8 29 +e3 +cc +95 +38. 8e +3e5+ +40 +7a 04 +ed 10 6e 12 2a ad bd f9 a6 8b d3 53 15 66 d6 90 33 83 +50 +1a 1b bc 66 fd da d0 46 e3 94 5a ac ef 87 3e 3d b6 5c 4c 23 34 cb 8f ae 0b 67 a1 1c d7tushar@tushar:~$ sudo btlejack -f ox5628bocb +BtleJack version 2.o +[i] Detected sniffers: +> Sniffer #o: fw version 2.o +ox5628b0cb.. +[i] Synchronizing with connection +Accesing address of Wearable +VCRCInit=0x9252c5 +Device +V channel Map = oxiffffffeoo +for Detecting Errorness packets +V Hop interval = 36 +of wearable device +V Hop increment = 6 +[i] Synchronized. +packet capture in progress +Data +06 +ob +07 +00 +04 +00 +08 +01 +00 +ff ff o0 2a +Data: +02 +10 +oc +00 +04 +00 +60 +oa +16 +006165663066653639 +Data +oa +08 +04 +00 +04 +00 +1b +3e +00 +oe +Data: +02 +07 +03 +00 +04 +00 +oa +38 +00 +Data: +02 +07 +03 +00 +04 +00 +oa +38 +00 +Captured +Data: +02 +07 +03 +00 +04 +00 +oa +38 +00 +Data +Data: +02 +07 +03 +00 +04 +00 +oa +3800 +Packets +Data: +02 +07 +03 +00 +04 +00 +oa +38 +00 +Data: +02 +07 +03 +00 +04 +00 +oa +38 +00 +Data: +02 +07 +03 +00 +04 +00 +oa +38 +00 +Data: +02 +07 +03 +00 +04 +00 +oa +38 +00 +Data: +02 +07 +03 +00 +04 +00 +oa +38 +00 +Data: +02 +07 +03 +00 +04 +00 +oa +38 +00 +LL +Data: +oa +12 +0e 00 04 00 0b 0c +51 1400 00 2a 1a000060 0100003) Blue Printing: Blueprinting gathers precise information +on the device’s model, manufacturer, unique identifier (IMEI), +and software version, with a focus on user privacy issues. This +exploit affects both traditional bluetooth and BLE. Although +the assault may not do significant damage, it is used to organise +future attacks on the victim’s device. Blueprinting is not a +serious assault, but it does expose personal information. Ac- +cording to the BLE standard, IoT-BLE devices must publicly +broadcast their GATT services so that an attacker can obtain +this data [17]. Furthermore, attackers can get data on the +quantity of devices deployed by a specific manufacturer. If +a gadget has a well-known security flaw, this attack can be +quite damaging. To carry out this assault, there are several +open-source tools accessible. These sources might gather data +about the bluetooth stack. Another powerful application for +executing this attack quickly is nRF Connect for smartphones, +which can be write services directly to the IoT-BLE devices. +Fig. 13. nRF Connect application +4) Device Fingerprinting: Device fingerprinting is used +to identify a specific device by utilizing data exclusive to +that device, such as the MAC address, UUID, advertising +packets, and GATT services. Device fingerprinting infringes +user privacy, as seen in fig. 12 above. The fixed MAC +addresses of many BLE devices can be used to identify +them. By analyzing IoT mobile applications and identifying +static UUIDs from ads, several BLE based IoT and Wearable +Devices are susceptible to fingerprinting [17]. +• UUIDs have a hierarchical structure and need to analyze +the set of values to determine the UUID hierarchy for +fingerprinting. Simultaneously, it pinpoints application- +level flaws such as inappropriate encryption usage. +• Sniffed advertising UUIDs leads to the fingerprinting +of IoT devices [13]. Therefore, a single UUID may be +utilized by several apps. Connecting to the gadget is +indeed required. +• The value set predictable outcome implementation level +weaknesses, which leads to the identification of devices +that seem to be susceptible to sniffer or unauthorised +access. +We need to fix an app-level vulnerability to stop this attack. +Developers must provide cryptographically secure features and +encrypt credentials. To disguise the UUIDs, implementing use +of encryption techniques since the broadcast signal must have +a channel level measurement [13]. As a result, the attacker can +only receive the annoying signal. +5) Blue Stumbling: Finding devices with known security +issues is a practice known as blue stumbling. It is a set- +up for later, more significant attacks rather than active ones. +Using tiny sniffers like the Microbit V2, an attacker sniffs for +susceptible devices in a crowded area to remain undetected. +The attacker’s target devices with security weaknesses. +When a BLE connection is not required, the device must be +kept invisible or in un-discoverable mode to prevent attackers +from discovering it. Unauthenticated devices must provide just +the most basic information. +6) DOS attack: Bluetooth-enabled peers can request and +receive echoes using the L2CAP protocol in a DoS attack. +DoS attack is possible with L2CAPping. This L2CAPping +allows seeing the established connection and the round trip +time with other Bluetooth-enabled devices. Attacks against +smartphones can maintain a minimum of about 10 meters. +Powerful transmissions for laptops can reach up to 100 meters. +Using standard tools like l2ping, which comes with the Linux +Bluex utils package. Several instructions in the l2ping program +allow hackers to specify the packet length. Hackers infect +Bluetooth-enabled gadgets with malicious programs, rendering +them useless to their users. The assault can potentially disrupt +the victim’s device’s regular operation and possibly damage +its functionality. The -s number option in the standard BlueZ +utility distribution’s l2ping allows users to select the packet +length for l2ping. Many devices use a packet size of 600 bytes. +Creates a packet of the requested size and transmits it to the +provided MAC address. As a result, the end device’s reaction +time grows longer and longer, and the attacked device’s +Bluetooth capability stops working. +Fig. 14. DOS attack using Adafruit +V. SUMMARY OF BLE ATTACKS +IoT devices and wearable gadgets that use the BLE protocol +have a variety of vulnerabilities. However, lousy protocol +design is to blame for many issues. Some BLE security policy +devices are subject to cryptographic attack pairing techniques +and BLE privacy vulnerabilities. The National Institute of +Standards and Technology (NIST) and other security systems +researchers have provided specific devices with considerably + +三 +Devices +DISCONNECT +Devices +DISCONNECT +MISMARTBAND4 +BONDED +ADVERTISER +MI SMART BAND 4 +X +EB:AC:08:56:79:28 +BONDED +ADVERTISER +EB:AC:08:56:79:28 +CONNECTED +CLIENT +SERVER +CONNECTED +CLIENT +SERVER +NOT BONDED +NOT BONDED +Unknown Characteristic +Unknown Service +UUiD:0000aaa2-0000-1000-8000-aabbccddeeff +UUID: 00001530-0000-3512-2118-0009af100700 +PRIMARY SERVICE +Properties:INDICATE,WRITE,WRITENORESPONSE +Descriptors: +Alert Notification Service +ClientCharacteristicConfiguration +UUID: 0x1811 +UUID:0x2902 +PRIMARY SERVICE +Value:Notifications and indicationsdisabled +Alert Services +New Alert +UserData +Writable via +UUID: 0x2A46 +UUID:0x181C +Nrf Connect +Properties: READ, WRITE +PRIMARY SERVICE +application +Descriptors: +Device +Characteristic User Description ++ +First Name +Developer +UUID: 0x2901 +UUID:0x2A8A +provide all +Properties:READ,WRITE +Alert Notification Control Point +77 +services +Value:Natenczas +UUID: 0x2A44 +should be in +Properties: NOTIFY, READ, WRITE +Last Name +read mode. +Descriptors: +UUID:0x2A90 +Client Characteristic Configuration +Properties:READ,WRITE +UUID: 0x2902 +Value: Wojski +Gender +Immediate Alert +UUID: 0x1802 +UUID:0x2A8C +PRIMARY SERVICE +Properties:READ,WRITE +Value: Male +Heart Rate +UUID: 0x180D +PRIMARY SERVICE +Unknown Service +UUID: OxFEEOTarget addr > +A1:B2:C3:D4:22:3Z +MAC address of Target Device +Packages size +V +600 +Targeting Packet Size +Threads count +300 +V +Threads count under Packetsolder versions of (BLE v4.0 and v4.1). As per the authors +analysis according to shown in fig. 15 mind map threats to +IoT-BLE across the three OSI levels (e.g., physical layer, +data link layer, and application layer). Physical layer attacks +accompany assaults in the radio spectrum. The bluetooth data +connection layer is where the majority of assaults happen. +Data transfer via data transfer layer, The attacker captures +the link-layer packet and can send the malicious packet via +link layer hence link-layer security is a significant problem +for BLE developers. Some vulnerabilities are characterized +as incorrectly implemented application-layer vulnerabilities by +the device maker or developer. +VI. FUTURE SCOPE +BLE-enabled smart wearables and IoT gadgets have become +a part of our everyday life. Data IoT device management and +access control will need to be highly secure. Also, connect to +IoT devices via the server rather than users. The gadget has +enhanced device management and user privacy dramatically. +This server is designed to prevent IoT devices from gaining +access to personal data. However, if the network/server is +compromised, all devices linked to it are in danger. So there +is a chance that researchers may start looking into employing +distributed type Blockchain technology to safeguard linked +IoT devices shortly. Industries can benefit from a BLE mesh +network. +VII. CONCLUSION +This paper investigates the effectiveness of IoT-BLE de- +vices. Creating a threat model using the STRIDE threat +modelling tool by analyzing and experimenting with IoT- +BLE wearable device’s attack surface and finding threats is +easy to evaluate. Sometimes, its severe only vulnerabilities +and different exploitation scenarios like denial of service +(DoS), elevation of privilege, information disclosure, spoofing, +tampering, and repudiation during the pairing and writable +services phase permanently secure this phase with appro- +priate pairing methods. Moreover, authentication techniques +make services UUIDs unreachable for other sources except +applications. This paper also gives a quantitative analysis to +provide a framework to address security concerns for IoT-BLE +wearable devices. BLE is a secure wireless communication +protocol, but only if implemented correctly. This investigative +analysis with practical proof thoughtfully and practically ana- +lyzes the IoT-BLE security procedures and identifying security +vulnerabilities during device pairing-related concerns, consid- +ering the security measures against the threats and assaults +determined after developing specific security measures from +threat modeling and the analysis given in this paper. Applying +security controls to the system may introduce additional issues, +including key management. As a result, it is crucial to repeat +the methods described in this work for the security system to +assess the security after rebuilding a system. +ACKNOWLEDGMENT +Authors would acknowledge the support of the Centre +of Excellence (CoE) in Complex and Nonlinear Dynamical +Systems(CNDS), VJTI, and the members for providing us +immense support and a platform for my project. +REFERENCES +[1] W. Albazrqaoe, J. Huang and G. 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Analyzing the Security of +Bluetooth Low Energy. 1-5. 10.23919/ELINFOCOM.2019.8706457. +[22] M. +E. +Garbelini, +S. +Chattopadhyay, +and +C. +Wang, +“Sweyn- +Tooth:Unleashing mayhem over Bluetooth low energy,” in LL Encryp- +tionProcedure, Channels, vol. 37, USENIX Assoc., 2020, pp. 39–40. + diff --git a/WtE2T4oBgHgl3EQfYQdR/content/tmp_files/load_file.txt b/WtE2T4oBgHgl3EQfYQdR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9ca9c2d77694f79596c397eac4dce13474a9c47 --- /dev/null +++ b/WtE2T4oBgHgl3EQfYQdR/content/tmp_files/load_file.txt @@ -0,0 +1,928 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf,len=927 +page_content='BLE Protocol in IoT Devices and Smart Wearable Devices: Security and Privacy Threats Tushar Nagrare*, Parul Sindhewad, Faruk Kazi Electronics and Telecommunication Engineering Department, VJTI, Mumbai, India Email: *tpnagrare m20@et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='vjti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='in Abstract—Bluetooth Low Energy (BLE) has become the pri- mary transmission media due to its extremely low energy consumption, good network scope, and data transfer speed for the Internet of Things (IoT) and smart wearable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' With the exponential boom of the Internet of Things (IoT) and the Bluetooth Low Energy (BLE) connection protocol, a requirement to discover defensive techniques to protect it with practical security analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Unfortunately, IoT-BLE is at risk of spoofing assaults where an attacker can pose as a gadget and provide its users a harmful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Furthermore, due to the simplified strategy of this protocol, there were many security and privacy vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Justifying this quantitative security analysis with STRIDE Methodology change to create a framework to deal with protection issues for the IoT-BLE sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Therefore, providing probable attack scenarios for various exposures in this analysis, and offer mitigating strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' In light of this authors performed STRIDE threat modeling to understand the attack surface for smart wearable devices supporting BLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The study evaluates different exploitation scenarios Denial of Service (DoS), Elevation of privilege, Information disclosure, spoofing, Tampering, and re- pudiation on MI Band, One plus Band, Boat Storm smartwatch, and Fire Bolt Invincible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Index Terms—Bluetooth Low Energy, STRIDE Model, Quan- titative Experimental Analysis, Wearable Device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' INTRODUCTION All IoT devices feature at least one sensor unit, enabling more direct integration between the natural environment and the computer system via communication protocols such as BLE, Wi-Fi, and ZigBee [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' BLE is an IoT commu- nication protocol that focused on low power needs, fewer channel hopping, and improved security over prior versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' BLE, often known as Bluetooth Smart (BS), is the most widely used IoT communication technology [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' It is a Wireless Personal Area Network (WPAN) era BLE initially included in the BCS (Bluetooth Core Specification) in June 2010 and has several superior features over regular bluetooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' IoT has been increasingly used in corporate systems, healthcare systems, army packages, beacons, novel household items, and various packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' There will be 15 billion connected IoT devices by 2022 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Nearly all modern operating systems support Bluetooth and BLE, including Windows 10, Linux, Android, and Mac OS [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' There are three steps to the BLE pairing process as shown in fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' First, both devices notify each other which pairing technique to employ and what the BLE device may do and anticipate in the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' A short-term key (STK) is generated and processed in stage two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' In order to generate the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' BLE Protocol Stack [25] STK, the two devices must agree on a temporary key (TK) that is combined with random integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The STK is not transferred between devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' BLE is now utilized in billions of devices, it is critical to look at security flaws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Wearable devices like MI band 4, Boat Storm smartwatch, One-plus Band and Fire bolt Invincible smart watch these all personal and industrial gadgets make our lives easier but they also posses high threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' BLE’s vulnerabilities might be fatal be- cause of its broad use in healthcare applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' We found that wearable devices like MI band 4, Boat Storm smartwatch, One plus Band and Fire bolt Invincible smart watch has insecure pairing, inappropriate authentication, and poor protocol imple- mentation exposes them to eavesdropping, pin cracking, and other attacks [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Security hazards include revealing personal information [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' They write Personal security risks, identifiable information, and critical infrastructure attacks the Internet of Things developing technology is worrisome for instance, accounting and financial data storage [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' He write Concerns around IoT trust and sensor integration include the secure IoT nodes are vulnerable because of threats to sensors[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Several studies on BLE security and privacy risks have been published independently, some researchers look at the protocol’s secu- rity design and perform specific attacks that take advantage of the protocol’s inadequate implementation [9], [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Additionally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' security researchers from academia and industry presented multiple attacks scenarios against IoT-BLE devices Enables user to Interact App Application with Application Interface with Profiles Generic Access Profile on Physical Layer Service framework for Generic Access Profile Data Communication Host Attribute Security Device Pairing Protocol Manager Host Controller Logica Link Control and Data Encapsulation Interface Adaption Protocol Link Layer Advertising,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Scaning and Maintaining Connections Controller Basic Radio Frequency Physical Layer TransmissionFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' BLE Connections Phases [3] by performing attacks on IoT-BLE device via attack tools such as btlejack, adafruit, ubertooth one at various security conferences [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' This study aims to evaluate wearable device threat surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' BLE Analysis proposed model This paper’s contributions can be summarised as follows: With the help of STRIDE threat analysis of wearable devices, several vulnerabilities identified in multiple well- known (most recent) versions of the BLE protocol are utilized to categorize BLE risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Present various examples of attacks on BLE wearable devices, along with the tools used to do them and discuss the lessons learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Suggest mitigation technique for observed attack threat surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' TOOLS USED FOR ATTACKING AND ANALYSING BLE This section details open-source security researchers that have created hardware and software tools to analyze various BLE vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Security researchers have used MITM, passive eavesdropping, and bluetooth encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' These tools demonstrate security vulnerabilities on standard IoT devices as shown the analysis based in the mind map as shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' This list of open source, free tools help security researchers, BLE developers, and testers build up low-cost application components, identify and assess vulnerabilities, and safeguard BLE-enabled IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' This section enlist various tools to perform attack on BLE wearable device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Hardware tools 1) Ubertooth: A free tool for exploring Bluetooth is called Ubertooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Ubertooth offers gear that enables passive inspec- tion of Bluetooth and BLE device communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The Jan- uary 2011-released Ubertooth One can detect and demodulate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='4GHz Frequency band transmissions with a bandwidth of 1MHz [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Ubertooth sniffs the data and displays a visual of the local traffic sorted by frequencies and transmission inten- sity in dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' This study approximates the overall Bluetooth traffic in the region, including distance diagnostics searching for items outside the purview of this project [4],[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The intensity signal strength on the channel frequency is used to structure the Ubertooth spectrum analyzer as shown in fig 12 the green lines show the strongest signal for that identified frequency, while the white lines show activities now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Wireless technology transmits data over the air, making it simple to intercept or tamper with these data packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 2) ADA-FRUIT BLE Sniffer: Adafruit is a Bluetooth Low Energy (BLE) package capturing device similar to Ubertooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Adafruit was created by Adafruit Industry[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' It has the ability to intercept sent data packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Analyze two BLE devices with this tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Data gathered via Wireshark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Software tools 1) Btlejack: Btlejack is used to spy, jam, and hijack Blue- tooth Low Energy devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' One or more of these devices are now supported, and one or more are running the special firmware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Based on the BBC Micro: Bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' This utility’s latest version (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='0) supports BLE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='0 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' However, only 1 Mbps uncoded PHY is supported by BLE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='0 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' BtleJack controls the connection by actively disconnecting the master and moving it about in the connection, taking and sniffing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Master ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Slave ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='EstablishedLLConnection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='SecurityRequest(optional) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Pairing Request ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Phasel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Pairing Responses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='DetermineTempraryKey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='PairingOverSMP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Phasell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='LegacyParingorSecureConnections ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Pairing Confirmation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='STK basedencrypted connection Eshtablished ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Key Distribution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Phase Il ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='KeyDistribution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Key DistributionSystematic LiteratureReview ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='QualitativeContentAnalysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Threats ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Vulnerabilities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Analysisof ThreatsandVulnerabilities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='ThreatModeling/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='STRIDEFramework ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Mapping2) Wireshark: Wireshark is a packet analyzer that is avail- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='able for free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' It is an open-source network protocol analyzer that frequently uses software for packet analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' To help with data gathering, it may collect information via Ethernet, Blue- tooth, USB ports, and other communication mediums Using a specific filter, Wireshark’s graphical user interface (GUI) users can see captured packets in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 4 of some wearable devices as where vendors can’t used encryption techniques data is transferred directly in plain text as seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Sniffed Packets in Wireshark 3) Btlejuice: Cauquil created BtleJuice, which he exhibited at the 2016 DefCon24 conference as a MITM attack against BLE devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Intercept score, intercept proxy, specialized web interface, python, and Node JS binding are the four components of Btlejuice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The key components are interception cores and proxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' These two components function indepen- dently and communicate via the Web Socket protocol btlejuice functions as a proxy between BLE accessories and mobile applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' SECURITY PREREQUISITES FROM STRIDE THREAT ANALYSIS Looking at the DFD of wearable devices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' we can derive se- curity requirements and evaluate the attack scenarios acquired from the analysis of wearable devices like MI band 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Boat Storm smartwatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' One plus Band,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' and Firebolt Invincible smartwatch since,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Determining security needs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' we consider the attack scenarios derived from the DFD diagram since the security requirement is meant to stop attackers from attaining their objectives,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' below are some security requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 1) Adding timestamp in the packet: Incorporating a times- tamp into the transmission for all data exchange and connec- tion processes, a timestamp should be included in the packet to avoid replay attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 2) Authentication: Even if an attacker edits a message, the message authentication mechanism can identify the change and prevent it from being inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Also, to avoid connecting to an attacker’s device, stealing accounts, and circumventing the login procedure, the user authentication process should be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 3) Encryption: An attacker’s ability to read and change the original communication is hampered by encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' As a result, message alteration and network packet sniffing should be prevented by the packet encryption procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Additionally, before saving any sensitive data in the database, it must be encrypted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The attacker won’t be able to access the data with even physical access to the storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 4) Traffic analysis and intrusion detection: Attack detection and traffic analysis It’s tough to stop excessive data transmis- sion completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' As a result, we must mitigate it on the server side using traffic analysis and penetration testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 5) Secure account management: Account management that is safe to avoid account theft, all users must maintain their accounts securely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The service provider should inform the user about the possibility of account theft and safeguard the data store.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The Microsoft threat modeling tool provides a threat mod- eling report based on our model as data flow diagram shown in fig 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' It generates 111 threats on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Based on prior research on smart band security,and identified 44 significant threats that may achieve the attacker’s aim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' also, given the practical analysis below in attack analysis section A and B with all attack scenario on IoT-BLE wearable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' ATTACK ANALYSIS OF WEARABLE DEVICE The security architecture of BLE is distinct from that of classic bluetooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Low power consumption, computationally restricted sensors, and connection with IoT devices are all supported by BLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' With BLE, we have to choose between performance, security, privacy concerns, and low power usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Different ways for secure connections are included in the BLE standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Device binding, link-layer encryption, and device safe-list Unfortunately, many IoT-BLE devices do not imple- ment these security methods properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' As a result, numerous security threats arise because many of these risks are caused by vulnerabilities in a common underlying architecture or protocol, the same mitigating procedure applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Some attacks are mutually beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' This section focuses on how to arrange and perform assaults using the threat model sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The user information is sent between the smartphone and the smart band as demonstrated in the DFD in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 5 smartphone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' also shown each threats to the specific BLE wearable devices (MI Band 4, Boat Storm Smart watch, Oneplus Band and Fire bolt invincible) as shown in the table identified threat for analysed wearable devices which derived from the STRIDE threat analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Examining the system with an emphasis on user data and the smart band service should be done to overcome the security issues in BLE devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Passive Attack The first step in any attack scenarios is passive listening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' As- sailants secretly monitor every communication between linked devices, resulting in a wide range of destructive attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' By interfering with data transmission somehow, we can eavesdrop on and intercept every conveyed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Passive eavesdropping attacks are particularly vulnerable since data is delivered wirelessly, and an attacker only needs an interceptor such as Micro bit V2 to intercept wireless communications of devices Wireshark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Packet2250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='eth0and/tmp/onepipe X oooo ooo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' = Reserved: 0X00 Header length: 24 DLT:147 Reserved: 36750c0000620900cf2e185d16091400 DLT: 147,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Payload: btle (Bluetooth Low Energy LinkLayer BluetoothLowEnergyLinkLayer Accesing Packets via Link Layer Access Address: ox8e89bed6 PacketHeader: 0x2304 (PDU Type: SCAN_RSP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='TxAdd: Public) Advertising Address: eb:ac:08:56:79:28 (eb:ac:08:56:79:28) scan Response Data:10094d6920536d6172742042616e6420340302e0fe0716e0fe82010000 CRC: 0xf8d1b9 0000 000018 0093000000 36750c 00 00 62 09 00 6u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' b 0010 cf 2e 18 5d16 09 14 00 d6 be 89 8e 04 23 28 79 井（ 0020 6 0 80 V MiSmart B 0030 61 6e 64 20 34 03 02 e0 fe 07 16 e0 fe 82 01 00 and 4 0040 o01f8b9d Adressing Packets Here seen details in plain text makers should use Encryption techniquesFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Data Flow Diagram PPIDIRECTCOMMUNICATION 1MasterDevice Slave Device PP Measured Data Secure Connection L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='PPI ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='EPPI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='(Request&Authentication) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Mi Band4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='BoatStorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='OneplusBand ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Fire-BoltInvincible ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Shows all the Data Communication Between Master and Slave Device ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Shows Specific Data Communicationof BLEbased Specific Wearable DevicesTABLE I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='IDENTIFIED THREATS FOR ANALYSED WEARABLE DEVICES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Denial of Services ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Elevation of Privilege ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Information Disclosure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Repudiation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Spoofing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Tampering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Mi Band 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Boat Storm Smart Watch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='One Plus Band ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Fire-Bolt Invincible ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Security Analysis Mind-map Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Attackers Testbed (MI Band 4, One plus Band and Fire bolt invincible) this IoT- BLE wearables are particularly vulnerable to this attack due to its predictable and obvious channel switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Passive Attack Most BLE systems like MI Band 4 and fire bolt invincible smart watch and Boat storm are vulnerable to this sort of attack due to the insufficient protection of the BLE standard and poor encryption algorithms, as well as several critical exchange mechanisms attackers to decode data easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='BruteForce ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Denial of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='BD ADDR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Service ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='NrF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='MITM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Attack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Printing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='connect ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Fire Bolt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='BTLEJack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Invincible ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Bluejacking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Spoofing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Attack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Ubertooth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Denial of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Adafruit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='One ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Service ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='BruteForce ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Attack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='BD_ADDR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Attack0x6e4d5787 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='BtleJack version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='0 [i] Detected sniffers: >Sniffer #0:fwversion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='0 withconnection0x6e4d5787 Attackers ox1fffffc01f Window hijacki itisallyourslo/ a678cf50df610a Ubertooth 67a11cd7 One Adafruit BLE Sniffer Micro bit V2 MI band One plus Fire bolt band InvincibleDevices are being paired Device A Device B Original Connection (Master) (Slave) Attacker sniffs Data AttackerFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Sniffing packets using Micro bit V2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Active Attacks Listening actively, the attacker disrupts communication and takes information in these assaults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' MITM and Replay are two active eavesdropping attack versions to compromise data integrity, attackers actively participate in the communication process in MITM [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' In contrast to MITM, a replay attack does not compromise the sender or recipient instead, the attacker captures the packet and re-transmits it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Active Attacks 1) Brute-Force BD-ADDR Attack: As previously stated, the bluetooth protocol employs Frequency Hopping Spread Spec- trum (FHSS) to avoid interference by hopping between unique channels of the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='4GHz ISM radio band [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The pseudo- random records utilised to clock this hopping behaviour are obtained from the ”main device” for BD-ADDR to sniff data from a device, you must account for hops, which necessitates knowing the BD-ADDR format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Bluetooth frames do not carry the entire BD-ADDR, but do include the Lower Address Part(LAP), which is a reduction of 24 bits (three bytes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The BD-ADDR is really a combination of the LAP and 1 byte that makes up the upper address part component (UAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Since the LAPs are broadcast in every frame, the Ubertooth One can capture them passively to calculate the UAP and LAP parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' After a while, the discovery of MAC ID’s acquired from the UAP and LAP sniffing of devices by using Ubertooth-Rx in scan mode (using the -z flag) for a few minutes and letting the gadgets communicate closely with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Sniffing LAP’s and UAP’s using ubertooth to capture MAC Id’s Ubertooth allows to find gadgets in both concealed and non-discoverable modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Ubertooth isn’t a full-fledged BTLE device here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' it’s just a sniffer that gathers LAP and UAP to form addresses and sends an inquiry to the appropriate BTLE device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 2) Bluetooth Man-In-The-Middle attack: There has been researching on MITM attack tactics in IoT systems for both traditional Bluetooth and BLE, and MITM is a common type of wireless communication threat [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' By placing oneself in the center of the BLE peripheral devices, an attacker may conduct a MITM attack upon them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The attacker intercepts and alters a packet supplied by several devices before being sent to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Neither device knows its data is blocked or modified by a suspicious device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' BLE device makers must rigorously adhere to BLE protocol binding and encryption requirements to prevent MITM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' It is also best to bypass using the matching approach without updating devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Secure connections are also recommended for developers since they offer far more robust cryptographic protection than older connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' If the central device (Mobile Phone) is aware that the matching device (MI Band 4) has I/O capabilities, the MITM flag should be supplied during pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Man in the Middle Attack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Original Connection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Device A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Device B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='(Master) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='(Slave) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Diconnected Original ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Connection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} 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+page_content='0 Allinterfacesshown [i] Detected sniffers: > Sniffer #o: fw version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='0 [i] Synchronizing with connection 0x6e4d5787 v CRCInit = 0xe2e0de V Channel Map = ox1fffffc01f Hop interval e- 12 v Hop increment =e80 Svnchronizedhijacking in progress 1 Connection successfuily hijacked,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' it is all yours o/ Taking over the ongoing connection btlejack> discover Discovering and Sniffing ongoing connection Spoofing and Sending unexpected data to the Command not found wearable device which accepted by slave device 1b c4 e4 a6 500+ 61 e De e2 93 8b 50 09 e6 of 8a 02 30 5d 97 fd d5 b7 e3 15 e 7459b 66 db d8 29 e3 cc 95 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 8e 3e5+ 40 7a 04 ed 10 6e 12 2a ad bd f9 a6 8b d3 53 15 66 d6 90 33 83 50 1a 1b bc 66 fd da d0 46 e3 94 5a ac ef 87 3e 3d b6 5c 4c 23 34 cb 8f ae 0b 67 a1 1c d7tushar@tushar:~$ sudo btlejack -f ox5628bocb BtleJack version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='o [i] Detected sniffers: > Sniffer #o: fw version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='o ox5628b0cb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='04 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='oa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='LL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Data: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='oa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='0e 00 04 00 0b 0c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='51 1400 00 2a 1a000060 0100003) Blue Printing: Blueprinting gathers precise information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='on the device’s model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' manufacturer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' unique identifier (IMEI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' and software version,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' with a focus on user privacy issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' This exploit affects both traditional bluetooth and BLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Although the assault may not do significant damage, it is used to organise future attacks on the victim’s device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Blueprinting is not a serious assault, but it does expose personal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Ac- cording to the BLE standard, IoT-BLE devices must publicly broadcast their GATT services so that an attacker can obtain this data [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Furthermore, attackers can get data on the quantity of devices deployed by a specific manufacturer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' If a gadget has a well-known security flaw, this attack can be quite damaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' To carry out this assault, there are several open-source tools accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' These sources might gather data about the bluetooth stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Another powerful application for executing this attack quickly is nRF Connect for smartphones, which can be write services directly to the IoT-BLE devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' nRF Connect application 4) Device Fingerprinting: Device fingerprinting is used to identify a specific device by utilizing data exclusive to that device, such as the MAC address, UUID, advertising packets, and GATT services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Device fingerprinting infringes user privacy, as seen in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 12 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The fixed MAC addresses of many BLE devices can be used to identify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' By analyzing IoT mobile applications and identifying static UUIDs from ads, several BLE based IoT and Wearable Devices are susceptible to fingerprinting [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' UUIDs have a hierarchical structure and need to analyze the set of values to determine the UUID hierarchy for fingerprinting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Simultaneously, it pinpoints application- level flaws such as inappropriate encryption usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Sniffed advertising UUIDs leads to the fingerprinting of IoT devices [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Therefore, a single UUID may be utilized by several apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Connecting to the gadget is indeed required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The value set predictable outcome implementation level weaknesses, which leads to the identification of devices that seem to be susceptible to sniffer or unauthorised access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' We need to fix an app-level vulnerability to stop this attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Developers must provide cryptographically secure features and encrypt credentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' To disguise the UUIDs, implementing use of encryption techniques since the broadcast signal must have a channel level measurement [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' As a result, the attacker can only receive the annoying signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 5) Blue Stumbling: Finding devices with known security issues is a practice known as blue stumbling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' It is a set- up for later, more significant attacks rather than active ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Using tiny sniffers like the Microbit V2, an attacker sniffs for susceptible devices in a crowded area to remain undetected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The attacker’s target devices with security weaknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' When a BLE connection is not required, the device must be kept invisible or in un-discoverable mode to prevent attackers from discovering it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Unauthenticated devices must provide just the most basic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 6) DOS attack: Bluetooth-enabled peers can request and receive echoes using the L2CAP protocol in a DoS attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' DoS attack is possible with L2CAPping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' This L2CAPping allows seeing the established connection and the round trip time with other Bluetooth-enabled devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Attacks against smartphones can maintain a minimum of about 10 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Powerful transmissions for laptops can reach up to 100 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Using standard tools like l2ping, which comes with the Linux Bluex utils package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Several instructions in the l2ping program allow hackers to specify the packet length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Hackers infect Bluetooth-enabled gadgets with malicious programs, rendering them useless to their users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The assault can potentially disrupt the victim’s device’s regular operation and possibly damage its functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The -s number option in the standard BlueZ utility distribution’s l2ping allows users to select the packet length for l2ping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Many devices use a packet size of 600 bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Creates a packet of the requested size and transmits it to the provided MAC address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' As a result, the end device’s reaction time grows longer and longer, and the attacked device’s Bluetooth capability stops working.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' DOS attack using Adafruit V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' SUMMARY OF BLE ATTACKS IoT devices and wearable gadgets that use the BLE protocol have a variety of vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' However, lousy protocol design is to blame for many issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Some BLE security policy devices are subject to cryptographic attack pairing techniques and BLE privacy vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The National Institute of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Standards and Technology (NIST) and other security systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='researchers have provided specific devices with considerably ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='三 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Devices ' metadata={'source': 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+page_content='Unknown Characteristic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Unknown Service ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='UUiD:0000aaa2-0000-1000-8000-aabbccddeeff ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='UUID: 00001530-0000-3512-2118-0009af100700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='PRIMARY SERVICE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='Properties:INDICATE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='WRITE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='WRITENORESPONSE Descriptors: Alert Notification Service ClientCharacteristicConfiguration UUID: 0x1811 UUID:0x2902 PRIMARY SERVICE Value:Notifications and indicationsdisabled Alert Services New Alert UserData Writable via UUID: 0x2A46 UUID:0x181C Nrf Connect Properties: READ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' WRITE PRIMARY SERVICE application Descriptors: Device Characteristic User Description + First Name Developer UUID: 0x2901 UUID:0x2A8A provide all Properties:READ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='WRITE Alert Notification Control Point 77 services Value:Natenczas UUID: 0x2A44 should be in Properties: NOTIFY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' READ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' WRITE Last Name read mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Descriptors: UUID:0x2A90 Client Characteristic Configuration Properties:READ,WRITE UUID: 0x2902 Value: Wojski Gender Immediate Alert UUID: 0x1802 UUID:0x2A8C PRIMARY SERVICE Properties:READ,WRITE Value: Male Heart Rate UUID: 0x180D PRIMARY SERVICE Unknown Service UUID: OxFEEOTarget addr > A1:B2:C3:D4:22:3Z MAC address of Target Device Packages size V 600 Targeting Packet Size Threads count 300 V Threads count under Packetsolder versions of (BLE v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='0 and v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' As per the authors analysis according to shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 15 mind map threats to IoT-BLE across the three OSI levels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=', physical layer, data link layer, and application layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Physical layer attacks accompany assaults in the radio spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The bluetooth data connection layer is where the majority of assaults happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Data transfer via data transfer layer, The attacker captures the link-layer packet and can send the malicious packet via link layer hence link-layer security is a significant problem for BLE developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Some vulnerabilities are characterized as incorrectly implemented application-layer vulnerabilities by the device maker or developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' FUTURE SCOPE BLE-enabled smart wearables and IoT gadgets have become a part of our everyday life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Data IoT device management and access control will need to be highly secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Also, connect to IoT devices via the server rather than users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' The gadget has enhanced device management and user privacy dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' This server is designed to prevent IoT devices from gaining access to personal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' However, if the network/server is compromised, all devices linked to it are in danger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' So there is a chance that researchers may start looking into employing distributed type Blockchain technology to safeguard linked IoT devices shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Industries can benefit from a BLE mesh network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' CONCLUSION This paper investigates the effectiveness of IoT-BLE de- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Creating a threat model using the STRIDE threat modelling tool by analyzing and experimenting with IoT- BLE wearable device’s attack surface and finding threats is easy to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Sometimes, its severe only vulnerabilities and different exploitation scenarios like denial of service (DoS), elevation of privilege, information disclosure, spoofing, tampering, and repudiation during the pairing and writable services phase permanently secure this phase with appro- priate pairing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Moreover, authentication techniques make services UUIDs unreachable for other sources except applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' This paper also gives a quantitative analysis to provide a framework to address security concerns for IoT-BLE wearable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' BLE is a secure wireless communication protocol, but only if implemented correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' This investigative analysis with practical proof thoughtfully and practically ana- lyzes the IoT-BLE security procedures and identifying security vulnerabilities during device pairing-related concerns, consid- ering the security measures against the threats and assaults determined after developing specific security measures from threat modeling and the analysis given in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Applying security controls to the system may introduce additional issues, including key management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' As a result, it is crucial to repeat the methods described in this work for the security system to assess the security after rebuilding a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' ACKNOWLEDGMENT Authors would acknowledge the support of the Centre of Excellence (CoE) in Complex and Nonlinear Dynamical Systems(CNDS), VJTI, and the members for providing us immense support and a platform for my project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' REFERENCES [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Albazrqaoe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Huang and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' Xing, ”A Practical Bluetooth Traffic Sniffing System: Design, Implementation, and Countermeasure,” in IEEE/ACM Transactions on Networking, vol.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=', 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} +page_content=' 39–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE2T4oBgHgl3EQfYQdR/content/2301.03852v1.pdf'} diff --git a/X9E1T4oBgHgl3EQfwAUv/vector_store/index.faiss b/X9E1T4oBgHgl3EQfwAUv/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..9bcded38bbfaefa70865a29b7d7bb49eace5b203 --- /dev/null +++ b/X9E1T4oBgHgl3EQfwAUv/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f17445d1bab23c89e1f98bb34e5fb3924506dfce9f8bfd65ea718147ff6a5aae +size 3473453 diff --git a/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf b/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf new file mode 100644 index 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Jan 2023 +A Complete Inference System for Skip-free +Guarded Kleene Algebra with Tests +Todd Schmid1 +, Tobias Kapp´e2,3 +, and Alexandra Silva4 +1 University College London, London, UK +2 Open University of the Netherlands +3 ILLC, University of Amsterdam, NL +4 Cornell University, Ithaca, NY, USA +Abstract. Guarded Kleene Algebra with Tests (GKAT) is a fragment +of Kleene Algebra with Tests (KAT) that was recently introduced to +reason efficiently about imperative programs. In contrast to KAT, GKAT +does not have an algebraic axiomatization, but relies on an analogue of +Salomaa’s axiomatization of Kleene Algebra. In this paper, we present an +algebraic axiomatization and prove two completeness results for a large +fragment of GKAT in the form of skip-free programs. +1 +Introduction +Kleene algebra with tests (KAT) [27] is a logic for reasoning about semantics and +equivalence of simple imperative programs. It extends Kleene Algebra (KA) with +Boolean control flow, which enables encoding of conditionals and while loops. +KAT has been applied to verification tasks. For example, it was used in proof- +carrying Java programs [25], in compiler optimization [29], and file systems [9]. +More recently, KAT was used for reasoning about packet-switched networks, +serving as a core to NetKAT [4] and Probabilistic NetKAT [13,45]. +The success of KAT in networking is partly due to its dual nature: it can be +used to both specify and verify network properties. Moreover, the implementa- +tions of NetKAT and ProbNetKAT were surprisingly competitive with state-of- +the-art tools [14,46]. Part of the surprise with the efficiency of these implemen- +tations is that the decision problem for equivalence in both KAT and NetKAT +is PSPACE-complete [30,4]. Further investigations [44] revealed that the tasks +performed in NetKAT only make use of a fragment of KAT. It turns out that +the difficulty of deciding equivalence in KAT can largely be attributed to the +non-deterministic nature of KAT programs. If one restricts to KAT programs +that operate deterministically with respect to Boolean control flow, the associ- +ated decision problem is almost linear. This fragment of KAT was first identified +in [31] and further explored as guarded Kleene algebra with tests (GKAT) [44]. +The study in [44] proved that the decision problem for GKAT programs is +almost linear, and proposed an axiomatization of equivalence. However, the ax- +iomatization suffered from a serious drawback: it included a powerful uniqueness +of solutions axiom (UA), which greatly encumbers algebraic reasoning in prac- +tice. In order to use (UA) to show that a pair of programs are equivalent, one + +needs to find a system of equations that they both satisfy. Even more worry- +ingly, the axiomatization contained a fixed-point axiom with a side condition +reminiscent of Salomaa’s axiomatization for regular expressions, which is known +to be non-algebraic and impair the use of the axiomatic reasoning in context (as +substitution of atomic programs is not sound anymore). The authors of [44] left +as open questions whether (UA) can be derived from the other GKAT axioms and +whether the non-algebraic side condition can be removed. Despite the attention +GKAT has received in recent literature [41,50,43], these questions remain open. +In the present work, we offer a partial answer to the questions posed in [44]. +We show that proving the validity of an equivalence in GKAT does not require +(UA) if the pair of programs in question are of a particular form, what we call +skip-free. This fragment of GKAT is expressive enough to capture a large class +of programs, and it also provides a better basis for algebraic reasoning: we show +that the side condition of the fixed-point axiom can be removed. Our inspiration +to look at this fragment came from recent work of Grabmayer and Fokkink’s on +the axiomatization of 1-free star expressions modulo bisimulation [16,15], an im- +portant stepping stone to solving a decades-open problem posed by Milner [34]. +In a nutshell, our contribution is to identify a large fragment of GKAT, what +we call the skip-free fragment, that admits an algebraic axiomatization. We ax- +iomatize both bisimilarity and language semantics and provide two completeness +proofs. The first proves completeness of skip-free GKAT modulo bisimulation [41], +via a reduction to completeness of Grabmayer and Fokkink’s system [16]. The +second proves completeness of skip-free GKAT w.r.t. language semantics via a +reduction to skip-free GKAT modulo bisimulation. We also show that equivalence +proofs of skip-free GKAT expressions (for both semantics) embed in full GKAT. +The next section contains an introduction to GKAT and an overview of the +open problems we tackle in the technical sections of the paper. +2 +Overview +In this section we provide an overview of our results. We start with a motivating +example of two imperative programs to discuss program equivalence as a verifi- +cation technology. We then show how GKAT can be used to solve this problem +and explore the open questions that we tackle in this paper. +Equivalence for Verification. +In the game Fizz! Buzz! [37], players sit in +a circle taking turns counting up from one. Instead of saying any number that +is a multiple of 3, players must say “fizz”, and multiples of 5 are replaced with +“buzz”. If the number is a multiple both 3 and 5, the player must say “fizz buzz”. +Imagine you are asked in a job interview to write a program that prints out +the first 100 rounds of a perfect game of Fizz! Buzz!. You write the function +fizzbuzz1 as given in Figure 1(i). Thinking about the interview later that day, +you look up a solution, and you find fizzbuzz2, depicted in Figure 1(ii). You +suspect that fizzbuzz2 should do the same thing as fizzbuzz1, and after thinking +it over for a few minutes, you realize your program could be transformed into the +reference solution by a series of transformations that do not change its semantics: + +def fizzbuzz1 = +(i) +n := 1; +while n ≤ 100 do +if 3|n then +if not 5|n then +print fizz; n++; +else +print fizzbuzz; n++; +else if 5|n then +print buzz; n++; +else +print n; n++; +print done!; +def fizzbuzz2 = +(ii) +n := 1; +while n ≤ 100 do +if 5|n and 3|n then +print fizzbuzz; +else if 3|n then +print fizz; +else if 5|n then +print buzz; +else +print n; +n++; +print done!; +Fig. 1. Two possible specifications of the ideal Fizz! Buzz! player. +1. Place the common action n++ at the end of the loop. +2. Replace not 5|n with 5|n and swap print fizz with print fizzbuzz. +3. Merge the nested branches of 3|n and 5|n into one. +Feeling somewhat more reassured, you ponder the three steps above. It seems +like their validity is independent of the actual tests and actions performed by the +code; for example, swapping the branches of an if - then - else - block while negat- +ing the test should be valid under any circumstances. This raises the question: +is there a family of primitive transformations that can be used to derive valid +ways of rearranging imperative programs? Furthermore, is there an algorithm to +decide whether two programs are equivalent under these laws? +Enter GKAT. Guarded Kleene Algebra with Tests (GKAT) [44] has been pro- +posed as a way of answering the questions above. Expressions in the language +of GKAT model skeletons of imperative programs, where the exact meaning of +tests and actions is abstracted. The laws of GKAT correspond to program trans- +formations that are valid regardless of the semantics of tests and actions. +Formally, GKAT expressions are captured by a two-level grammar, generated +by a finite set of tests T and a finite set of actions Σ, as follows: +BExp ∋ a, b ::= 0 | 1 | t ∈ T | a ∨ b | a ∧ b | a +GExp ∋ e, f ::= p ∈ Σ | b | e +b f | e · f | e(b) +BExp is the set of Boolean expressions, built from 0 (false), 1 (true), and primitive +tests from T , and composed using ∨ (or), ∧ (and) and +(not). GExp is the set +of GKAT expressions, built from tests (assert statements) and primitive actions +p ∈ Σ. Here, e +b f is a condensed way of writing ‘if b then e else f’, and e(b) +is shorthand for ‘while b do e’; the operator · models sequential composition. By +convention, the sequence operator · takes precedence over the operator +b. +Example 2.1. Abbreviating statements of the form print foo by simply writing +foo, Figure 1(i) can be rendered as the GKAT expression +(n := 1) · +�(fizz · n++ +5|n fizzbuzz · n++) +3|n +(buzz · n++ +5|n n · n++) +�(n ≤ 100) +· done! +(1) + +Similarly, the program in Figure 1(ii) gives the GKAT expression +(n := 1) · ((fizzbuzz +5|n ∧ 3|n (fizz +3|n (buzz +5|n n))) · n++)(n ≤ 100) · done! (2) +Semantics. A moment ago, we stated that GKAT equivalences are intended to +witness program equivalence, regardless of how primitive tests and actions are +interpreted. We make this more precise by recalling the relational semantics of +GKAT programs [44].5 The intuition behind this semantics is that if the possible +states of the machine being programmed are modelled by some set S, then tests +are predicates on S (comprised of all states where the test succeeds), and actions +are relations on S (encoding the changes in state affected by the action). +Definition 2.2 ([44]). A (relational) interpretation is a triple σ = (S, eval, sat) +where S is a set, eval : Σ → P(S × S) and sat : T → P(S). Each relational +interpretation σ gives rise to a semantics �−�σ : GExp → P(S × S), as follows: +�0�σ = ∅ +�a�σ = �1�σ \ �a�σ +�1�σ = {(s, s) : s ∈ S} +�p�σ = eval(p) +�t�σ = {(s, s) : s ∈ sat(t)} +�e +a f�σ = �a�σ ◦ �e�σ ∪ �a�σ ◦ �f�σ +�a ∧ b�σ = �a�σ ∩ �b�σ +�e · f�σ = �e�σ ◦ �f�σ +�a ∨ b�σ = �a�σ ∪ �b�σ +�e(a)�σ = (�a�σ ◦ �e�σ)∗ ◦ �a�σ +Here we use ◦ for relation composition and ∗ for reflexive transitive closure. +Remark 2.3. If eval(p) is a partial function for every p ∈ Σ, then so is �e�σ for +each e. The above therefore also yields a semantics in terms of partial functions. +The relation �e�σ contains the possible pairs of start and end states of the +program e. For instance, the input-output relation of �e +a f� consists of the +pairs in �e�σ (resp. �f�σ) where the start state satisfies a (resp. violates a). +Example 2.4. We could model the states of the machine running Fizz! Buzz! as +pairs (m, ℓ), where m is the current value of the counter n, and ℓ is a list of +words printed so far; the accompanying maps sat and eval are given by: +sat(k|n) = {(m, ℓ) ∈ S : m ≡ 0 mod k} +sat(n ≤ k) = {(m, ℓ) ∈ S : m ≤ k} +eval(n++) = {((m, ℓ), (m + 1, ℓ) : (m, ℓ) ∈ S} +eval(n := k) = {((m, ℓ), (k, ℓ)) : (m, ℓ) ∈ S)} +eval(w) = {((m, ℓ), (m, ℓw)) : (m, ℓ) ∈ S} +(w ∈ {fizz, buzz, fizzbuzz}) +eval(n) = {((m, ℓ), (m, ℓm)) : (m, ℓ) ∈ S} +For instance, the interpretation of n++ connects states of the form (m, ℓ) to states +of the form (m + 1, ℓ)—incrementing the counter by one, and leaving the output +unchanged. Similarly, print statements append the given string to the output. +5 A probabilistic semantics in terms of sub-Markov kernels is also possible [44]. + +On the one hand, this parameterized semantics shows that programs in the +GKAT syntax can be given a semantics that corresponds to the intended meaning +of their actions and tests. On the other hand, it allows us to quantify over all +possible interpretations, and thus abstract from the meaning of the primitives. +As it happens, two expressions have the same relational semantics under any +interpretation if and only if they have the same language semantics [44], i.e., in +terms of languages of guarded strings as used in KAT [27]. Since equivalence un- +der the language semantics is efficiently decidable [44], so is equivalence under +all relational interpretations. The decision procedure in [44] uses bisimulation +and known results from automata theory. These techniques are good for mecha- +nization but hide the algebraic structure of programs that plays. To expose this, +algebraic laws of GKAT program equivalence were studied. +Program transformations. GKAT programs are (generalized) regular expres- +sions, which are intuitive to reason about and for which many syntactic equiv- +alences are known and explored. In [44], a set of sound axioms e ≡ f such that +�e�σ = �f�σ for all σ was proposed, and it was shown that these can be used to +prove a number of useful facts about programs. For instance, the following two +equivalences are axioms of GKAT: +e · g +b f · g ≡ (e +b f) · g +f +b e ≡ e +b f +The first of these says that common code at the tail end of branches can be +factored out, while the second says that the code in branches of a conditional +can be swapped, as long as we negate the test. Returning to our running example, +if we apply the first law to (1) three times (once for each guarded choice), +n := 1 · +�� +(fizzbuzz +5|n fizz) +3|n +(buzz +5|n n) +� +· n++ +�(n ≤ 100) +· done! +(3) +Finally, we can apply (e +a f) +b (g +a h) ≡ e +a∧b (f +b (g +a h)), which is +provable from the axioms of GKAT, to transform (3) into (2). +Being able to transform one GKAT program into another using the axioms of +GKAT is useful, but the question arises: do the axioms capture all equivalences +that hold? More specifically, are the axioms of GKAT powerful enough to prove +that e ≡ f whenever �e�σ = �f�σ holds for all σ? +In [44], a partial answer to the above question is provided: if we extend the +laws of GKAT with the uniqueness axiom (UA), then the resulting set of axioms +is sound and complete w.r.t. the language semantics. The problem with this is +that (UA) is not really a single axiom, but rather an axiom scheme, which makes +both its presentation and application somewhat unwieldy. +To properly introduce (UA), we need the following notion. +Definition 2.5. A left-affine system is defined by expressions e1,1, . . . , en,n ∈ +GExp and f1, . . . , fn ∈ GExp, along with tests b1,1, . . . , bn,n ∈ BExp. A sequence +of expressions s1, . . . , sn ∈ GExp is said to be a solution to this system if +ei,1 · s1 +bi,1 ei,2 · s2 +bi,2 · · · +bi,n−1 ei,n +bi,n fi ≡ si +(∀i ≤ n) +Here, the operations +bi,j associate to the right. + +Guarded Union +Sequencing +Loops +x = x +b x +x = x +1 y +x +b y = y +¯b x +x +b (y +c z) = (x +b y) +b∨c z +0x = 0 +x0 +(†) += 0 +x(yz) = (xy)z +(x +b y)z = xz +b yz +x(b)y = x(x(b)y) +b y +z = xz +b y +z = x(b)y +Fig. 2. Axioms for language semantics skip-free GKAT (in addition to Boolean algebra +axioms for tests, see Fig. 3). If the axiom marked † is omitted the above axiomatize a +finer semantics, bisimilarity. +A left-affine system called guarded if no ei,j that appears in the system suc- +cessfully terminates after reading an atomic test. In other words, each coefficient +denotes a productive program, meaning it must execute some action before suc- +cessfully terminating—we refer to Section 7.3 for more details. +Stated fully, (UA) says that if expressions s1, . . . , sn and t1, . . . , tn are solutions +to the same guarded left-affine system, then si ≡ ti for 1 ≤ i ≤ n. +On top of the infinitary nature of (UA), the side condition demanding guard- +edness prevents purely algebraic reasoning: replacing action symbols in a valid +GKAT equation with arbitrary GKAT expressions might yield an invalid equa- +tion! The situation is analogous to the empty word property used by Salomaa [39] +to axiomatize equivalence of regular expressions. The side condition of guarded- +ness appearing in (UA) is inherited from another axiom of GKAT, the fixed-point +axiom, which in essence is the unary version of this axiom scheme and explicitly +defines the solution of one guarded left-affine equation as a while loop. +g ≡ eg +b f =⇒ g ≡ e(b)f +if e is guarded. +Remark 2.6. Part of the problem of the uniqueness axiom is that the case for +general n does not seem to follow easily from the case where n = 1. The problem +here is that, unlike the analogous situation for Kleene algebra, there is no general +method to transform a left-affine system with n + 1 unknowns into one with n +unknowns [31], even if this is possible in certain cases [44]. +The open questions. We are motivated by two open questions from [44]: +– First, can the uniqueness axiom be eliminated? The other axioms of GKAT +contain the instantiation of (UA) for n = 1, which has so far been sufficient +in all handwritten proofs of equivalence that we know. Yet (UA) seems to be +necessary to complete both known completeness proofs. +– Second, can we eliminate the guardedness side condition? Kozen [26] showed +that Salomaa’s axiomatization is subsumed by a set of axioms that together +imply existence and uniqueness of least solutions to systems of equations, +but this approach has not yet borne fruit in GKAT. +This paper. Our main contribution is to show that, in a particular fragment +of GKAT, both questions can be answered in the positive (see Figure 2). +In Section 3, we present what we call the skip-free fragment of GKAT, con- +sisting of programs that do not contain assert statements in the body (other than + +assert false); in other words, Boolean statements are restricted to control state- +ments. For this fragment, we show that the axiom scheme (UA) can be avoided +entirely. In fact, this is true for language semantics (as first introduced in [44]) +as well as for the bisimulation semantics of [41]. +In Section 4, we provide a bridge to a recent result in process algebra. In the +80s, Milner offered an alternative interpretation of regular expressions [34], as +what he called star behaviours. Based on work of Salomaa from the 1960s [39], +Milner proposed a sound axiomatization of the algebra of star behaviours, but +left completeness an open problem. After 38 years, it was recently solved by +Clemens Grabmayer [15] following up on his joint work with Fokkink showing +that a suitable restriction of Milner’s axioms is complete for the one-free fragment +of regular expressions modulo bisimulation [16]. We leverage their work with an +interesting embedding of skip-free GKAT into the one-free regular expressions. +This leads to two completeness results. In Section 5, we start by focusing +on the bisimulation semantics of the skip-free fragment, and then in Section 6 +expand our argument to its language semantics. More precisely, we first provide +a reduction of the completeness of skip-free GKAT up to bisimulation to the +completeness of Grabmayer and Fokkink’s 1-free regular expressions modulo +bisimulation [16]. We then provide a reduction of the completeness of skip-free +GKAT modulo language semantics to the completeness of skip-free GKAT modulo +bisimulation via a technique inspired by the tree pruning approach of [41]. +Finally, in Section 7, we connect our semantics of skip-free GKAT expressions +to the established semantics of full GKAT. We also connect the syntactic proofs +between skip-free GKAT expressions in both our axiomatization and the existing +one. In conjunction with the results of Sections 5 and 6, the results in Section 7 +make a significant step towards answering the question of whether the axioms +of GKAT give a complete description of program equivalence, in the positive. +Proofs are deferred to the appendices. +3 +Introducing Skip-free GKAT +The axiom scheme (UA) can be avoided entirely in a certain fragment of GKAT, +both for determining bisimilarity and language equivalence. In this section, we +give a formal description of the expressions in this fragment and their semantics. +Skip-free expressions. The fragment of GKAT in focus is the one that excludes +sub-programs that may accept immediately, without performing any action. Since +these programs can be “skipped” under certain conditions, we call the fragment +that avoids them skip-free. Among others, it prohibits sub-programs of the form +assert b for b ̸= false, but also while false do p, which is equivalent to assert true. +Definition 3.1. Given a set Σ of atomic actions, the set GExp− of skip-free +GKAT expressions is given by the grammar +GExp− ∋ e1, e2 ::= 0 | p ∈ Σ | e1 +b e2 | e1 · e2 | e(b) +1 e2 +where b ranges over the Boolean algebra expressions BExp. + +x ∨ 0 = x +x ∨ ¯x = 1 +x ∨ y = y ∨ x +x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) +x ∧ 1 = x +x ∧ ¯x = 0 +x ∧ y = y ∧ x +x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) +Fig. 3. The axioms of Boolean algebra [20]. +Unlike full GKAT, in skip-free GKAT the loop construct is treated as a binary +operation, analogous to Kleene’s original star operation [24], which was also +binary. This helps us avoid loops of the form e(b), which can be skipped when b +does not hold. The expression e(b) +1 e2 corresponds to e(b) +1 +· e2 in GKAT. +Example 3.2. Using the same notational shorthand as in Example 2.1, the block +of code in Figure 1(ii) can be cast as the skip-free GKAT expression +(n := 1) · ((fizzbuzz +3|n∧5|n (fizz +3|n (buzz +5|n n))) · n++)(n ≤ 100)done! +Note how we use a skip-free loop of the form e1(b)e2 instead of the looping +construct e(b) +1 +before concatenating with e2, as was done for GKAT. +3.1 +Skip-free Semantics +There are three natural ways to interpret skip-free GKAT expressions: as au- +tomata, as behaviours, and as languages.6 After a short note on Boolean algebra, +we shall begin with the automaton interpretation, also known as the small-step +semantics, from which the other two can be derived. +Boolean algebra. To properly present our automata, we need to introduce one +more notion. Boolean expressions BExp are a syntax for elements of a Boolean +algebra, an algebraic structure satisfying the equations in Fig. 3. When a Boolean +algebra is freely generated from a finite set of basic tests (T in the case of +BExp), it has a finite set At of nonzero minimal elements called atoms. Atoms +are in one-to-one correspondence with sets of tests, and the Boolean algebra is +isomorphic to P(At), the sets of subsets of At, equipped with ∨ = ∪, ∧ = ∩, +and (−) = At \ (−). In the context of programming, one can think of an atom +as a complete description of the machine state, saying which tests are true and +which are false. We will denote atoms by the Greek letters α and β, sometimes +with indices. Given a Boolean expression b ∈ BExp and an atom α ∈ At we say +that α entails b, written α ≤ b, whenever α ∨ b = 1, or equivalently α ∨ b = b. +Automata. Throughout the paper, we use the notation • + S where S is a set +and • is a symbol to denote the disjoint union (coproduct) of {•} and S. +The small-step semantics of a skip-free GKAT expression uses a special type +of deterministic automaton. A skip-free automaton is a pair (X, h), where X is +a set of states and h: X → (⊥ + Σ × (✓ + X))At is a transition structure. At +every x ∈ X and for any α ∈ At, one of three things can happen: +6 We will connect these to the relational semantics from Definition 2.2 in Section 7. + +p +α|p +−−→ ✓ +e1 +α|p +−−→ e′ +α ≤ b +e1 +b e2 +α|p +−−→ e′ +e2 +α|p +−−→ e′ +α ̸≤ b +e1 +b e2 +α|p +−−→ e′ +e1 +α|p +−−→ e′ +e1e2 +α|p +−−→ e′e2 +e1 +α|p +−−→ ✓ +e1e2 +α|p +−−→ e2 +e1 +α|p +−−→ e′ +α ≤ b +e(b) +1 e2 +α|p +−−→ e′(e(b) +1 e2) +e1 +α|p +−−→ ✓ +α ≤ b +e(b) +1 e2 +α|p +−−→ e(b) +1 e2 +e2 +α|p +−−→ e′ +α ̸≤ b +e(b) +1 e2 +α|p +−−→ e′ +Fig. 4. The small-step semantics of skip-free GKAT expressions. +1. h(x)(α) = (p, y), which we write as x +α|p +−−→ y, means the state x under α +makes a transition to a new state y, after performing the action p; +2. h(x)(α) = (p, ✓), which we write x +α|p +−−→ ✓, means the state x under α +successfully terminates with action p; +3. h(x)(α) = ⊥, which we write x ↓ α, means the state x under α terminates +with failure. Often we will leave these outputs implicit. +Definition 3.3 (Automaton of expressions). +We equip the set GExp− of +all skip-free GKAT expressions with an automaton structure (GExp−, ∂) given in +Fig. 4, representing step-by-step execution. Given e ∈ GExp−, we denote the set +of states reachable from e by ⟨e⟩ and call this the small-step semantics of e. +The small-step semantics of skip-free GKAT expressions is inspired by Brzo- +zowski’s derivatives [8], which provide an automata-theoretic description of the +step-by-step execution of a regular expression. Our first lemma tells us that, like +regular expressions, skip-free GKAT expressions correspond to finite automata. +Lemma 3.4. For any e ∈ GExp−, ⟨e⟩ has finitely many states. +Example 3.5. The automaton that arises from the program fizzbuzz2 is below, +with a = n ≤ 100, b = 3|n, and c = 5|n. The expression e is the same as in +Example 3.2, e1 is the same as e but without the action n := 0 in front, and +e2 = n++ · e1. We also adopt the convention of writing x b|p +−−→ x′ where b ∈ BExp +to represent all transitions x α|p +−−→ x′ where α ≤ b. +e +e1 +e2 +✓ +1 | n := 1 +abc | fizzbuzz, +ab¯c | fizz, +a¯bc | buzz, +a¯b¯c | n +1 | n++ +¯a | done! +The automaton interpretation of a skip-free GKAT expression (its small-step +semantics) provides an intuitive visual depiction of the details of its execution. +This is a useful view on the operational semantics of expressions, but sometimes +one might want to have a more precise description of the global behaviour of the +program. The remaining two interpretations of skip-free GKAT expressions aim +to capture two denotational semantics of expressions: one finer, bisimilarity, that +makes a distinction on the branching created by how its states respond to atomic + +tests, which actions can be performed, and when successful termination and +crashes occur; another coarser, language semantics, that assigns a language of +traces to each expression capturing all sequences of actions that lead to successful +termination. The key difference between these two semantics will be their ability +to distinguish programs that crash early in the execution versus programs that +crash later—this will become evident in the axiomatizations of both semantics. +We start by presenting the language semantics as this is the more traditional +one associated with GKAT (and regular) expressions. +Language semantics. Formally, a (skip-free) guarded trace is a nonempty +string of the form α1p1 · · · αnpn, where each αi ∈ At and pi ∈ Σ. Intuitively, +each αi captures the state of program variables needed to execute program ac- +tion pi and the execution of each pi except the last yields a new program state +αi+1. A skip-free guarded language is a set of guarded traces. +Skip-free guarded languages should be thought of as sets of strings denoting +successfully terminating computations. +Definition 3.6 (Language acceptance). In a skip-free automaton (X, h) with +a state x ∈ X, the language accepted by x is the skip-free guarded language +L(x, (X, h)) = {α1p1 · · · αnpn | x α1|p1 +−−−→ x1 −→ · · · −→ xn +αn|pn +−−−−→ ✓} +If (X, h) is clear from context, we will simply write L(x) instead of L(x, (X, h)). +If L(x) = L(y), we write x ∼L y and say that x and y are language equivalent. +Each skip-free GKAT expression is a state in the automaton of expressions +(Definition 3.3) and therefore accepts a language. The language accepted by a +skip-free GKAT expression is the set of successful runs of the program it denotes. +Analogously to GKAT, we can describe this language inductively. +Lemma 3.7. Given an expression e ∈ GExp−, the language accepted by e in +(GExp−, ∂), i.e., L(e) = L(e, (GExp−, ∂)) can be characterized as follows: +L(0) = ∅ +L(p) = {αp | α ∈ At} +L(e1 +b e2) = bL(e1) ∪ ¯bL(e2) +L(e1 · e2) = L(e1) · L(e2) +L(e(b) +1 e2) = +� +n∈N +(bL(e1))n · ¯bL(e2) +Here, we write bL = {αpw ∈ L | α ≤ b} and L1 · L2 = {wx : w ∈ L1, x ∈ L2}, +while L0 = {ǫ} (where ǫ denotes the empty word) and Ln+1 = L · Ln. +Lemma 3.7 provides a way of computing the language of an expression e +without having to generate the automaton for e. +Bisimulation semantics. Another, finer, notion of equivalence that we can +associate with skip-free automata is bisimilarity. +Definition 3.8. Given skip-free automata (X, h) and (Y, k), a bisimulation is +a relation R ⊆ X × Y such that for any x R y, α ∈ At and p ∈ Σ: + +1. x ↓ α if and only if y ↓ α, +2. x α|p +−−→ ✓ if and only if y +α|p +−−→ ✓, and +3. for any x′ R y′, x α|p +−−→ x′ if and only if y +α|p +−−→ y′. +We call x and y bisimilar if x R y for some bisimulation R and write x ↔ y. +In a fixed skip-free automaton (X, h), we define ↔ ⊆ X × X is the largest +bisimulation, called bisimilarity. This is an equivalence relation.7 The bisimilarity +equivalence class of a state is often called its behaviour. +Example 3.9. In the automaton below, x1 and x2 are bisimilar. This is witnessed +by the bisimulation {(x1, x2), (x2, x2)}. +x1 +x2 +✓ +a | p +¯a | q +a | p +¯a | q +We can also use bisimulations to witness language equivalence. +Lemma 3.10. Let e1, e2 ∈ GExp−. If e1 ↔ e2, then L(e1) = L(e2). +The converse of Lemma 3.10 is not true. Consider, for example, the program +p(1)q that repeats the atomic action p ∈ Σ indefinitely, never reaching q. Since +L(p(1)q) = +� +n∈N +L(p)n · ∅ = ∅ = L(0) +we know that p(1)q ∼L 0. But p(1)q and 0 are not bisimilar, since Fig. 4 tells us +that p(1)q +α|p +−−→ p(1)q and 0 ↓ α, which together refute Definition 3.8.1. +3.2 +Axioms +Next, we give an inference system for bisimilarity and language equivalence con- +sisting of equations and equational inference rules. The axioms of skip-free GKAT +are given in Fig. 2. They include the equation (†), which says that early deadlock +is the same as late deadlock. This is sound with respect to the language interpre- +tation, meaning that (†) is true if x is replaced with a skip-free guarded language, +but it is not sound with respect to the bisimulation semantics. For example, the +expressions p · 0 and 0 are not bisimilar for any p ∈ Σ. Interestingly, this is the +only axiomatic difference between bisimilarity and language equivalence. +Remark 3.11. The underlying logical structure of our inference systems is equa- +tional logic [6], meaning that provable equivalence is an equivalence relation that +is preserved by the algebraic operations. +7 This follows directly from seeing skip-free automata as a special type of coalgebra and +the fact that the functor involved preserves weak pullbacks [38]. In fact, coalgebra +has been an indispensable tool in the production of the current paper, guiding us to +the correct definitions and simplifying many of the proofs. + +Given expressions e1, e2 ∈ GExp−, we write e1 ≡† e2 and say that e1 and +e2 are ≡†-equivalent if the equation e1 = e2 can be derived from the axioms in +Fig. 2 without the axiom marked (†). We write e1 ≡ e2 and say that e1 and e2 +are ≡-equivalent if e1 = e2 can be derived from the whole set of axioms in Fig. 2. +The axioms in Fig. 2 are sound with respect to the respective semantics they +axiomatize. The only axiom that is not sound w.r.t. bisimilarity is e · 0 ≡ 0, as +this would relate automata with different transition structure (as e may permit +some action to be performed, and this is observable in the bisimulation). +Theorem 3.12 (Soundness). For any e1, e2 ∈ GExp−, +1. If e1 ≡† e2, then e1 ↔ e2. +2. If e1 ≡ e2, then e1 ∼L e2. +We consider the next two results, which are jointly converse to Theorem 3.12, +to be the main theorems of this paper. They state that the axioms in Fig. 2 are +complete for bisimilarity and language equivalence respectively, i.e., they describe +a complete set of program transformations for skip-free GKAT. +Theorem 3.13 (Completeness I). If e1 ↔ e2, then e1 ≡† e2. +Theorem 3.14 (Completeness II). If e1 ∼L e2, then e1 ≡ e2. +We prove Theorem 3.13 in Section 5 by drawing a formal analogy between +skip-free GKAT and a recent study of regular expressions in the context of process +algebra [16]. We include a short overview of this recent work in the next section. +We delay the proof of Theorem 3.14 to Section 6, which uses a separate +technique based on the pruning method introduced in [41]. +4 +1-free Star Expressions +Regular expressions were introduced by Kleene [24] as a syntax for the algebra +of regular events. Milner offered an alternative interpretation of regular expres- +sions [34], as what he called star behaviours. Based on work of Salomaa [39], +Milner proposed a sound axiomatization of the algebra of star behaviours, but +left completeness an open problem. After nearly 40 years of active research from +the process algebra community, a solution was finally found by Grabmayer [15]. +A few years before this result, Grabmayer and Fokkink proved that a suit- +able restriction of Milner’s axioms gives a complete inference system for the +behaviour interpretation of a fragment of regular expressions, called the one- +free fragment [16]. In this section, we give a quick overview of Grabmayer and +Fokkink’s one-free fragment [16], slightly adapted to use an alphabet that will +be suitable to later use in one of the completeness proofs of skip-free GKAT. + +αp +αp +−−→ ✓ +r1 +αp +−−→ r′ +r1 + r2 +αp +−−→ r′ +r2 +αp +−−→ r′ +r1 + r2 +αp +−−→ r′ +r1 +αp +−−→ r′ +r1r2 +αp +−−→ r′r2 +r1 +αp +−−→ ✓ +r1r2 +αp +−−→ r2 +r1 +αp +−−→ r′ +r1 ∗ r2 +αp +−−→ r′(r1 ∗ r2) +r1 +αp +−−→ ✓ +r1 ∗ r2 +αp +−−→ r1 ∗ r2 +r2 +αp +−−→ x +r1 ∗ r2 +αp +−−→ x +Fig. 5. The small-step semantics of one-free star expressions. +Syntax. In the process algebra literature [34,16,15], regular expressions gen- +erated by a fixed alphabet A are called star expressions, and denote labelled +transition systems (LTSs) with labels drawn from A. As was mentioned in Sec- +tion 3, skip-free automata can be seen as certain LTSs where the labels are +atomic test/atomic action pairs. In Section 5, we encode skip-free GKAT expres- +sions as one-free regular expressions and skip-free automata as LTSs with labels +drawn from At · Σ. We instantiate the construction from [16] of the set of star +expressions generated by the label set At · Σ. +Definition 4.1. The set SExp of one-free star expressions is given by +SExp ∋ r1, r2 ::= 0 | αp ∈ At · Σ | r1 + r2 | r1r2 | r1 ∗ r2 +Semantics. The semantics of SExp is now an instance of the labelled transition +systems that originally appeared in [16], with atomic test/atomic action pairs +as labels and a (synthetic) output state ✓ denoting successful termination. +For the rest of this paper, we call a pair (S, t) a labelled transition system +when S is a set of states and t: S → P(At·Σ ×(✓+S)) is a transition structure. +We write x αp +−−→ y if (αp, y) ∈ t(x) and x αp +−−→ ✓ if (αp, ✓) ∈ t(x). +The set SExp can be given the structure of a labelled transition system +(SExp, τ), defined in Fig. 5. If r ∈ SExp, we write ⟨r⟩ for the transition sys- +tem obtained by restricting τ to the one-free star expressions reachable from r +and call ⟨r⟩ the small-step semantics of r. +The bisimulation interpretation of one-free star expressions is subtler than +the bisimulation interpretation of skip-free GKAT expressions. The issue is that +labelled transition systems (LTSs) are nondeterministic in general: it is possible +for an LTS to have both a x +αp +−−→ y and a x αq +−→ z transition for p ̸= q or y ̸= z. +The appropriate notion of bisimilarity for LTSs can be given as follows. +Definition 4.2. Given labelled transition systems (S, t) and (T, u), a bisimula- +tion between them is a relation R ⊆ S × T s.t. for any x R y and αp ∈ At · Σ, +1. x αp +−−→ ✓ if and only if y αp +−−→ ✓, +2. if x αp +−−→ x′, then there exist x′ R y′ such that y +αp +−−→ y′, and +3. if y αp +−−→ y′, then there exist x′ R y′ such that x αp +−−→ x′. +As before, we denote the largest bisimulation by ↔. We call x and y bisimilar +and write x ↔ y if x R y for some bisimulation R. + +Union +Sequencing +Loops +x = x + x +x = x + 0 +x + y = y + x +x + (y + z) = (x + y) + z +0x = 0 +x(yz) = (xy)z +(x + y)z = xz + yz +x ∗ y = x(x ∗ y) + y +z = xz + y +z = x ∗ y +Fig. 6. Axioms for equivalence for one-free star expressions. +The following closure properties of bisimulations of LTSs are useful later. +They also imply that bisimilarity is an equivalence relation. Like in the skip-free +case, the bisimilarity equivalence class of a state is called its behaviour. +Lemma 4.3. Let (S, t), (T, u), and (U, v) be labelled transition systems. Fur- +thermore, let R1, R2 ⊆ S × T and R3 ⊆ T × U be bisimulations. Then Rop +1 += +{(y, x) | x R1 y}, R1 ∪ R2 and R1 ◦ R3 are bisimulations. +Axiomatization. We follow [16], where it was shown that the axiomatization +found in Fig. 6 is complete with respect to bisimilarity for one-free star expres- +sions. Given a pair r1, r2 ∈ SExp, we write r1 ≡∗ r2 and say that r1 and r2 are +≡∗-equivalent if the equation r1 = r2 can be derived from the axioms in Fig. 6. +The next result is crucial to the next section, where we prove that the axioms +of ≡† are complete with respect to bisimilarity in skip-free GKAT. +Theorem 4.4 ([16, Theorem. 7.1]). r1 ↔ r2 if and only if r1 ≡∗ r2. +5 +Completeness for Skip-free Bisimulation GKAT +This section is dedicated to the proof of our first completeness result, Theo- +rem 3.13, which says that the axioms of Fig. 2 (excluding †) are complete with +respect to bisimilarity in skip-free GKAT. Our proof strategy is a reduction of +our completeness result to the completeness result for SExp (Theorem 4.4). +The key objects of interest in the reduction are a pair of translations: one +translation turns skip-free GKAT expressions into one-free star expressions and +maintains bisimilarity, and the other translation turns (certain) one-free star +expressions into skip-free GKAT expressions and maintains provable bisimilarity. +We first discuss the translation between automata and labelled transition sys- +tems, which preserves and reflects bisimilarity. We then introduce the syntactic +translations and present the completeness proof. +5.1 +Transforming skip-free automata to labelled transition systems +We can easily transform a skip-free automaton into an LTS by essentially turning +α|p +−−→ transitions into +αp +−−→ transitions. This can be formalized, as follows. +Definition 5.1. Given a set X, we define grphX : (⊥ + Σ × (✓ + X))At → +P(At·Σ ×(✓+X)) to be grphX(θ) = {(αp, x) | θ(α) = (p, x)}. Given a skip-free +automaton (X, h), we define grph∗(X, h) = (X, grphX ◦ h) + +The function grphX is injective: as its name suggests, grphX(θ) is essentially +the graph of θ when viewed as a partial function from At to Σ × (✓ + X). This +implies that the transformation grph∗ of skip-free automata into LTSs preserves +and reflects bisimilarity. +Lemma 5.2. Let x, y ∈ X, and (X, h) be a skip-free automaton. Then x ↔ y +in (X, h) if and only if x ↔ y in grph∗(X, h). +Leading up to the proof of Theorem 3.13, we also need to undo the effect of +grph∗ on skip-free automata with a transformation that takes every LTS of the +form grph∗(X, h) to its underlying skip-free automaton (X, h). +The LTSs that can be written in the form grph∗(X, h) for some skip-free +automaton (X, h) can be described as follows. Call a set U ∈ P(At·Σ ×(✓+X)) +graph-like if whenever (αp, x) ∈ U and (αq, y) ∈ U, then p = q and x = y. An +LTS (S, t) is deterministic if t(s) is graph-like for every s ∈ S. +Lemma 5.3. An LTS (S, t) is deterministic if and only if (S, t) = grph∗(X, h) +for some skip-free automaton (X, h). +Proof. For any set X, let PdetX = {U ∈ P(At · Σ × (✓ + X)) | U is graph-like} +and define funcX : PdetX → P(At · Σ × (✓ + X)) by +funcX(U)(α) = (p, x) if (αp, x) ∈ U +Let (S, t) be a deterministic LTS. Then func∗(S, t) = (S, funcS ◦ t) is a skip-free +automaton such that grph∗func∗(S, t) = (S, t), because grphS ◦ funcS = idPdet(S). +Remark 5.4. As mentioned in Footnote 7, there is a coalgebraic outlook in many +of the technical details in the present paper. For the interested reader, the maps +grph and func are actually natural transformations between the functors whose +coalgebras correspond to skip-free automata and labelled transitions, and are +furthermore inverse to one another. This implies that grph∗ and func∗ witness +an isomorphism between the categories of skip-free automata and LTSs. +5.2 +Translating Syntax +We can mimic the transformation of skip-free automata into deterministic la- +belled transition systems and vice-versa by a pair of syntactic translations going +back and forth between skip-free GKAT expressions and certain one-free star +expressions. Similar to how only some labelled transition systems can be turned +into skip-free automata, only some one-free star expressions have corresponding +skip-free GKAT expressions—the deterministic ones. +The definition of deterministic expressions requires the following notation: +given a test b ∈ BExp, we define b · r inductively on r ∈ SExp as follows: +b · 0 = 0 +b · αp = +� +αp +α ≤ b +0 +α ̸≤ b +b · (r1 + r2) = b · r1 + b · r2 +b · (r1r2) = (b · r1)r2 +b · (r1 ∗ r2) = (b · r1)(r1 ∗ r2) + b · r2 +for any αp ∈ At · Σ and r1, r2 ∈ SExp. + +Definition 5.5. The set of deterministic one-free star expressions is the small- +est subset Det ⊆ SExp such that 0 ∈ Det and αp ∈ Det for any α ∈ At and p ∈ Σ, +and for any r1, r2 ∈ Det, and b ∈ BExp, b·r1+¯b·r2, r1r2, and (b·r1)∗(¯b·r2) ∈ Det. +From GExp− to Det. We can now present the translations of skip-free expres- +sions to deterministic one-free star expressions. +Definition 5.6. We define the translation function gtr : GExp− → Det by +gtr(0) = 0 +gtr(p) = +� +α∈At +αp +gtr(e1 +b e2) = b · gtr(e1) + ¯b · gtr(e2) +gtr(e1 · e2) = gtr(e1) gtr(e2) +gtr(e(b) +1 e2) = (b · e1) ∗ (¯b · e2) +for any b ∈ BExp, p ∈ Σ, e1, e2 ∈ GExp. +Remark 5.7. In Definition 5.6, we make use of a generalized sum � +α∈At. Tech- +nically, this requires we fix an enumeration of At ahead of time, say At = +{α1, . . . , αn}, at which point we can define � +α∈At rα = rα1 + · · · + rαn. Of +course, + is commutative and associative up to ≡∗, the actual ordering of this +sum does not matter as far as equivalence is concerned. +The most prescient feature of this translation is that it respects bisimilarity. +Lemma 5.8. The graph of the translation function gtr is a bisimulation of la- +belled transition systems between grph∗(GExp−, ∂) and (SExp, τ). Consequently, +if e1 ↔ e2 in grph∗(GExp−, ∂), then gtr(e1) ↔ gtr(e2) in (SExp, τ). +From Det to GExp−. Next we translate deterministic one-free star expressions +into skip-free GKAT expressions. +Lemma 5.9. For any e ∈ GExp−, gtr(e) ∈ Det. +We would now like to define a back translation function rtg : Det → GExp− +by induction on its argument. Looking at Definition 5.5, one might be tempted +to write rtg(b·r1 +¯b·r2) = rtg(r1)+b rtg(r2), but the fact of the matter is that it +is possible for there to be distinct b, c ∈ BExp such that b·r1+¯b·r2 = c·r1 +¯c·r2, +even when b and c have different atoms. +Definition 5.10. Say that r1, r2 ∈ SExp are separated by b ∈ BExp if r1 = b·r1 +and r2 = ¯b · r2. If such a b exists we say that r1 and r2 are separated. +Another way to define Det is therefore to say that Det is the smallest subset +of SExp containing 0 and At · Σ that is closed under sequential composition and +closed under unions and stars of separated one-free star expressions. +Suppose r1 and r2 are separated. Since there are only finitely many Boolean +expressions up to equivalence, there is a maximal (weakest) test b(r1, r2) ∈ BExp +such that r1 and r2 are separated by b(r1, r2). + +Definition 5.11. The back translation rtg : Det → GExp− is defined by +rtg(0) = 0 +rtg(αp) = p +α 0 +rtg(r1 + r2) = rtg(r1) +b(r1,r2) rtg(r2) +rtg(r1r2) = rtg(r1) · rtg(r2) +rtg(r1 ∗ r2) = rtg(r1)(b(r1,r2)) rtg(r2) +for any r1, r2 ∈ SExp. In the union and star cases, we may use that r1 and r2 +are separated (by definition of Det), so that b(r1, r2) is well-defined. +The most prescient property of rtg is that it preserves provable equivalence. +Lemma 5.12. Let r1, r2 ∈ Det. If r1 ≡∗ r2, then rtg(r1) ≡† rtg(r2). +The last fact needed in the proof of completeness is that, up to provable +equivalence, every skip-free GKAT expression is equivalent to its back-translation. +Lemma 5.13. For any e ∈ GExp−, e ≡† rtg(gtr(e)). +We are now ready to prove Theorem 3.13, that provable bisimilarity is com- +plete with respect to behavioural equivalence in skip-free GKAT. +Theorem 3.13 (Completeness I). If e1 ↔ e2, then e1 ≡† e2. +Proof. Let e1, e2 ∈ GExp be a bisimilar pair of skip-free GKAT expressions. +By Lemma 5.2, e1 and e2 are bisimilar in grph∗(GExp−, ∂). By Lemmas 4.3 +and 5.8, the translation gtr : grph∗(GExp−, ∂) → (SExp, τ) preserves bisimilar- +ity, so gtr(e1) and gtr(e2) are bisimilar in (SExp, τ) as well. By Theorem 4.4, +gtr(e1) ≡∗ gtr(e2). Therefore, by Lemma 5.12, rtg(gtr(e1)) ≡† rtg(gtr(e2)). Fi- +nally, by Lemma 5.13, we have e1 ≡† rtg(gtr(e1)) ≡† rtg(gtr(e2)) ≡† e2. +6 +Completeness for Skip-free GKAT +The previous section establishes that ≡†-equivalence coincides with bisimilarity +for skip-free GKAT expressions by reducing the completeness problem of skip- +free GKAT up to bisimilarity to a solved completeness problem, namely that of +one-free star expressions up to bisimilarity. In this section we prove a complete- +ness result for skip-free GKAT up to language equivalence. We show this can be +achieved by reducing it to the completeness problem of skip-free GKAT up to +bisimilarity, which we just solved in the previous section. +Despite bisimilarity being a less traditional equivalence in the context of +Kleene algebra, this reduction simplifies the completeness proof greatly, and +justifies the study of bisimilarity in the pursuit of completeness for GKAT. +The axiom x · 0 = 0 (which is the only difference between skip-free GKAT up +to language equivalence and skip-free GKAT up to bisimilarity) indicates that the +only semantic difference between bisimilarity and language equivalence in skip- +free GKAT is early termination. This motivates our reduction to skip-free GKAT +up to bisimilarity below, which involves reducing each skip-free expression to an +expression representing only the successfully terminating branches of execution. + +Now let us turn to the formal proof of Theorem 3.14, which says that if +e, f ∈ GExp− are such that L(e) = L(f), then e ≡ f. In a nutshell, our strategy is +to produce two terms ⌊e⌋, ⌊f⌋ ∈ GExp− such that e ≡ ⌊e⌋, f ≡ ⌊f⌋ and ⌊e⌋ ↔ ⌊f⌋ +in (GExp−, ∂). The latter property tells us that ⌊e⌋ ≡† ⌊f⌋ by Theorem 3.13, +which allows us to conclude e ≡ f. The expression ⌊e⌋ can be thought of as the +early termination version of e, obtained by pruning the branches of its execution +that cannot end in successful termination. +To properly define the transformation ⌊−⌋ on expressions, we need the notion +of a dead state in a skip-free automaton, analogous to a similar notion from [44]. +Definition 6.1. Let (X, h) be a skip-free automaton. The set D(X, h) is the +largest subset of X such for all x ∈ D(X, h) and α ∈ At, either h(x)(α) = ⊥ or +h(x)(α) ∈ Σ × D(X, h). When x ∈ D(X, h), x is dead; otherwise, it is live. +In the sequel, we say e ∈ GExp− is dead when e is a dead state in (GExp−, ∂), +i.e., when e ∈ D(GExp−, ∂). Whether e is dead can be determined by a simple +depth-first search, since e can reach only finitely many expressions by ∂. The +axioms of skip-free GKAT can also tell when a skip-free expression is dead. +Lemma 6.2. Let e ∈ GExp. If e is dead, then e ≡ 0. +We are now ready to define ⌊−⌋, the transformation on expressions promised +above. The intuition here is to prune the dead subterms of e by recursive descent; +whenever we find a part that will inevitably lead to an expression that is never +going to lead to acceptance, we set it to 0. +Definition 6.3. Let e ∈ GExp− and a ∈ BExp. In the sequel we use ae as a +shorthand for e +a 0. We furthermore define ⌊e⌋ inductively, as follows +⌊0⌋ = 0 +⌊p⌋ = p +⌊e1 +a e2⌋ = ⌊e1⌋ +a ⌊e2⌋ +⌊e1 · e2⌋ = +� +0 +e2 is dead +⌊e1⌋ · ⌊e2⌋ +otherwise +⌊e1 +(b)e2⌋ = +� +0 +be2 is dead +⌊e1⌋(b)⌊e2⌋ +otherwise +The transformation defined above yields a term that is ≡-equivalent to e, +provided that we include the early termination axiom e · 0 ≡ 0. The proof is a +simple induction on e, using Lemma 6.2. +Lemma 6.4. For any e ∈ GExp−, e ≡ ⌊e⌋. +It remains to show that if L(e) = L(f), then ⌊e⌋ and ⌊f⌋ are bisimilar. To +this end, we need to relate the language semantics of e and f to their behaviour. +As a first step, we note that behaviour that never leads to acceptance can be +pruned from a skip-free automaton by removing transitions into dead states. +Definition 6.5. Let (X, h) be a skip-free automaton. Define ⌊h⌋ : X → GX by +⌊h⌋(x)(α) = +� +⊥ +h(x)(α) = (p, x′), x′ is dead +h(x)(α) +otherwise + +Moreover, language equivalence of two states in a skip-free automaton implies +bisimilarity of those states, but only in the pruned version of that skip-free +automaton. The proof works by showing that the relation on X that connects +states with the same language is, in fact, a bisimulation in (X, ⌊h⌋). +Lemma 6.6. Let (X, h) be a skip-free automaton, and let x, y ∈ X; we have +L(x, (X, h)) = L(y, (X, h)) =⇒ (X, ⌊h⌋), x ↔ (X, ⌊h⌋), y +The final intermediate property relates the behaviour of to states in the +pruned skip-free automaton of expressions to the syntactic skip-free automaton. +Lemma 6.7. The graph {(e, ⌊e⌋) | e ∈ GExp−} of ⌊−⌋ is a bisimulation of +skip-free automata between (GExp−, ⌊∂⌋) and (GExp−, ∂). +We now have all the ingredients necessary to prove Theorem 3.14. +Theorem 3.14 (Completeness II). If e1 ∼L e2, then e1 ≡ e2. +Proof. If e1 ∼L e2, then by definition L(e1) = L(e2). By Lemma 6.6, e1 ↔ e2 in +(GExp−, ⌊∂⌋), which by Lemma 6.7 implies that ⌊e1⌋ ↔ ⌊e2⌋ in (GExp−, ∂). From +Theorem 3.13 we know that ⌊e1⌋ ≡† ⌊e2⌋, and therefore e1 ≡ e2 by Lemma 6.4. +7 +Relation to GKAT +So far we have seen the technical development of skip-free GKAT without much +reference to the original development of GKAT as it was presented in [44] and [41]. +In this section, we make the case that the semantics of skip-free GKAT is merely +a simplified version of the semantics of GKAT, and that the two agree on which +expressions are equivalent after embedding skip-free GKAT into GKAT. More +precisely, we identify the bisimulation and language semantics of skip-free GKAT +given in Section 3 with instances of the existing bisimulation [41] and lan- +guage [44] semantics of GKAT proper. The main takeaway is that two skip-free +GKAT expressions are equivalent in our semantics precisely when they are equiv- +alent when interpreted as proper GKAT expressions in the existing semantics. +7.1 +Bisimulation semantics +To connect the bisimulation semantics of skip-free GKAT to GKAT at large, we +start by recalling the latter. To do this, we need to define GKAT automata. +Definition 7.1. A (GKAT) automaton is a pair (X, d) such that X is a set and +d : X → (⊥ + ✓ + Σ × X)At is a function called the transition function. We +write x +α|p +−−→ y to denote d(x)(α) = (p, y), x ⇒ α to denote d(x)(α) = ✓, and +x ↓ α if d(x)(α) is undefined. + +α ≤ b +b ⇒ a +α ≤ b +e1 ⇒ α +e1 +b e2 ⇒ α +α ≤ ¯b +e2 ⇒ α +e1 +b e2 ⇒ a +α ≤ b +e1 +α|p +−−→ e′ +e1 +b e2 +α|p +−−→ e′ +α ≤ ¯b +e2 +α|p +−−→ e′ +e1 +b e2 +α|p +−−→ e′ +p +α|p +−−→ 1 +e ⇒ α +e2 ⇒ α +e1 · e2 ⇒ a +e ⇒ α +f +α|p +−−→ e′ +e1 · e2 +α|p +−−→ e′ +e +α|p +−−→ e′ +e1 · e2 +α|p +−−→ e′ � e2 +α ≤ b +e +α|p +−−→ e′ +e(b) +α|p +−−→ e′ � e(b) +α ≤ ¯b +e(b) ⇒ a +Fig. 7. The transition function δ : GExp → (⊥ + ✓ + Σ × GExp)At defined inductively. +Here, e1�e2 is e2 when e = 1 and e1·e2 otherwise, b ∈ BExp, p ∈ Σ, and e, e′, ei ∈ GExp. +Automata can be equipped with their own notion of bisimulation.8 +Definition 7.2. Given automata (X, h) and (Y, k), a bisimulation between them +is a relation R ⊆ X × Y such that if x R y, α ∈ At and p ∈ Σ,: +1. if h(x)(α) = ⊥, then k(y)(α) = ⊥; and +2. if h(x)(α) = ✓, then k(y)(α) = ✓; and +3. if h(x)(α) = (p, x′), then k(y)(α) = (p, y′) such that x′ R y′. +We call x and y bisimilar and write x ↔ y if x R y for some bisimulation R. +Remark 7.3. The properties listed above are implications, but it is not hard to +show that if all three properties hold for R, then so do all of their symmetric +counterparts. For instance, if k(y)(α) = (p, y′), then certainly h(x)(α) must be +of the form (q, x′), which then implies that q = p while x′ R y′. +Two GKAT expressions are bisimilar when they are bisimilar as states in the +syntactic automaton [41], (GExp, δ), summarised in Fig. 7. +Remark 7.4. The definition of δ given above diverges slightly from the definition +in [41]. Fortunately, this does not make a difference in terms of the bisimulation +semantics: two expressions are bisimilar in (GExp, δ) if and only if they are +bisimilar in the original semantics. We refer to ?? for a detailed account. +There is a fairly easy way to convert a skip-free automaton into a GKAT +automaton: simply reroute all accepting transitions into a new state ⊤, that +accepts immediately, and leave the other transitions the same. +Definition 7.5. Given a skip-free automaton (X, d), we define the automaton +embed(X, d) = (X + ⊤, ˜d), where ˜d is defined by +˜d(x)(α) = + + + + + +✓ +x = ⊤ +(p, ⊤) +d(x)(α) = (p, ✓) +d(x)(α) +otherwise +8 As in previous sections, automata can be studied as coalgebras for a given functor +and the notions below are instances of general abstract notions [18,38]. + +We can show that two states are bisimilar in a skip-free automaton if and +only if these same states are bisimilar in the corresponding GKAT automaton. +Lemma 7.6. Let (X, d) be a skip-free automaton, and let x, y ∈ X. +x ↔ y in (X, d) ⇐⇒ x ↔ y in embed(X, d) +The syntactic skip-free automaton (GExp−, ∂) can of course be converted to +a GKAT automaton in this way. It turns out that there is a very natural way of +correlating this automaton to the syntactic GKAT automaton (GExp, δ). +Lemma 7.7. The relation {(e, e) : e ∈ GExp−} ∪ {(⊤, 1)} is a bisimulation +between embed(GExp−, ∂) and (GExp, δ). +We now have everything to relate the bisimulation semantics of skip-free +GKAT expressions to the bisimulation semantics of GKAT expressions at large. +Lemma 7.8. Let e, f ∈ GExp−. The following holds: +e ↔ f in (GExp−, ∂) ⇐⇒ e ↔ f in (GExp, δ) +Proof. We derive using Lemmas 7.6 and 7.7, as follows: since the graph of embed +is a bisimulation, e ↔ f in (GExp−, ∂) iff e ↔ f in embed(GExp−, ∂) if and only +if e ↔ f in (GExp, δ). In the last step, we use the fact that if R is a bisimulation +(of automata) between (X, h) and (Y, k), and S is a bisimulation between (Y, k) +and (Z, ℓ), then R ◦ S is a bisimulation between (X, h) and (Z, ℓ). +7.2 +Language semantics +We now recall the language semantics of GKAT, which is defined in terms of +guarded strings [30], i.e., words in the set At · (Σ · At)∗, where atoms and actions +alternate. In GKAT, successful termination occurs with a trailing associated test, +representing the state of the machine at termination. In an execution of the +sequential composition of two programs e · f, the test trailing the execution of e +needs to match up with an input test compatible with f, otherwise the program +crashes at the end of executing e. The following operations on languages of +guarded strings record this behaviour by matching the ends of traces on the left +with the beginnings of traces on the right. +Definition 7.9. For L, K ⊆ At · (Σ · At)∗, define L ⋄ K = {wαx : wα ∈ L, αx ∈ +K} and L(∗) = � +n∈N L(n), where L(n) is defined inductively by setting L(0) = At +and L(n+1) = L ⋄ L(n). +The language semantics of a GKAT expression is now defined in terms of the +composition operators above, as follows. +Definition 7.10. We define �L : GExp → P(At·(Σ·At)∗) inductively, as follows: +�L(b) = {α ∈ At | α ≤ b} +�L(p) = {αpβ | α, β ∈ At} +�L(e · f) = �L(e) ⋄ �L(f) +�L(e +b f) = �L(b) ⋄ �L(e) ∪ �L(b) ⋄ �L(f) +�L(e(b)) = ( �L(b) ⋄ �L(e))(∗) ⋄ �L(b) + +Guarded Union +Sequencing +Loops +x = x +b x +x +b y = y +¯b x +x +b (y +c z) = (x +b y) +b∨c z +x +b y = bx +b y +(x +b y)z = xz +b yz +x(yz) = (xy)z +0x = 0 +x0 +(†) += 0 +1x = x +x1 = x +xx(b) +b 1 = x(b) +(x +a 1)(b) = (ax)(b) +z = xz +b y +E(x) = 0 +z = x(b)y +Fig. 8. +Axioms for language semantics GKAT (without the Boolean algebra axioms +for tests). The function E : GExp → BExp is defined below. If the axiom marked (†) is +omitted, the above axiomatizes bisimilarity. +This semantics is connected to the relational semantics from Definition 2.2: +Theorem 7.11 ([44]). For e, f ∈ GExp, we have �L(e) = �L(f) if and only if +�e�σ = �f�σ for all relational interpretations σ +Moreover, since skip-free GKAT expressions are also GKAT expressions, this +means that we now have two language interpretations of the former, given by �L +and L. Fortunately, one can easily be expressed in terms of the other. +Lemma 7.12. For e ∈ GExp−, it holds that �L(e) = L(e) · At. +As an easy consequence of the above, we find that the two semantics must +identify the same skip-free GKAT-expressions. +Lemma 7.13. For e, f ∈ GExp−, we have L(e) = L(f) iff �L(e) = �L(f). +By Theorem 3.14, these properties imply that ≡ also axiomatizes relational +equivalence of skip-free GKAT-expressions, as a result. +Corollary 7.14. Let e, f ∈ GExp−, we have e ≡ f if and only if �e�σ = �f�σ +for all relational interpretations σ. +7.3 +Equivalences +Finally, we relate equivalences as proved for skip-free GKAT expressions to those +provable for GKAT expressions, showing that proofs of equivalence for skip-free +GKAT expressions can be replayed in the larger calculus, without (UA). +The axioms of GKAT as presented in [44,41] are provided in Figure 8. We +write e ≈† f when e = f is derivable from the axioms in Figure 8 with the +exception of (†), and e ≈ f when e = f is derivable from the full set. +The last axiom of GKAT is not really a single axiom, but rather an axiom +scheme, parameterized by the function E : GExp → BExp defined as follows: +E(b) = b +E(p) = 0 +E(e +b f) = (b ∧ E(e)) ∨ (b ∧ E(f)) +E(e · f) = E(e) ∧ E(f) +E(e(b)) = b +The function E models the analogue of Salomaa’s empty word property [39]: we +say e is guarded when E(b) is equivalent to 0 by to the laws of Boolean algebra. +Notice that as GKAT expressions, skip-free GKAT expressions are always guarded. + +Since skip-free GKAT expressions are also GKAT expressions, we have four +notions of equivalence for GKAT expressions: as skip-free expressions or GKAT +expressions in general, either with or without (†). These are related as follows. +Theorem 7.15. Let e, f ∈ GExp−. Then (1) e ≈† f if and only if e ≡† f, and +(2) e ≈ f if and only if e ≡ f. +Proof. For the forward direction of (1), we note that if e ≈† f, then e ↔ f in +(GExp, δ) by Theorem 3.12. By Lemma 7.8, e ↔ f in (GExp−, δ) and therefore +e ≡† f by Theorem 3.13. Conversely, note that any proof of e = f by the axioms +of Figure 2 can be replayed using the rules from Figure 8. In particular, the +guardedness condition required for the last skip-free GKAT axiom using the last +GKAT axiom is always true, because E(g) ≈† 0 for any g ∈ GExp−. +The proof of the second claim is similar, but uses Theorem 3.13 instead. +8 +Related Work +This paper fits into a larger research program focused on understanding the +logical and algebraic content of programming. Kleene’s paper introducing the +algebra of regular languages [24] was a foundational contribution to this re- +search program, containing an algebraic account of mechanical programming +and some of its sound equational laws. The paper also contained an interesting +completeness problem: give a complete description of the equations satisfied by +the algebra of regular languages. Salomaa was the first to provide a sound and +complete axiomatization of language equivalence for regular expressions [39]. +The axiomatization in op. cit. included an inference rule with a side condition +that prevented it from being algebraic in the sense that the validity of an equa- +tion is not preserved when substituting letters for arbitrary regular expressions. +Nevertheless, this inspired axiomatizations of several variations and extensions +of Kleene algebra [48,44,43], as well as Milner’s axiomatization of the algebra of +star behaviours [34]. The side condition introduced by Salomaa is often called +the empty word property, an early version of a concept from process theory called +guardedness9 that is also fundamental to the theory of iteration [7]. +Our axiomatization of skip-free GKAT is algebraic due to the lack of a guard- +edness side-condition (it is an equational Horn theory [33]). This is particularly +desirable because it allows for an abundance of other models of the axioms. +Kozen proposed an algebraic axiomatization of Kleene algebra that is sound +and complete for language equivalence [26], which has become the basis for a +number of axiomatizations of other Kleene algebra variants [14,21,22,49] includ- +ing Kleene algebra with tests [27]. KAT also has a plethora of relational models, +which are desirable for reasons we hinted at in Section 2. +9 This is a different use of the word “guarded” than in “guarded Kleene algebra with +tests”. In the context of process theory, a recursive specification is guarded if every +of its function calls occurs within the scope of an operation. + +GKAT is a fragment of KAT, first identified in [31]. It was later given a sound +and complete axiomatization in [44], although the axiomatization is neither al- +gebraic nor finite (it includes (UA), an axiom scheme, shorthand for infinitely +many axioms). It was later shown that dropping x · 0 = 0 (called (S3) in [44]) +from this axiomatization gives a sound and complete axiomatization of bisim- +ilarity [41]. The inspiration for our pruning technique is also in [41], where a +reduction of the language equivalence case to the bisimilarity case is discussed. +Despite the existence of an algebraic axiomatization of language equivalence +in KAT, GKAT has resisted algebraic axiomatization so far. Skip-free GKAT hap- +pens to be a fragment of GKAT in which every expression is guarded, thus +eliminating the need for the side condition in Fig. 8 and allowing for an alge- +braic axiomatization. An inequational axiomatization resembling that of KAT +might be gleaned from the recent preprint [40], but we have not investigated this +carefully. The GKAT axioms for bisimilarity of ground terms can also likely be +obtained from the small-step semantics of GKAT using [1,2,3], but unfortunately +this does not appear to help with the larger completeness problem. +The idea of reducing one completeness problem in Kleene algebra to another +is common in Kleene algebra; for instance, it is the core behind the completeness +proof of KAT [30]. Cohen also reduced weak Kleene algebra as an axiomatization +of star expressions up to simulation to monodic trees [11], whose completeness +was conjectured by Takai and Furusawa [47]. Grabmayer’s solution to the com- +pleteness problem of regular expressions modulo bisimulation [15] can also be +seen as a reduction to the one-free case [16], since his crystallization procedure +produces an automaton that can be solved using the technique found in op. cit. +Other instances of reductions include [10,4,12,49,21,23,32,36,28]. Recent work +has started to study reductions and their compositionality properties [12,22,35]. +9 +Discussion +We continue the study of efficient fragments of Kleene Algebra with Tests (KAT) +initiated in [44], where the authors introduced Guarded Kleene Algebra with +Tests (GKAT) and provided an efficient decision procedure for equivalence. They +also proposed a candidate axiomatization, but left open two questions. +– The first question concerned the existence of an algebraic axiomatization, +which is an axiomatization that is closed under substitution—i.e., where one +can prove properties about a certain program p and then use p as a variable +in the context of a larger program, being able to substitute as needed. This +is essential to enable compositional analysis. +– The second question left open in [44] was whether an axiomatization that +did not require an axiom scheme was possible. Having a completeness proof +that does not require an axiom scheme to reason about mutually dependent +loops is again essential for scalability: we should be able to axiomatize single +loops and generalize this behaviour to multiple, potentially, nested loops. + +In this paper, we identified a large fragment of GKAT, which we call skip-free +GKAT (GKAT−), that can be axiomatized algebraically without relying on an ax- +iom scheme. We show how the axiomatization works well for two types of equiva- +lence: bisimilarity and language equivalence, by proving completeness results for +both semantics. Having the two semantics is interesting from a verification point +of view as it gives access to different levels of precision when analyzing program +behaviour, but also enables a layered approach to the completeness proofs. +We provide a reduction of the completeness proof for language semantics to +the one for bisimilarity. Moreover, the latter is connected to a recently solved [15] +problem proposed by Milner. This approach enabled two things: it breaks down +the completeness proofs and reuse some of the techniques, while also highlighting +the exact difference between the two equivalences (captured by the axiom e·0 ≡ 0 +which does not hold for bisimilarity). We also showed that proofs of equivalence +in skip-free GKAT transfer without any loss to proofs of equivalence in GKAT. +There are several directions for future work. The bridge between process +algebra and Kleene algebra has not been exploited to its full potential. The +fact that we could reuse results by Grabmayer and Fokkink [15,16] was a major +step towards completeness. An independent proof would have been much more +complex and very likely required the development of technical tools resembling +those in [15,16]. We hope the results in this paper can be taken further and more +results can be exchanged between the two communities to solve open problems. +The completeness problem for full GKAT remains open, but our completeness +results for skip-free GKAT are encouraging. We believe they show a path towards +studying whether an algebraic axiomatization can be devised or a negative re- +sult can be proved. A first step in exploring a completeness result would be +to try extending Grabmayer’s completeness result [15] to a setting with output +variables—this is a non-trivial exploration, but we are hopeful will yield new +tools for completeness. As mentioned in the introduction, NetKAT [4] (and its +probabilistic variants [13,45]) have been one of the most successful extensions of +KAT. We believe the step from (skip-free) GKAT to a guarded version of NetKAT +is also a worthwhile exploration. Following [17], we hope to be able to explore +these extensions in a modular and parametric way. +Acknowledgements A. Silva and T. Schmid were partially funded by ERC +grant Autoprobe (grant agreement 101002697). T. Kapp´e was supported by the +EU’s Horizon 2020 research and innovation programme under Marie Sk�lodowska- +Curie grant agreement No. 101027412 (VERLAN). +References +1. 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Zetzsche, S., Silva, A., Sammartino, M.: Guarded Kleene algebra with tests: Au- +tomata learning (2022). https://doi.org/10.48550/arXiv.2204.14153 + diff --git a/ZdFIT4oBgHgl3EQfkiuG/content/tmp_files/load_file.txt b/ZdFIT4oBgHgl3EQfkiuG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..012f7c959cabb3ffd5843a9f950b619ea6db23f9 --- /dev/null +++ b/ZdFIT4oBgHgl3EQfkiuG/content/tmp_files/load_file.txt @@ -0,0 +1,1326 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf,len=1325 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='11301v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='LO] 26 Jan 2023 A Complete Inference System for Skip-free Guarded Kleene Algebra with Tests Todd Schmid1 , Tobias Kapp´e2,3 , and Alexandra Silva4 1 University College London, London, UK 2 Open University of the Netherlands 3 ILLC, University of Amsterdam, NL 4 Cornell University, Ithaca, NY, USA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Guarded Kleene Algebra with Tests (GKAT) is a fragment of Kleene Algebra with Tests (KAT) that was recently introduced to reason efficiently about imperative programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In contrast to KAT, GKAT does not have an algebraic axiomatization, but relies on an analogue of Salomaa’s axiomatization of Kleene Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In this paper, we present an algebraic axiomatization and prove two completeness results for a large fragment of GKAT in the form of skip-free programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 1 Introduction Kleene algebra with tests (KAT) [27] is a logic for reasoning about semantics and equivalence of simple imperative programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' It extends Kleene Algebra (KA) with Boolean control flow, which enables encoding of conditionals and while loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' KAT has been applied to verification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For example, it was used in proof- carrying Java programs [25], in compiler optimization [29], and file systems [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' More recently, KAT was used for reasoning about packet-switched networks, serving as a core to NetKAT [4] and Probabilistic NetKAT [13,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The success of KAT in networking is partly due to its dual nature: it can be used to both specify and verify network properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Moreover, the implementa- tions of NetKAT and ProbNetKAT were surprisingly competitive with state-of- the-art tools [14,46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Part of the surprise with the efficiency of these implemen- tations is that the decision problem for equivalence in both KAT and NetKAT is PSPACE-complete [30,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Further investigations [44] revealed that the tasks performed in NetKAT only make use of a fragment of KAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' It turns out that the difficulty of deciding equivalence in KAT can largely be attributed to the non-deterministic nature of KAT programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If one restricts to KAT programs that operate deterministically with respect to Boolean control flow, the associ- ated decision problem is almost linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This fragment of KAT was first identified in [31] and further explored as guarded Kleene algebra with tests (GKAT) [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The study in [44] proved that the decision problem for GKAT programs is almost linear, and proposed an axiomatization of equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' However, the ax- iomatization suffered from a serious drawback: it included a powerful uniqueness of solutions axiom (UA), which greatly encumbers algebraic reasoning in prac- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In order to use (UA) to show that a pair of programs are equivalent, one needs to find a system of equations that they both satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Even more worry- ingly, the axiomatization contained a fixed-point axiom with a side condition reminiscent of Salomaa’s axiomatization for regular expressions, which is known to be non-algebraic and impair the use of the axiomatic reasoning in context (as substitution of atomic programs is not sound anymore).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The authors of [44] left as open questions whether (UA) can be derived from the other GKAT axioms and whether the non-algebraic side condition can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Despite the attention GKAT has received in recent literature [41,50,43], these questions remain open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In the present work, we offer a partial answer to the questions posed in [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We show that proving the validity of an equivalence in GKAT does not require (UA) if the pair of programs in question are of a particular form, what we call skip-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This fragment of GKAT is expressive enough to capture a large class of programs, and it also provides a better basis for algebraic reasoning: we show that the side condition of the fixed-point axiom can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Our inspiration to look at this fragment came from recent work of Grabmayer and Fokkink’s on the axiomatization of 1-free star expressions modulo bisimulation [16,15], an im- portant stepping stone to solving a decades-open problem posed by Milner [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In a nutshell, our contribution is to identify a large fragment of GKAT, what we call the skip-free fragment, that admits an algebraic axiomatization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We ax- iomatize both bisimilarity and language semantics and provide two completeness proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The first proves completeness of skip-free GKAT modulo bisimulation [41], via a reduction to completeness of Grabmayer and Fokkink’s system [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The second proves completeness of skip-free GKAT w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' language semantics via a reduction to skip-free GKAT modulo bisimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We also show that equivalence proofs of skip-free GKAT expressions (for both semantics) embed in full GKAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The next section contains an introduction to GKAT and an overview of the open problems we tackle in the technical sections of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 2 Overview In this section we provide an overview of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We start with a motivating example of two imperative programs to discuss program equivalence as a verifi- cation technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We then show how GKAT can be used to solve this problem and explore the open questions that we tackle in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Equivalence for Verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In the game Fizz!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Buzz!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' [37], players sit in a circle taking turns counting up from one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Instead of saying any number that is a multiple of 3, players must say “fizz”, and multiples of 5 are replaced with “buzz”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If the number is a multiple both 3 and 5, the player must say “fizz buzz”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Imagine you are asked in a job interview to write a program that prints out the first 100 rounds of a perfect game of Fizz!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Buzz!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='. You write the function fizzbuzz1 as given in Figure 1(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Thinking about the interview later that day, you look up a solution, and you find fizzbuzz2, depicted in Figure 1(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' You suspect that fizzbuzz2 should do the same thing as fizzbuzz1, and after thinking it over for a few minutes, you realize your program could be transformed into the reference solution by a series of transformations that do not change its semantics: def fizzbuzz1 = (i) n := 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' while n ≤ 100 do if 3|n then if not 5|n then print fizz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' n++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' else print fizzbuzz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' n++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' else if 5|n then print buzz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' n++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' else print n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' n++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' print done!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' def fizzbuzz2 = (ii) n := 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' while n ≤ 100 do if 5|n and 3|n then print fizzbuzz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' else if 3|n then print fizz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' else if 5|n then print buzz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' else print n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' n++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' print done!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Two possible specifications of the ideal Fizz!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Buzz!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Place the common action n++ at the end of the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Replace not 5|n with 5|n and swap print fizz with print fizzbuzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Merge the nested branches of 3|n and 5|n into one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Feeling somewhat more reassured, you ponder the three steps above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' It seems like their validity is independent of the actual tests and actions performed by the code;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' for example, swapping the branches of an if - then - else - block while negat- ing the test should be valid under any circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This raises the question: is there a family of primitive transformations that can be used to derive valid ways of rearranging imperative programs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Furthermore, is there an algorithm to decide whether two programs are equivalent under these laws?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Enter GKAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Guarded Kleene Algebra with Tests (GKAT) [44] has been pro- posed as a way of answering the questions above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Expressions in the language of GKAT model skeletons of imperative programs, where the exact meaning of tests and actions is abstracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The laws of GKAT correspond to program trans- formations that are valid regardless of the semantics of tests and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Formally, GKAT expressions are captured by a two-level grammar, generated by a finite set of tests T and a finite set of actions Σ, as follows: BExp ∋ a, b ::= 0 | 1 | t ∈ T | a ∨ b | a ∧ b | a GExp ∋ e, f ::= p ∈ Σ | b | e +b f | e · f | e(b) BExp is the set of Boolean expressions, built from 0 (false), 1 (true), and primitive tests from T , and composed using ∨ (or), ∧ (and) and (not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' GExp is the set of GKAT expressions, built from tests (assert statements) and primitive actions p ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Here, e +b f is a condensed way of writing ‘if b then e else f’, and e(b) is shorthand for ‘while b do e’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' the operator · models sequential composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' By convention, the sequence operator · takes precedence over the operator +b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Abbreviating statements of the form print foo by simply writing foo, Figure 1(i) can be rendered as the GKAT expression (n := 1) · �(fizz · n++ +5|n fizzbuzz · n++) +3|n (buzz · n++ +5|n n · n++) �(n ≤ 100) done!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' (1) Similarly, the program in Figure 1(ii) gives the GKAT expression (n := 1) · ((fizzbuzz +5|n ∧ 3|n (fizz +3|n (buzz +5|n n))) · n++)(n ≤ 100) · done!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' (2) Semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' A moment ago, we stated that GKAT equivalences are intended to witness program equivalence, regardless of how primitive tests and actions are interpreted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We make this more precise by recalling the relational semantics of GKAT programs [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='5 The intuition behind this semantics is that if the possible states of the machine being programmed are modelled by some set S, then tests are predicates on S (comprised of all states where the test succeeds), and actions are relations on S (encoding the changes in state affected by the action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='2 ([44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' A (relational) interpretation is a triple σ = (S, eval, sat) where S is a set, eval : Σ → P(S × S) and sat : T → P(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Each relational interpretation σ gives rise to a semantics �−�σ : GExp → P(S × S), as follows: �0�σ = ∅ �a�σ = �1�σ \\ �a�σ �1�σ = {(s, s) : s ∈ S} �p�σ = eval(p) �t�σ = {(s, s) : s ∈ sat(t)} �e +a f�σ = �a�σ ◦ �e�σ ∪ �a�σ ◦ �f�σ �a ∧ b�σ = �a�σ ∩ �b�σ �e · f�σ = �e�σ ◦ �f�σ �a ∨ b�σ = �a�σ ∪ �b�σ �e(a)�σ = (�a�σ ◦ �e�σ)∗ ◦ �a�σ Here we use ◦ for relation composition and ∗ for reflexive transitive closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If eval(p) is a partial function for every p ∈ Σ, then so is �e�σ for each e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The above therefore also yields a semantics in terms of partial functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The relation �e�σ contains the possible pairs of start and end states of the program e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For instance, the input-output relation of �e +a f� consists of the pairs in �e�σ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' �f�σ) where the start state satisfies a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' violates a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We could model the states of the machine running Fizz!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Buzz!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' as pairs (m, ℓ), where m is the current value of the counter n, and ℓ is a list of words printed so far;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' the accompanying maps sat and eval are given by: sat(k|n) = {(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ℓ) ∈ S : m ≡ 0 mod k} sat(n ≤ k) = {(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ℓ) ∈ S : m ≤ k} eval(n++) = {((m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ℓ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' (m + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ℓ) : (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ℓ) ∈ S} eval(n := k) = {((m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ℓ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ℓ)) : (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ℓ) ∈ S)} eval(w) = {((m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ℓ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ℓw)) : (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ℓ) ∈ S} (w ∈ {fizz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' buzz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' fizzbuzz}) eval(n) = {((m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ℓ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ℓm)) : (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ℓ) ∈ S} For instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' the interpretation of n++ connects states of the form (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ℓ) to states of the form (m + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ℓ)—incrementing the counter by one,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' and leaving the output unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Similarly, print statements append the given string to the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 5 A probabilistic semantics in terms of sub-Markov kernels is also possible [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' On the one hand, this parameterized semantics shows that programs in the GKAT syntax can be given a semantics that corresponds to the intended meaning of their actions and tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' On the other hand, it allows us to quantify over all possible interpretations, and thus abstract from the meaning of the primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' As it happens, two expressions have the same relational semantics under any interpretation if and only if they have the same language semantics [44], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', in terms of languages of guarded strings as used in KAT [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Since equivalence un- der the language semantics is efficiently decidable [44], so is equivalence under all relational interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The decision procedure in [44] uses bisimulation and known results from automata theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' These techniques are good for mecha- nization but hide the algebraic structure of programs that plays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' To expose this, algebraic laws of GKAT program equivalence were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Program transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' GKAT programs are (generalized) regular expres- sions, which are intuitive to reason about and for which many syntactic equiv- alences are known and explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In [44], a set of sound axioms e ≡ f such that �e�σ = �f�σ for all σ was proposed, and it was shown that these can be used to prove a number of useful facts about programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For instance, the following two equivalences are axioms of GKAT: e · g +b f · g ≡ (e +b f) · g f +b e ≡ e +b f The first of these says that common code at the tail end of branches can be factored out, while the second says that the code in branches of a conditional can be swapped, as long as we negate the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Returning to our running example, if we apply the first law to (1) three times (once for each guarded choice), n := 1 · �� (fizzbuzz +5|n fizz) +3|n (buzz +5|n n) � n++ �(n ≤ 100) done!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' (3) Finally, we can apply (e +a f) +b (g +a h) ≡ e +a∧b (f +b (g +a h)), which is provable from the axioms of GKAT, to transform (3) into (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Being able to transform one GKAT program into another using the axioms of GKAT is useful, but the question arises: do the axioms capture all equivalences that hold?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' More specifically, are the axioms of GKAT powerful enough to prove that e ≡ f whenever �e�σ = �f�σ holds for all σ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In [44], a partial answer to the above question is provided: if we extend the laws of GKAT with the uniqueness axiom (UA), then the resulting set of axioms is sound and complete w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' the language semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The problem with this is that (UA) is not really a single axiom, but rather an axiom scheme, which makes both its presentation and application somewhat unwieldy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' To properly introduce (UA), we need the following notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' A left-affine system is defined by expressions e1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' , en,n ∈ GExp and f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' , fn ∈ GExp, along with tests b1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' , bn,n ∈ BExp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' A sequence of expressions s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' , sn ∈ GExp is said to be a solution to this system if ei,1 · s1 +bi,1 ei,2 · s2 +bi,2 · · · +bi,n−1 ei,n +bi,n fi ≡ si (∀i ≤ n) Here, the operations +bi,j associate to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Guarded Union Sequencing Loops x = x +b x x = x +1 y x +b y = y +¯b x x +b (y +c z) = (x +b y) +b∨c z 0x = 0 x0 (†) = 0 x(yz) = (xy)z (x +b y)z = xz +b yz x(b)y = x(x(b)y) +b y z = xz +b y z = x(b)y Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Axioms for language semantics skip-free GKAT (in addition to Boolean algebra axioms for tests, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If the axiom marked † is omitted the above axiomatize a finer semantics, bisimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' A left-affine system called guarded if no ei,j that appears in the system suc- cessfully terminates after reading an atomic test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In other words, each coefficient denotes a productive program, meaning it must execute some action before suc- cessfully terminating—we refer to Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='3 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Stated fully, (UA) says that if expressions s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' , sn and t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' , tn are solutions to the same guarded left-affine system, then si ≡ ti for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' On top of the infinitary nature of (UA), the side condition demanding guard- edness prevents purely algebraic reasoning: replacing action symbols in a valid GKAT equation with arbitrary GKAT expressions might yield an invalid equa- tion!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The situation is analogous to the empty word property used by Salomaa [39] to axiomatize equivalence of regular expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The side condition of guarded- ness appearing in (UA) is inherited from another axiom of GKAT, the fixed-point axiom, which in essence is the unary version of this axiom scheme and explicitly defines the solution of one guarded left-affine equation as a while loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' g ≡ eg +b f =⇒ g ≡ e(b)f if e is guarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Part of the problem of the uniqueness axiom is that the case for general n does not seem to follow easily from the case where n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The problem here is that, unlike the analogous situation for Kleene algebra, there is no general method to transform a left-affine system with n + 1 unknowns into one with n unknowns [31], even if this is possible in certain cases [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We are motivated by two open questions from [44]: – First, can the uniqueness axiom be eliminated?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The other axioms of GKAT contain the instantiation of (UA) for n = 1, which has so far been sufficient in all handwritten proofs of equivalence that we know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Yet (UA) seems to be necessary to complete both known completeness proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' – Second, can we eliminate the guardedness side condition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Kozen [26] showed that Salomaa’s axiomatization is subsumed by a set of axioms that together imply existence and uniqueness of least solutions to systems of equations, but this approach has not yet borne fruit in GKAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Our main contribution is to show that, in a particular fragment of GKAT, both questions can be answered in the positive (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In Section 3, we present what we call the skip-free fragment of GKAT, con- sisting of programs that do not contain assert statements in the body (other than assert false);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' in other words, Boolean statements are restricted to control state- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For this fragment, we show that the axiom scheme (UA) can be avoided entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In fact, this is true for language semantics (as first introduced in [44]) as well as for the bisimulation semantics of [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In Section 4, we provide a bridge to a recent result in process algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In the 80s, Milner offered an alternative interpretation of regular expressions [34], as what he called star behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Based on work of Salomaa from the 1960s [39], Milner proposed a sound axiomatization of the algebra of star behaviours, but left completeness an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' After 38 years, it was recently solved by Clemens Grabmayer [15] following up on his joint work with Fokkink showing that a suitable restriction of Milner’s axioms is complete for the one-free fragment of regular expressions modulo bisimulation [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We leverage their work with an interesting embedding of skip-free GKAT into the one-free regular expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This leads to two completeness results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In Section 5, we start by focusing on the bisimulation semantics of the skip-free fragment, and then in Section 6 expand our argument to its language semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' More precisely, we first provide a reduction of the completeness of skip-free GKAT up to bisimulation to the completeness of Grabmayer and Fokkink’s 1-free regular expressions modulo bisimulation [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We then provide a reduction of the completeness of skip-free GKAT modulo language semantics to the completeness of skip-free GKAT modulo bisimulation via a technique inspired by the tree pruning approach of [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Finally, in Section 7, we connect our semantics of skip-free GKAT expressions to the established semantics of full GKAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We also connect the syntactic proofs between skip-free GKAT expressions in both our axiomatization and the existing one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In conjunction with the results of Sections 5 and 6, the results in Section 7 make a significant step towards answering the question of whether the axioms of GKAT give a complete description of program equivalence, in the positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Proofs are deferred to the appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 3 Introducing Skip-free GKAT The axiom scheme (UA) can be avoided entirely in a certain fragment of GKAT, both for determining bisimilarity and language equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In this section, we give a formal description of the expressions in this fragment and their semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Skip-free expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The fragment of GKAT in focus is the one that excludes sub-programs that may accept immediately, without performing any action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Since these programs can be “skipped” under certain conditions, we call the fragment that avoids them skip-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Among others, it prohibits sub-programs of the form assert b for b ̸= false, but also while false do p, which is equivalent to assert true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Given a set Σ of atomic actions, the set GExp− of skip-free GKAT expressions is given by the grammar GExp− ∋ e1, e2 ::= 0 | p ∈ Σ | e1 +b e2 | e1 · e2 | e(b) 1 e2 where b ranges over the Boolean algebra expressions BExp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' x ∨ 0 = x x ∨ ¯x = 1 x ∨ y = y ∨ x x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) x ∧ 1 = x x ∧ ¯x = 0 x ∧ y = y ∧ x x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The axioms of Boolean algebra [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Unlike full GKAT, in skip-free GKAT the loop construct is treated as a binary operation, analogous to Kleene’s original star operation [24], which was also binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This helps us avoid loops of the form e(b), which can be skipped when b does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The expression e(b) 1 e2 corresponds to e(b) 1 e2 in GKAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Using the same notational shorthand as in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1, the block of code in Figure 1(ii) can be cast as the skip-free GKAT expression (n := 1) · ((fizzbuzz +3|n∧5|n (fizz +3|n (buzz +5|n n))) · n++)(n ≤ 100)done!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Note how we use a skip-free loop of the form e1(b)e2 instead of the looping construct e(b) 1 before concatenating with e2, as was done for GKAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1 Skip-free Semantics There are three natural ways to interpret skip-free GKAT expressions: as au- tomata, as behaviours, and as languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='6 After a short note on Boolean algebra, we shall begin with the automaton interpretation, also known as the small-step semantics, from which the other two can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Boolean algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' To properly present our automata, we need to introduce one more notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Boolean expressions BExp are a syntax for elements of a Boolean algebra, an algebraic structure satisfying the equations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' When a Boolean algebra is freely generated from a finite set of basic tests (T in the case of BExp), it has a finite set At of nonzero minimal elements called atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Atoms are in one-to-one correspondence with sets of tests, and the Boolean algebra is isomorphic to P(At), the sets of subsets of At, equipped with ∨ = ∪, ∧ = ∩, and (−) = At \\ (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In the context of programming, one can think of an atom as a complete description of the machine state, saying which tests are true and which are false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We will denote atoms by the Greek letters α and β, sometimes with indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Given a Boolean expression b ∈ BExp and an atom α ∈ At we say that α entails b, written α ≤ b, whenever α ∨ b = 1, or equivalently α ∨ b = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Throughout the paper, we use the notation • + S where S is a set and • is a symbol to denote the disjoint union (coproduct) of {•} and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The small-step semantics of a skip-free GKAT expression uses a special type of deterministic automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' A skip-free automaton is a pair (X, h), where X is a set of states and h: X → (⊥ + Σ × (✓ + X))At is a transition structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' At every x ∈ X and for any α ∈ At, one of three things can happen: 6 We will connect these to the relational semantics from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='2 in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' p α|p −−→ ✓ e1 α|p −−→ e′ α ≤ b e1 +b e2 α|p −−→ e′ e2 α|p −−→ e′ α ̸≤ b e1 +b e2 α|p −−→ e′ e1 α|p −−→ e′ e1e2 α|p −−→ e′e2 e1 α|p −−→ ✓ e1e2 α|p −−→ e2 e1 α|p −−→ e′ α ≤ b e(b) 1 e2 α|p −−→ e′(e(b) 1 e2) e1 α|p −−→ ✓ α ≤ b e(b) 1 e2 α|p −−→ e(b) 1 e2 e2 α|p −−→ e′ α ̸≤ b e(b) 1 e2 α|p −−→ e′ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The small-step semantics of skip-free GKAT expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' h(x)(α) = (p, y), which we write as x α|p −−→ y, means the state x under α makes a transition to a new state y, after performing the action p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' h(x)(α) = (p, ✓), which we write x α|p −−→ ✓, means the state x under α successfully terminates with action p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' h(x)(α) = ⊥, which we write x ↓ α, means the state x under α terminates with failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Often we will leave these outputs implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='3 (Automaton of expressions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We equip the set GExp− of all skip-free GKAT expressions with an automaton structure (GExp−, ∂) given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 4, representing step-by-step execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Given e ∈ GExp−, we denote the set of states reachable from e by ⟨e⟩ and call this the small-step semantics of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The small-step semantics of skip-free GKAT expressions is inspired by Brzo- zowski’s derivatives [8], which provide an automata-theoretic description of the step-by-step execution of a regular expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Our first lemma tells us that, like regular expressions, skip-free GKAT expressions correspond to finite automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For any e ∈ GExp−, ⟨e⟩ has finitely many states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The automaton that arises from the program fizzbuzz2 is below, with a = n ≤ 100, b = 3|n, and c = 5|n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The expression e is the same as in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='2, e1 is the same as e but without the action n := 0 in front, and e2 = n++ · e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We also adopt the convention of writing x b|p −−→ x′ where b ∈ BExp to represent all transitions x α|p −−→ x′ where α ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' e e1 e2 ✓ 1 | n := 1 abc | fizzbuzz, ab¯c | fizz, a¯bc | buzz, a¯b¯c | n 1 | n++ ¯a | done!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The automaton interpretation of a skip-free GKAT expression (its small-step semantics) provides an intuitive visual depiction of the details of its execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This is a useful view on the operational semantics of expressions, but sometimes one might want to have a more precise description of the global behaviour of the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The remaining two interpretations of skip-free GKAT expressions aim to capture two denotational semantics of expressions: one finer, bisimilarity, that makes a distinction on the branching created by how its states respond to atomic tests, which actions can be performed, and when successful termination and crashes occur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' another coarser, language semantics, that assigns a language of traces to each expression capturing all sequences of actions that lead to successful termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The key difference between these two semantics will be their ability to distinguish programs that crash early in the execution versus programs that crash later—this will become evident in the axiomatizations of both semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We start by presenting the language semantics as this is the more traditional one associated with GKAT (and regular) expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Language semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Formally, a (skip-free) guarded trace is a nonempty string of the form α1p1 · · · αnpn, where each αi ∈ At and pi ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Intuitively, each αi captures the state of program variables needed to execute program ac- tion pi and the execution of each pi except the last yields a new program state αi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' A skip-free guarded language is a set of guarded traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Skip-free guarded languages should be thought of as sets of strings denoting successfully terminating computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='6 (Language acceptance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In a skip-free automaton (X, h) with a state x ∈ X, the language accepted by x is the skip-free guarded language L(x, (X, h)) = {α1p1 · · · αnpn | x α1|p1 −−−→ x1 −→ · · · −→ xn αn|pn −−−−→ ✓} If (X, h) is clear from context, we will simply write L(x) instead of L(x, (X, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If L(x) = L(y), we write x ∼L y and say that x and y are language equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Each skip-free GKAT expression is a state in the automaton of expressions (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='3) and therefore accepts a language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The language accepted by a skip-free GKAT expression is the set of successful runs of the program it denotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Analogously to GKAT, we can describe this language inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Given an expression e ∈ GExp−, the language accepted by e in (GExp−, ∂), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', L(e) = L(e, (GExp−, ∂)) can be characterized as follows: L(0) = ∅ L(p) = {αp | α ∈ At} L(e1 +b e2) = bL(e1) ∪ ¯bL(e2) L(e1 · e2) = L(e1) · L(e2) L(e(b) 1 e2) = � n∈N (bL(e1))n · ¯bL(e2) Here, we write bL = {αpw ∈ L | α ≤ b} and L1 · L2 = {wx : w ∈ L1, x ∈ L2}, while L0 = {ǫ} (where ǫ denotes the empty word) and Ln+1 = L · Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='7 provides a way of computing the language of an expression e without having to generate the automaton for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Bisimulation semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Another, finer, notion of equivalence that we can associate with skip-free automata is bisimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Given skip-free automata (X, h) and (Y, k), a bisimulation is a relation R ⊆ X × Y such that for any x R y, α ∈ At and p ∈ Σ: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' x ↓ α if and only if y ↓ α, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' x α|p −−→ ✓ if and only if y α|p −−→ ✓, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' for any x′ R y′, x α|p −−→ x′ if and only if y α|p −−→ y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We call x and y bisimilar if x R y for some bisimulation R and write x ↔ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In a fixed skip-free automaton (X, h), we define ↔ ⊆ X × X is the largest bisimulation, called bisimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='7 The bisimilarity equivalence class of a state is often called its behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In the automaton below, x1 and x2 are bisimilar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This is witnessed by the bisimulation {(x1, x2), (x2, x2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' x1 x2 ✓ a | p ¯a | q a | p ¯a | q We can also use bisimulations to witness language equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Let e1, e2 ∈ GExp−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If e1 ↔ e2, then L(e1) = L(e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The converse of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='10 is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Consider, for example, the program p(1)q that repeats the atomic action p ∈ Σ indefinitely, never reaching q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Since L(p(1)q) = � n∈N L(p)n · ∅ = ∅ = L(0) we know that p(1)q ∼L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' But p(1)q and 0 are not bisimilar, since Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 4 tells us that p(1)q α|p −−→ p(1)q and 0 ↓ α, which together refute Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='2 Axioms Next, we give an inference system for bisimilarity and language equivalence con- sisting of equations and equational inference rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The axioms of skip-free GKAT are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' They include the equation (†), which says that early deadlock is the same as late deadlock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This is sound with respect to the language interpre- tation, meaning that (†) is true if x is replaced with a skip-free guarded language, but it is not sound with respect to the bisimulation semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For example, the expressions p · 0 and 0 are not bisimilar for any p ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Interestingly, this is the only axiomatic difference between bisimilarity and language equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The underlying logical structure of our inference systems is equa- tional logic [6], meaning that provable equivalence is an equivalence relation that is preserved by the algebraic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 7 This follows directly from seeing skip-free automata as a special type of coalgebra and the fact that the functor involved preserves weak pullbacks [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In fact, coalgebra has been an indispensable tool in the production of the current paper, guiding us to the correct definitions and simplifying many of the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Given expressions e1, e2 ∈ GExp−, we write e1 ≡† e2 and say that e1 and e2 are ≡†-equivalent if the equation e1 = e2 can be derived from the axioms in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 2 without the axiom marked (†).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We write e1 ≡ e2 and say that e1 and e2 are ≡-equivalent if e1 = e2 can be derived from the whole set of axioms in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The axioms in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 2 are sound with respect to the respective semantics they axiomatize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The only axiom that is not sound w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' bisimilarity is e · 0 ≡ 0, as this would relate automata with different transition structure (as e may permit some action to be performed, and this is observable in the bisimulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='12 (Soundness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For any e1, e2 ∈ GExp−, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If e1 ≡† e2, then e1 ↔ e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If e1 ≡ e2, then e1 ∼L e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We consider the next two results, which are jointly converse to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='12, to be the main theorems of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' They state that the axioms in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 2 are complete for bisimilarity and language equivalence respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', they describe a complete set of program transformations for skip-free GKAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='13 (Completeness I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If e1 ↔ e2, then e1 ≡† e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='14 (Completeness II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If e1 ∼L e2, then e1 ≡ e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='13 in Section 5 by drawing a formal analogy between skip-free GKAT and a recent study of regular expressions in the context of process algebra [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We include a short overview of this recent work in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We delay the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='14 to Section 6, which uses a separate technique based on the pruning method introduced in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 4 1-free Star Expressions Regular expressions were introduced by Kleene [24] as a syntax for the algebra of regular events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Milner offered an alternative interpretation of regular expres- sions [34], as what he called star behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Based on work of Salomaa [39], Milner proposed a sound axiomatization of the algebra of star behaviours, but left completeness an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' After nearly 40 years of active research from the process algebra community, a solution was finally found by Grabmayer [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' A few years before this result, Grabmayer and Fokkink proved that a suit- able restriction of Milner’s axioms gives a complete inference system for the behaviour interpretation of a fragment of regular expressions, called the one- free fragment [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In this section, we give a quick overview of Grabmayer and Fokkink’s one-free fragment [16], slightly adapted to use an alphabet that will be suitable to later use in one of the completeness proofs of skip-free GKAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' αp αp −−→ ✓ r1 αp −−→ r′ r1 + r2 αp −−→ r′ r2 αp −−→ r′ r1 + r2 αp −−→ r′ r1 αp −−→ r′ r1r2 αp −−→ r′r2 r1 αp −−→ ✓ r1r2 αp −−→ r2 r1 αp −−→ r′ r1 ∗ r2 αp −−→ r′(r1 ∗ r2) r1 αp −−→ ✓ r1 ∗ r2 αp −−→ r1 ∗ r2 r2 αp −−→ x r1 ∗ r2 αp −−→ x Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The small-step semantics of one-free star expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In the process algebra literature [34,16,15], regular expressions gen- erated by a fixed alphabet A are called star expressions, and denote labelled transition systems (LTSs) with labels drawn from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' As was mentioned in Sec- tion 3, skip-free automata can be seen as certain LTSs where the labels are atomic test/atomic action pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In Section 5, we encode skip-free GKAT expres- sions as one-free regular expressions and skip-free automata as LTSs with labels drawn from At · Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We instantiate the construction from [16] of the set of star expressions generated by the label set At · Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The set SExp of one-free star expressions is given by SExp ∋ r1, r2 ::= 0 | αp ∈ At · Σ | r1 + r2 | r1r2 | r1 ∗ r2 Semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The semantics of SExp is now an instance of the labelled transition systems that originally appeared in [16], with atomic test/atomic action pairs as labels and a (synthetic) output state ✓ denoting successful termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For the rest of this paper, we call a pair (S, t) a labelled transition system when S is a set of states and t: S → P(At·Σ ×(✓+S)) is a transition structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We write x αp −−→ y if (αp, y) ∈ t(x) and x αp −−→ ✓ if (αp, ✓) ∈ t(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The set SExp can be given the structure of a labelled transition system (SExp, τ), defined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If r ∈ SExp, we write ⟨r⟩ for the transition sys- tem obtained by restricting τ to the one-free star expressions reachable from r and call ⟨r⟩ the small-step semantics of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The bisimulation interpretation of one-free star expressions is subtler than the bisimulation interpretation of skip-free GKAT expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The issue is that labelled transition systems (LTSs) are nondeterministic in general: it is possible for an LTS to have both a x αp −−→ y and a x αq −→ z transition for p ̸= q or y ̸= z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The appropriate notion of bisimilarity for LTSs can be given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Given labelled transition systems (S, t) and (T, u), a bisimula- tion between them is a relation R ⊆ S × T s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' for any x R y and αp ∈ At · Σ, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' x αp −−→ ✓ if and only if y αp −−→ ✓, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' if x αp −−→ x′, then there exist x′ R y′ such that y αp −−→ y′, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' if y αp −−→ y′, then there exist x′ R y′ such that x αp −−→ x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' As before, we denote the largest bisimulation by ↔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We call x and y bisimilar and write x ↔ y if x R y for some bisimulation R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Union Sequencing Loops x = x + x x = x + 0 x + y = y + x x + (y + z) = (x + y) + z 0x = 0 x(yz) = (xy)z (x + y)z = xz + yz x ∗ y = x(x ∗ y) + y z = xz + y z = x ∗ y Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Axioms for equivalence for one-free star expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The following closure properties of bisimulations of LTSs are useful later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' They also imply that bisimilarity is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Like in the skip-free case, the bisimilarity equivalence class of a state is called its behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Let (S, t), (T, u), and (U, v) be labelled transition systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Fur- thermore, let R1, R2 ⊆ S × T and R3 ⊆ T × U be bisimulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Then Rop 1 = {(y, x) | x R1 y}, R1 ∪ R2 and R1 ◦ R3 are bisimulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Axiomatization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We follow [16], where it was shown that the axiomatization found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 6 is complete with respect to bisimilarity for one-free star expres- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Given a pair r1, r2 ∈ SExp, we write r1 ≡∗ r2 and say that r1 and r2 are ≡∗-equivalent if the equation r1 = r2 can be derived from the axioms in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The next result is crucial to the next section, where we prove that the axioms of ≡† are complete with respect to bisimilarity in skip-free GKAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='4 ([16, Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' r1 ↔ r2 if and only if r1 ≡∗ r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 5 Completeness for Skip-free Bisimulation GKAT This section is dedicated to the proof of our first completeness result, Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='13, which says that the axioms of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 2 (excluding †) are complete with respect to bisimilarity in skip-free GKAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Our proof strategy is a reduction of our completeness result to the completeness result for SExp (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The key objects of interest in the reduction are a pair of translations: one translation turns skip-free GKAT expressions into one-free star expressions and maintains bisimilarity, and the other translation turns (certain) one-free star expressions into skip-free GKAT expressions and maintains provable bisimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We first discuss the translation between automata and labelled transition sys- tems, which preserves and reflects bisimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We then introduce the syntactic translations and present the completeness proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1 Transforming skip-free automata to labelled transition systems We can easily transform a skip-free automaton into an LTS by essentially turning α|p −−→ transitions into αp −−→ transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This can be formalized, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Given a set X, we define grphX : (⊥ + Σ × (✓ + X))At → P(At·Σ ×(✓+X)) to be grphX(θ) = {(αp, x) | θ(α) = (p, x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Given a skip-free automaton (X, h), we define grph∗(X, h) = (X, grphX ◦ h) The function grphX is injective: as its name suggests, grphX(θ) is essentially the graph of θ when viewed as a partial function from At to Σ × (✓ + X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This implies that the transformation grph∗ of skip-free automata into LTSs preserves and reflects bisimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Let x, y ∈ X, and (X, h) be a skip-free automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Then x ↔ y in (X, h) if and only if x ↔ y in grph∗(X, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Leading up to the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='13, we also need to undo the effect of grph∗ on skip-free automata with a transformation that takes every LTS of the form grph∗(X, h) to its underlying skip-free automaton (X, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The LTSs that can be written in the form grph∗(X, h) for some skip-free automaton (X, h) can be described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Call a set U ∈ P(At·Σ ×(✓+X)) graph-like if whenever (αp, x) ∈ U and (αq, y) ∈ U, then p = q and x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' An LTS (S, t) is deterministic if t(s) is graph-like for every s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' An LTS (S, t) is deterministic if and only if (S, t) = grph∗(X, h) for some skip-free automaton (X, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For any set X, let PdetX = {U ∈ P(At · Σ × (✓ + X)) | U is graph-like} and define funcX : PdetX → P(At · Σ × (✓ + X)) by funcX(U)(α) = (p, x) if (αp, x) ∈ U Let (S, t) be a deterministic LTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Then func∗(S, t) = (S, funcS ◦ t) is a skip-free automaton such that grph∗func∗(S, t) = (S, t), because grphS ◦ funcS = idPdet(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' As mentioned in Footnote 7, there is a coalgebraic outlook in many of the technical details in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For the interested reader, the maps grph and func are actually natural transformations between the functors whose coalgebras correspond to skip-free automata and labelled transitions, and are furthermore inverse to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This implies that grph∗ and func∗ witness an isomorphism between the categories of skip-free automata and LTSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='2 Translating Syntax We can mimic the transformation of skip-free automata into deterministic la- belled transition systems and vice-versa by a pair of syntactic translations going back and forth between skip-free GKAT expressions and certain one-free star expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Similar to how only some labelled transition systems can be turned into skip-free automata, only some one-free star expressions have corresponding skip-free GKAT expressions—the deterministic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The definition of deterministic expressions requires the following notation: given a test b ∈ BExp, we define b · r inductively on r ∈ SExp as follows: b · 0 = 0 b · αp = � αp α ≤ b 0 α ̸≤ b b · (r1 + r2) = b · r1 + b · r2 b · (r1r2) = (b · r1)r2 b · (r1 ∗ r2) = (b · r1)(r1 ∗ r2) + b · r2 for any αp ∈ At · Σ and r1, r2 ∈ SExp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The set of deterministic one-free star expressions is the small- est subset Det ⊆ SExp such that 0 ∈ Det and αp ∈ Det for any α ∈ At and p ∈ Σ, and for any r1, r2 ∈ Det, and b ∈ BExp, b·r1+¯b·r2, r1r2, and (b·r1)∗(¯b·r2) ∈ Det.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' From GExp− to Det.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We can now present the translations of skip-free expres- sions to deterministic one-free star expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We define the translation function gtr : GExp− → Det by gtr(0) = 0 gtr(p) = � α∈At αp gtr(e1 +b e2) = b · gtr(e1) + ¯b · gtr(e2) gtr(e1 · e2) = gtr(e1) gtr(e2) gtr(e(b) 1 e2) = (b · e1) ∗ (¯b · e2) for any b ∈ BExp, p ∈ Σ, e1, e2 ∈ GExp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='6, we make use of a generalized sum � α∈At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Tech- nically, this requires we fix an enumeration of At ahead of time, say At = {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' , αn}, at which point we can define � α∈At rα = rα1 + · · · + rαn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Of course, + is commutative and associative up to ≡∗, the actual ordering of this sum does not matter as far as equivalence is concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The most prescient feature of this translation is that it respects bisimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The graph of the translation function gtr is a bisimulation of la- belled transition systems between grph∗(GExp−, ∂) and (SExp, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Consequently, if e1 ↔ e2 in grph∗(GExp−, ∂), then gtr(e1) ↔ gtr(e2) in (SExp, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' From Det to GExp−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Next we translate deterministic one-free star expressions into skip-free GKAT expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For any e ∈ GExp−, gtr(e) ∈ Det.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We would now like to define a back translation function rtg : Det → GExp− by induction on its argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Looking at Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='5, one might be tempted to write rtg(b·r1 +¯b·r2) = rtg(r1)+b rtg(r2), but the fact of the matter is that it is possible for there to be distinct b, c ∈ BExp such that b·r1+¯b·r2 = c·r1 +¯c·r2, even when b and c have different atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Say that r1, r2 ∈ SExp are separated by b ∈ BExp if r1 = b·r1 and r2 = ¯b · r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If such a b exists we say that r1 and r2 are separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Another way to define Det is therefore to say that Det is the smallest subset of SExp containing 0 and At · Σ that is closed under sequential composition and closed under unions and stars of separated one-free star expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Suppose r1 and r2 are separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Since there are only finitely many Boolean expressions up to equivalence, there is a maximal (weakest) test b(r1, r2) ∈ BExp such that r1 and r2 are separated by b(r1, r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The back translation rtg : Det → GExp− is defined by rtg(0) = 0 rtg(αp) = p +α 0 rtg(r1 + r2) = rtg(r1) +b(r1,r2) rtg(r2) rtg(r1r2) = rtg(r1) · rtg(r2) rtg(r1 ∗ r2) = rtg(r1)(b(r1,r2)) rtg(r2) for any r1, r2 ∈ SExp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In the union and star cases, we may use that r1 and r2 are separated (by definition of Det), so that b(r1, r2) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The most prescient property of rtg is that it preserves provable equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Let r1, r2 ∈ Det.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If r1 ≡∗ r2, then rtg(r1) ≡† rtg(r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The last fact needed in the proof of completeness is that, up to provable equivalence, every skip-free GKAT expression is equivalent to its back-translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For any e ∈ GExp−, e ≡† rtg(gtr(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We are now ready to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='13, that provable bisimilarity is com- plete with respect to behavioural equivalence in skip-free GKAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='13 (Completeness I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If e1 ↔ e2, then e1 ≡† e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Let e1, e2 ∈ GExp be a bisimilar pair of skip-free GKAT expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='2, e1 and e2 are bisimilar in grph∗(GExp−, ∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' By Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='8, the translation gtr : grph∗(GExp−, ∂) → (SExp, τ) preserves bisimilar- ity, so gtr(e1) and gtr(e2) are bisimilar in (SExp, τ) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='4, gtr(e1) ≡∗ gtr(e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Therefore, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='12, rtg(gtr(e1)) ≡† rtg(gtr(e2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Fi- nally, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='13, we have e1 ≡† rtg(gtr(e1)) ≡† rtg(gtr(e2)) ≡† e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 6 Completeness for Skip-free GKAT The previous section establishes that ≡†-equivalence coincides with bisimilarity for skip-free GKAT expressions by reducing the completeness problem of skip- free GKAT up to bisimilarity to a solved completeness problem, namely that of one-free star expressions up to bisimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In this section we prove a complete- ness result for skip-free GKAT up to language equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We show this can be achieved by reducing it to the completeness problem of skip-free GKAT up to bisimilarity, which we just solved in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Despite bisimilarity being a less traditional equivalence in the context of Kleene algebra, this reduction simplifies the completeness proof greatly, and justifies the study of bisimilarity in the pursuit of completeness for GKAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The axiom x · 0 = 0 (which is the only difference between skip-free GKAT up to language equivalence and skip-free GKAT up to bisimilarity) indicates that the only semantic difference between bisimilarity and language equivalence in skip- free GKAT is early termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This motivates our reduction to skip-free GKAT up to bisimilarity below, which involves reducing each skip-free expression to an expression representing only the successfully terminating branches of execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Now let us turn to the formal proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='14, which says that if e, f ∈ GExp− are such that L(e) = L(f), then e ≡ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In a nutshell, our strategy is to produce two terms ⌊e⌋, ⌊f⌋ ∈ GExp− such that e ≡ ⌊e⌋, f ≡ ⌊f⌋ and ⌊e⌋ ↔ ⌊f⌋ in (GExp−, ∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The latter property tells us that ⌊e⌋ ≡† ⌊f⌋ by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='13, which allows us to conclude e ≡ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The expression ⌊e⌋ can be thought of as the early termination version of e, obtained by pruning the branches of its execution that cannot end in successful termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' To properly define the transformation ⌊−⌋ on expressions, we need the notion of a dead state in a skip-free automaton, analogous to a similar notion from [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Let (X, h) be a skip-free automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The set D(X, h) is the largest subset of X such for all x ∈ D(X, h) and α ∈ At, either h(x)(α) = ⊥ or h(x)(α) ∈ Σ × D(X, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' When x ∈ D(X, h), x is dead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' otherwise, it is live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In the sequel, we say e ∈ GExp− is dead when e is a dead state in (GExp−, ∂), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', when e ∈ D(GExp−, ∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Whether e is dead can be determined by a simple depth-first search, since e can reach only finitely many expressions by ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The axioms of skip-free GKAT can also tell when a skip-free expression is dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Let e ∈ GExp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If e is dead, then e ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We are now ready to define ⌊−⌋, the transformation on expressions promised above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The intuition here is to prune the dead subterms of e by recursive descent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' whenever we find a part that will inevitably lead to an expression that is never going to lead to acceptance, we set it to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Let e ∈ GExp− and a ∈ BExp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In the sequel we use ae as a shorthand for e +a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We furthermore define ⌊e⌋ inductively, as follows ⌊0⌋ = 0 ⌊p⌋ = p ⌊e1 +a e2⌋ = ⌊e1⌋ +a ⌊e2⌋ ⌊e1 · e2⌋ = � 0 e2 is dead ⌊e1⌋ · ⌊e2⌋ otherwise ⌊e1 (b)e2⌋ = � 0 be2 is dead ⌊e1⌋(b)⌊e2⌋ otherwise The transformation defined above yields a term that is ≡-equivalent to e, provided that we include the early termination axiom e · 0 ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The proof is a simple induction on e, using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For any e ∈ GExp−, e ≡ ⌊e⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' It remains to show that if L(e) = L(f), then ⌊e⌋ and ⌊f⌋ are bisimilar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' To this end, we need to relate the language semantics of e and f to their behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' As a first step, we note that behaviour that never leads to acceptance can be pruned from a skip-free automaton by removing transitions into dead states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Let (X, h) be a skip-free automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Define ⌊h⌋ : X → GX by ⌊h⌋(x)(α) = � ⊥ h(x)(α) = (p, x′), x′ is dead h(x)(α) otherwise Moreover, language equivalence of two states in a skip-free automaton implies bisimilarity of those states, but only in the pruned version of that skip-free automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The proof works by showing that the relation on X that connects states with the same language is, in fact, a bisimulation in (X, ⌊h⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Let (X, h) be a skip-free automaton, and let x, y ∈ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' we have L(x, (X, h)) = L(y, (X, h)) =⇒ (X, ⌊h⌋), x ↔ (X, ⌊h⌋), y The final intermediate property relates the behaviour of to states in the pruned skip-free automaton of expressions to the syntactic skip-free automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The graph {(e, ⌊e⌋) | e ∈ GExp−} of ⌊−⌋ is a bisimulation of skip-free automata between (GExp−, ⌊∂⌋) and (GExp−, ∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We now have all the ingredients necessary to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='14 (Completeness II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If e1 ∼L e2, then e1 ≡ e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If e1 ∼L e2, then by definition L(e1) = L(e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='6, e1 ↔ e2 in (GExp−, ⌊∂⌋), which by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='7 implies that ⌊e1⌋ ↔ ⌊e2⌋ in (GExp−, ∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='13 we know that ⌊e1⌋ ≡† ⌊e2⌋, and therefore e1 ≡ e2 by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 7 Relation to GKAT So far we have seen the technical development of skip-free GKAT without much reference to the original development of GKAT as it was presented in [44] and [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In this section, we make the case that the semantics of skip-free GKAT is merely a simplified version of the semantics of GKAT, and that the two agree on which expressions are equivalent after embedding skip-free GKAT into GKAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' More precisely, we identify the bisimulation and language semantics of skip-free GKAT given in Section 3 with instances of the existing bisimulation [41] and lan- guage [44] semantics of GKAT proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The main takeaway is that two skip-free GKAT expressions are equivalent in our semantics precisely when they are equiv- alent when interpreted as proper GKAT expressions in the existing semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1 Bisimulation semantics To connect the bisimulation semantics of skip-free GKAT to GKAT at large, we start by recalling the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' To do this, we need to define GKAT automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' A (GKAT) automaton is a pair (X, d) such that X is a set and d : X → (⊥ + ✓ + Σ × X)At is a function called the transition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We write x α|p −−→ y to denote d(x)(α) = (p, y), x ⇒ α to denote d(x)(α) = ✓, and x ↓ α if d(x)(α) is undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' α ≤ b b ⇒ a α ≤ b e1 ⇒ α e1 +b e2 ⇒ α α ≤ ¯b e2 ⇒ α e1 +b e2 ⇒ a α ≤ b e1 α|p −−→ e′ e1 +b e2 α|p −−→ e′ α ≤ ¯b e2 α|p −−→ e′ e1 +b e2 α|p −−→ e′ p α|p −−→ 1 e ⇒ α e2 ⇒ α e1 · e2 ⇒ a e ⇒ α f α|p −−→ e′ e1 · e2 α|p −−→ e′ e α|p −−→ e′ e1 · e2 α|p −−→ e′ � e2 α ≤ b e α|p −−→ e′ e(b) α|p −−→ e′ � e(b) α ≤ ¯b e(b) ⇒ a Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The transition function δ : GExp → (⊥ + ✓ + Σ × GExp)At defined inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Here, e1�e2 is e2 when e = 1 and e1·e2 otherwise, b ∈ BExp, p ∈ Σ, and e, e′, ei ∈ GExp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Automata can be equipped with their own notion of bisimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='8 Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Given automata (X, h) and (Y, k), a bisimulation between them is a relation R ⊆ X × Y such that if x R y, α ∈ At and p ∈ Σ,: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' if h(x)(α) = ⊥, then k(y)(α) = ⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' if h(x)(α) = ✓, then k(y)(α) = ✓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' if h(x)(α) = (p, x′), then k(y)(α) = (p, y′) such that x′ R y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We call x and y bisimilar and write x ↔ y if x R y for some bisimulation R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The properties listed above are implications, but it is not hard to show that if all three properties hold for R, then so do all of their symmetric counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For instance, if k(y)(α) = (p, y′), then certainly h(x)(α) must be of the form (q, x′), which then implies that q = p while x′ R y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Two GKAT expressions are bisimilar when they are bisimilar as states in the syntactic automaton [41], (GExp, δ), summarised in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The definition of δ given above diverges slightly from the definition in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Fortunately, this does not make a difference in terms of the bisimulation semantics: two expressions are bisimilar in (GExp, δ) if and only if they are bisimilar in the original semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We refer to ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' for a detailed account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' There is a fairly easy way to convert a skip-free automaton into a GKAT automaton: simply reroute all accepting transitions into a new state ⊤, that accepts immediately, and leave the other transitions the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Given a skip-free automaton (X, d), we define the automaton embed(X, d) = (X + ⊤, ˜d), where ˜d is defined by ˜d(x)(α) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ✓ x = ⊤ (p, ⊤) d(x)(α) = (p, ✓) d(x)(α) otherwise 8 As in previous sections, automata can be studied as coalgebras for a given functor and the notions below are instances of general abstract notions [18,38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We can show that two states are bisimilar in a skip-free automaton if and only if these same states are bisimilar in the corresponding GKAT automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Let (X, d) be a skip-free automaton, and let x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' x ↔ y in (X, d) ⇐⇒ x ↔ y in embed(X, d) The syntactic skip-free automaton (GExp−, ∂) can of course be converted to a GKAT automaton in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' It turns out that there is a very natural way of correlating this automaton to the syntactic GKAT automaton (GExp, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The relation {(e, e) : e ∈ GExp−} ∪ {(⊤, 1)} is a bisimulation between embed(GExp−, ∂) and (GExp, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We now have everything to relate the bisimulation semantics of skip-free GKAT expressions to the bisimulation semantics of GKAT expressions at large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Let e, f ∈ GExp−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The following holds: e ↔ f in (GExp−, ∂) ⇐⇒ e ↔ f in (GExp, δ) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We derive using Lemmas 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='7, as follows: since the graph of embed is a bisimulation, e ↔ f in (GExp−, ∂) iff e ↔ f in embed(GExp−, ∂) if and only if e ↔ f in (GExp, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In the last step, we use the fact that if R is a bisimulation (of automata) between (X, h) and (Y, k), and S is a bisimulation between (Y, k) and (Z, ℓ), then R ◦ S is a bisimulation between (X, h) and (Z, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='2 Language semantics We now recall the language semantics of GKAT, which is defined in terms of guarded strings [30], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', words in the set At · (Σ · At)∗, where atoms and actions alternate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In GKAT, successful termination occurs with a trailing associated test, representing the state of the machine at termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In an execution of the sequential composition of two programs e · f, the test trailing the execution of e needs to match up with an input test compatible with f, otherwise the program crashes at the end of executing e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The following operations on languages of guarded strings record this behaviour by matching the ends of traces on the left with the beginnings of traces on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For L, K ⊆ At · (Σ · At)∗, define L ⋄ K = {wαx : wα ∈ L, αx ∈ K} and L(∗) = � n∈N L(n), where L(n) is defined inductively by setting L(0) = At and L(n+1) = L ⋄ L(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The language semantics of a GKAT expression is now defined in terms of the composition operators above, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We define �L : GExp → P(At·(Σ·At)∗) inductively, as follows: �L(b) = {α ∈ At | α ≤ b} �L(p) = {αpβ | α, β ∈ At} �L(e · f) = �L(e) ⋄ �L(f) �L(e +b f) = �L(b) ⋄ �L(e) ∪ �L(b) ⋄ �L(f) �L(e(b)) = ( �L(b) ⋄ �L(e))(∗) ⋄ �L(b) Guarded Union Sequencing Loops x = x +b x x +b y = y +¯b x x +b (y +c z) = (x +b y) +b∨c z x +b y = bx +b y (x +b y)z = xz +b yz x(yz) = (xy)z 0x = 0 x0 (†) = 0 1x = x x1 = x xx(b) +b 1 = x(b) (x +a 1)(b) = (ax)(b) z = xz +b y E(x) = 0 z = x(b)y Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Axioms for language semantics GKAT (without the Boolean algebra axioms for tests).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The function E : GExp → BExp is defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' If the axiom marked (†) is omitted, the above axiomatizes bisimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This semantics is connected to the relational semantics from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='2: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='11 ([44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For e, f ∈ GExp, we have �L(e) = �L(f) if and only if �e�σ = �f�σ for all relational interpretations σ Moreover, since skip-free GKAT expressions are also GKAT expressions, this means that we now have two language interpretations of the former, given by �L and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Fortunately, one can easily be expressed in terms of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For e ∈ GExp−, it holds that �L(e) = L(e) · At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' As an easy consequence of the above, we find that the two semantics must identify the same skip-free GKAT-expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For e, f ∈ GExp−, we have L(e) = L(f) iff �L(e) = �L(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='14, these properties imply that ≡ also axiomatizes relational equivalence of skip-free GKAT-expressions, as a result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Let e, f ∈ GExp−, we have e ≡ f if and only if �e�σ = �f�σ for all relational interpretations σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='3 Equivalences Finally, we relate equivalences as proved for skip-free GKAT expressions to those provable for GKAT expressions, showing that proofs of equivalence for skip-free GKAT expressions can be replayed in the larger calculus, without (UA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The axioms of GKAT as presented in [44,41] are provided in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We write e ≈† f when e = f is derivable from the axioms in Figure 8 with the exception of (†), and e ≈ f when e = f is derivable from the full set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The last axiom of GKAT is not really a single axiom, but rather an axiom scheme, parameterized by the function E : GExp → BExp defined as follows: E(b) = b E(p) = 0 E(e +b f) = (b ∧ E(e)) ∨ (b ∧ E(f)) E(e · f) = E(e) ∧ E(f) E(e(b)) = b The function E models the analogue of Salomaa’s empty word property [39]: we say e is guarded when E(b) is equivalent to 0 by to the laws of Boolean algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Notice that as GKAT expressions, skip-free GKAT expressions are always guarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Since skip-free GKAT expressions are also GKAT expressions, we have four notions of equivalence for GKAT expressions: as skip-free expressions or GKAT expressions in general, either with or without (†).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' These are related as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Let e, f ∈ GExp−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Then (1) e ≈† f if and only if e ≡† f, and (2) e ≈ f if and only if e ≡ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' For the forward direction of (1), we note that if e ≈† f, then e ↔ f in (GExp, δ) by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='8, e ↔ f in (GExp−, δ) and therefore e ≡† f by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Conversely, note that any proof of e = f by the axioms of Figure 2 can be replayed using the rules from Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In particular, the guardedness condition required for the last skip-free GKAT axiom using the last GKAT axiom is always true, because E(g) ≈† 0 for any g ∈ GExp−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The proof of the second claim is similar, but uses Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='13 instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 8 Related Work This paper fits into a larger research program focused on understanding the logical and algebraic content of programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Kleene’s paper introducing the algebra of regular languages [24] was a foundational contribution to this re- search program, containing an algebraic account of mechanical programming and some of its sound equational laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The paper also contained an interesting completeness problem: give a complete description of the equations satisfied by the algebra of regular languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Salomaa was the first to provide a sound and complete axiomatization of language equivalence for regular expressions [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The axiomatization in op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' included an inference rule with a side condition that prevented it from being algebraic in the sense that the validity of an equa- tion is not preserved when substituting letters for arbitrary regular expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Nevertheless, this inspired axiomatizations of several variations and extensions of Kleene algebra [48,44,43], as well as Milner’s axiomatization of the algebra of star behaviours [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The side condition introduced by Salomaa is often called the empty word property, an early version of a concept from process theory called guardedness9 that is also fundamental to the theory of iteration [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Our axiomatization of skip-free GKAT is algebraic due to the lack of a guard- edness side-condition (it is an equational Horn theory [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This is particularly desirable because it allows for an abundance of other models of the axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Kozen proposed an algebraic axiomatization of Kleene algebra that is sound and complete for language equivalence [26], which has become the basis for a number of axiomatizations of other Kleene algebra variants [14,21,22,49] includ- ing Kleene algebra with tests [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' KAT also has a plethora of relational models, which are desirable for reasons we hinted at in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 9 This is a different use of the word “guarded” than in “guarded Kleene algebra with tests”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In the context of process theory, a recursive specification is guarded if every of its function calls occurs within the scope of an operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' GKAT is a fragment of KAT, first identified in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' It was later given a sound and complete axiomatization in [44], although the axiomatization is neither al- gebraic nor finite (it includes (UA), an axiom scheme, shorthand for infinitely many axioms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' It was later shown that dropping x · 0 = 0 (called (S3) in [44]) from this axiomatization gives a sound and complete axiomatization of bisim- ilarity [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The inspiration for our pruning technique is also in [41], where a reduction of the language equivalence case to the bisimilarity case is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Despite the existence of an algebraic axiomatization of language equivalence in KAT, GKAT has resisted algebraic axiomatization so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Skip-free GKAT hap- pens to be a fragment of GKAT in which every expression is guarded, thus eliminating the need for the side condition in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 8 and allowing for an alge- braic axiomatization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' An inequational axiomatization resembling that of KAT might be gleaned from the recent preprint [40], but we have not investigated this carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The GKAT axioms for bisimilarity of ground terms can also likely be obtained from the small-step semantics of GKAT using [1,2,3], but unfortunately this does not appear to help with the larger completeness problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The idea of reducing one completeness problem in Kleene algebra to another is common in Kleene algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' for instance, it is the core behind the completeness proof of KAT [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Cohen also reduced weak Kleene algebra as an axiomatization of star expressions up to simulation to monodic trees [11], whose completeness was conjectured by Takai and Furusawa [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Grabmayer’s solution to the com- pleteness problem of regular expressions modulo bisimulation [15] can also be seen as a reduction to the one-free case [16], since his crystallization procedure produces an automaton that can be solved using the technique found in op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Other instances of reductions include [10,4,12,49,21,23,32,36,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Recent work has started to study reductions and their compositionality properties [12,22,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 9 Discussion We continue the study of efficient fragments of Kleene Algebra with Tests (KAT) initiated in [44], where the authors introduced Guarded Kleene Algebra with Tests (GKAT) and provided an efficient decision procedure for equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' They also proposed a candidate axiomatization, but left open two questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' – The first question concerned the existence of an algebraic axiomatization, which is an axiomatization that is closed under substitution—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', where one can prove properties about a certain program p and then use p as a variable in the context of a larger program, being able to substitute as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This is essential to enable compositional analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' – The second question left open in [44] was whether an axiomatization that did not require an axiom scheme was possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Having a completeness proof that does not require an axiom scheme to reason about mutually dependent loops is again essential for scalability: we should be able to axiomatize single loops and generalize this behaviour to multiple, potentially, nested loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In this paper, we identified a large fragment of GKAT, which we call skip-free GKAT (GKAT−), that can be axiomatized algebraically without relying on an ax- iom scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We show how the axiomatization works well for two types of equiva- lence: bisimilarity and language equivalence, by proving completeness results for both semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Having the two semantics is interesting from a verification point of view as it gives access to different levels of precision when analyzing program behaviour, but also enables a layered approach to the completeness proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We provide a reduction of the completeness proof for language semantics to the one for bisimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Moreover, the latter is connected to a recently solved [15] problem proposed by Milner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' This approach enabled two things: it breaks down the completeness proofs and reuse some of the techniques, while also highlighting the exact difference between the two equivalences (captured by the axiom e·0 ≡ 0 which does not hold for bisimilarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We also showed that proofs of equivalence in skip-free GKAT transfer without any loss to proofs of equivalence in GKAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' There are several directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The bridge between process algebra and Kleene algebra has not been exploited to its full potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The fact that we could reuse results by Grabmayer and Fokkink [15,16] was a major step towards completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' An independent proof would have been much more complex and very likely required the development of technical tools resembling those in [15,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We hope the results in this paper can be taken further and more results can be exchanged between the two communities to solve open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' The completeness problem for full GKAT remains open, but our completeness results for skip-free GKAT are encouraging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We believe they show a path towards studying whether an algebraic axiomatization can be devised or a negative re- sult can be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' A first step in exploring a completeness result would be to try extending Grabmayer’s completeness result [15] to a setting with output variables—this is a non-trivial exploration, but we are hopeful will yield new tools for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' As mentioned in the introduction, NetKAT [4] (and its probabilistic variants [13,45]) have been one of the most successful extensions of KAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' We believe the step from (skip-free) GKAT to a guarded version of NetKAT is also a worthwhile exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Following [17], we hope to be able to explore these extensions in a modular and parametric way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Acknowledgements A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Silva and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Schmid were partially funded by ERC grant Autoprobe (grant agreement 101002697).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Kapp´e was supported by the EU’s Horizon 2020 research and innovation programme under Marie Sk�lodowska- Curie grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 101027412 (VERLAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Aceto, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=': Deriving complete inference systems for a class of GSOS lan- guages generation regular behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In: CONCUR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 449–464 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1007/978-3-540-48654-1_33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Aceto, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Caltais, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Goriac, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Ing´olfsd´ottir, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=': Axiomatizing GSOS with pred- icates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In: SOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 1–15 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='4204/EPTCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 343–355 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1145/2676726.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='2677011 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Grabmayer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=': Milner’s proof system for regular expressions modulo bisimilarity is complete: Crystallization: Near-collapsing process graph in- terpretations of regular expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In: LICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 34:1–34:13 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1145/3531130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='3532430 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Grabmayer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Fokkink, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' : A complete proof system for 1-free reg- ular expressions modulo bisimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In: LICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 465–478 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1145/3373718.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In: PLDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 594–608 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1145/3519939.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='3523722 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Gumm, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' : Functors for coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Algebra Universalis 45 (11 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1007/s00012-001-8156-x 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Gumm, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' : Elements of the general theory of coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' LUATCS’99, Rand Afrikaans University, Johannesburg (1999) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Huntington, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' : Sets of independent postulates for the algebra of logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Transactions of the American Mathematical Society 5(3), 288–309 (1904).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1090/S0002-9947-1904-1500675-4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Kapp´e, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Brunet, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Rot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Silva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Wagemaker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Zanasi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=': Kleene algebra with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In: CONCUR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 41:1–41:16 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='4230/LIPIcs.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Zanasi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=': Concurrent Kleene algebra with observations: From hypotheses to completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In: FOSSACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 381–400 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1007/978-3-030-45231-5_20 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Kapp´e, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Brunet, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Silva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Zanasi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=': Concurrent Kleene al- gebra: Free model and completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In: ESOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 856–882 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1007/978-3-319-89884-1_30 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Kleene, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=': Representation of events in nerve nets and finite automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Automata studies 34, 3–41 (1956) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Kot, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Kozen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=': Kleene algebra and bytecode verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Notes Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 141(1), 221–236 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='entcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='028 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Kozen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=': A completeness theorem for Kleene algebras and the algebra of regular events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 110(2), 366–390 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1006/inco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1037 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Kozen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=': Kleene algebra with tests and commutativity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In: TACAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 14–33 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1007/3-540-61042-1_35 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Kozen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Mamouras, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=': Kleene algebra with equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In: ICALP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 280–292 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1007/978-3-662-43951-7_24 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Kozen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Patron, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=': Certification of compiler optimizations using Kleene algebra with tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In: CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 568–582 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1007/3-540-44957-4_38 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Kozen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Smith, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=': Kleene algebra with tests: Completeness and decidability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In: CSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 244–259 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='1007/3-540-63172-0_43 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Kozen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Tseng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' : The B¨ohm-Jacopini theorem is false, propositionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In: MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} 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Games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Learning Development Aids (2002) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' Rutten, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' : Universal coalgebra: a theory of systems.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Foster, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Hsu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Kozen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Silva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=': Scalable verification of probabilistic networks.' metadata={'source': 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+page_content=', Kapp´e, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Rot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Silva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=': Par- tially observable concurrent Kleene algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' In: CONCUR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' 20:1–20:22 (2020).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Silva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=', Sammartino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=': Guarded Kleene algebra with tests: Au- tomata learning (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFIT4oBgHgl3EQfkiuG/content/2301.11301v1.pdf'} +page_content='48550/arXiv.' metadata={'source': 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+NeuralMind, Brazil +UNICAMP, Brazil +Luiz Bonifacio +NeuralMind, Brazil +UNICAMP, Brazil +Hugo Abonizio +NeuralMind, Brazil +UNICAMP, Brazil +Marzieh Fadaee +Zeta Alpha, Netherlands +Roberto Lotufo +NeuralMind, Brazil +UNICAMP, Brazil +Jakub Zavrel +Zeta Alpha, Netherlands +Rodrigo Nogueira +NeuralMind, Brazil +UNICAMP, Brazil +Zeta Alpha, Netherlands +Abstract +Recently, InPars introduced a method to efficiently use large language models +(LLMs) in information retrieval tasks: via few-shot examples, an LLM is induced +to generate relevant queries for documents. +These synthetic query-document +pairs can then be used to train a retriever. +However, InPars and, more re- +cently, Promptagator, rely on proprietary LLMs such as GPT-3 and FLAN to +generate such datasets. In this work we introduce InPars-v2, a dataset genera- +tor that uses open-source LLMs and existing powerful rerankers to select syn- +thetic query-document pairs for training. A simple BM25 retrieval pipeline fol- +lowed by a monoT5 reranker finetuned on InPars-v2 data achieves new state- +of-the-art results on the BEIR benchmark. To allow researchers to further im- +prove our method, we open source the code, synthetic data, and finetuned models: +https://github.com/zetaalphavector/inPars/tree/master/tpu +1 +Introduction and Background +Data augmentation has been a reliable tool to improve the effectiveness of AI models in the face +of the scarcity of high-quality in-domain training data, which is a common problem in practical +applications. Previous work by Bonifacio et al. [1] and Dai et al. [2] successfully leveraged the +few-shot capabilities of LLMs to generate reliable synthetic training data for information retrieval +models. These training data helped their models achieve state-of-the-art (SOTA) results on the BEIR +benchmark [6]. +Bonifacio et al. [1] propose InPars where they generate queries from documents in the corpus using +LLMs. Similarly to Bonifacio et al. [1], the recently published Promptagator [2] model also feeds +prompts to LLMs in order to generate alternative queries for a given document in an unsupervised +manner. It differs primarily from InPars in that it uses dataset-specific prompts, a larger LLM to +generate queries, and a fully trainable retrieval pipeline with smaller models. +This work extends the method of Bonifacio et al. [1] by using a reranker as a filtering mechanism +to select the best synthetically generated examples and further improving retrieval effectiveness +on BEIR. We also use an open-source query generator as opposed to the proprietary one used by +Preprint. Under review. + +Bonifacio et al. and provide the source code and data to reproduce our results on TPUs. We refer to +Bonifacio et al. [1] model as Inpars-v1 and the model presented in this paper as Inpars-v2. +2 +Methodology +In this section, we explain the experiments we performed and how they differ from InPars-v1 [1]. +To generate synthetic queries, we use the open-source GPT-J [8] with 6B parameters to replace +OpenAI’s curie model used in InPars-v1. For each dataset in the BEIR benchmark, we sample 100k +documents from its corpus and generate one synthetic query per document using GPT-J prompted +with 3 examples from MS MARCO. We use greedy decoding and the “gbq” prompt template from +InPars-v1. Some corpora in BEIR such as ArguAna [7] have less than 100k documents. In these +cases, we generate as many synthetic queries as there are documents in the corpus. It takes on +average 30 hours on an A100 GPU to generate 100k queries. +Once the synthetic queries are generated, we apply a filtering step to select query-document pairs +that are more likely to be relevant to each other. In InPars-v1, this filtering step consisted of selecting +the top 10k query-document pairs with the highest log probabilities of generating a query given the +3-shot examples and the document as input. In InPars-v2, we use monoT5-3B [4] already finetuned +on MS MARCO for one epoch1 to estimate a relevancy score for each of the 100k query-document +pairs. Then, we keep only the top 10k pairs with the highest scores as our positive query-document +pairs for training. It takes approximately 1.5 hours to score 100k query-document pairs on a TPU +v3-8. It should take twice as much on a A100. +To obtain negatives (i.e., non-relevant) query-document pairs, we randomly sample one document +from the top 1000 retrieved by BM25 when issued the synthetic query. Thus, our training set consists +of 10k positive query-document pairs and 10k negative query-document pairs. +The rerankers are finetuned in the same manner as in InPars-v1: monoT5-3B is finetuned on MS +MARCO for one epoch and then further finetuned for one epoch on the synthetic data. We use the +Adafactor optimizer [5] with a constant learning rate of 1e-3. Each batch has 64 positive and 64 +negative query-document pairs randomly sampled from the training dataset. We finetune one model +on each synthetic dataset from BEIR, that is, we end up with 18 different rerankers, one per dataset, +which are then evaluated on the corresponding test sets. Finetuning on each synthetic dataset takes +less around 10 minutes on a TPU v3-8. +Evaluation is performed using the following pipeline: first we use Pyserini’s [3] flat indexes2 to +retrieve a thousand documents for each query using BM25 with default parameters (k1=0.9, b=0.4), +for each dataset. Then we use the finetuned monoT5-3B models to rerank these documents. +3 +Results +Table 1 presents results for BM25 (2nd column), monoT5-3B finetuned on MS MARCO (3rd col- +umn), monoT5-3b finetuned on MS MARCO and further finetuned on InPars-v1 (4th column), and +monoT5-3B finetuned on MS MARCO and then finetuned on InPars-v2 data (5th column). Com- +pared to InPars-v1, our approach is substantially better on TREC-News, Climate-FEVER, Robust +and Touche. Additionally, we compare our method with Promptagator [2] and RankT5 [10]. Taking +into account the average of all BEIR datasets, these results represent a new state of the art on BEIR. +Promptagator and RankT5 strive on datasets that monoT5 and InPars-v2 cannot even surpass BM25, +such as Touche and ArguAna. Note that these datasets focus on argument retrieval, which is slightly +different from other datasets in the BEIR benchmark. As a result, they benefit from using cus- +tom prompts.3 Promptagator does this without using supervised data from MS MARCO and using +smaller T5 models with 110M parameters for the retrieval and reranking steps. +1https://huggingface.co/castorini/monot5-3b-msmarco-10k +2As opposed to the multifield index. +3In preliminary experiments, we also observed an improvement of more than 10 nDCG@10 points on +ArguAna by using a dataset-specific prompt to generate synthetic queries. More details and results on the full +BEIR benchmark will appear in an upcoming paper. +2 + +BM25 +monoT5-3B +PrGator +RankT5 +MARCO ++InPars-v1 ++InPars-v2 +TREC-Covid +0.594 +0.801 +0.846 +0.846 +0.762 +0.823 +Robust +0.407 +0.615 +0.610 +0.632 +- +- +FiQA +0.236 +0.509 +0.492 +0.509 +0.494 +0.493 +DBPedia +0.318 +0.472 +0.494 +0.498 +0.434 +0.459 +SciDocs +0.149 +0.197 +0.206 +0.208 +0.201 +0.191 +SciFact +0.678 +0.774 +0.774 +0.774 +0.731 +0.760 +NFCorpus +0.321 +0.383 +0.385 +0.385 +0.370 +0.399 +BioASQ +0.522 +0.566 +0.607 +0.595 +- +0.579 +Natural Questions +0.305 +0.625 +0.625 +0.638 +- +0.647 +HotpotQA +0.633 +0.760 +0.790 +0.791 +0.736 +0.753 +TREC-News +0.395 +0.477 +0.458 +0.490 +- +- +Quora +0.788 +0.835 +0.874 +0.845 +- +0.819 +FEVER +0.651 +0.848 +0.852 +0.872 +0.866 +0.848 +Climate-FEVER +0.165 +0.288 +0.287 +0.323 +0.241 +0.275 +Signal +0.328 +0.302 +0.319 +0.308 +- +0.319 +ArguAna +0.397 +0.379 +0.371 +0.369 +0.630 +0.406 +Touche +0.442 +0.309 +0.260 +0.291 +0.381 +0.486 +CQADupstack +0.302 +0.449 +0.449 +0.448 +- +- +Avg +0.431 +0.538 +0.544 +0.551 +- +- +Avg PrGator +0.417 +0.520 +0.523 +0.533 +0.531 +0.536 +Table 1: nDCG@10 on BEIR. “Avg PrGator” is the average of datasets reported by Promptagator. +Promptagator uses a proprietary model, FLAN [9], to generate synthetic queries. The RankT5 model +is a modified version of the monoT5 reranker, but its checkpoint and code are not published. In this +work, we make the code, models, and data open-source and publicly available. +4 +Conclusion +In this work, we presented InPars-v2, an improved version of InPars [1] that uses a publicly available +language model to generate queries and a better query-document pair selection process. Our results +show that we achieve effectiveness on par with the state of the art on BEIR. The synthetic data and +finetuned models were publicly released. +Acknowledgments +This research was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo +(FAPESP) (project id 2022/01640-2). We also thank Centro Nacional de Processamento de Alto +Desempenho (CENAPAD-SP) and Google Cloud for computing credits. +References +[1] L. Bonifacio, H. Abonizio, M. Fadaee, and R. Nogueira. Inpars: Data augmentation for infor- +mation retrieval using large language models. arXiv preprint arXiv:2202.05144, 2022. +[2] Z. Dai, V. Y. Zhao, J. Ma, Y. Luan, J. Ni, J. Lu, A. Bakalov, K. Guu, K. B. Hall, and +M.-W. Chang. +Promptagator: Few-shot dense retrieval from 8 examples. +arXiv preprint +arXiv:2209.11755, 2022. +[3] J. Lin, X. Ma, S.-C. Lin, J.-H. Yang, R. Pradeep, and R. Nogueira. Pyserini: An easy-to-use +python toolkit to support replicable ir research with sparse and dense representations. arXiv +preprint arXiv:2102.10073, 2021. +[4] R. Nogueira, Z. Jiang, R. Pradeep, and J. Lin. Document ranking with a pretrained sequence- +to-sequence model. In Proceedings of the 2020 Conference on Empirical Methods in Natural +Language Processing: Findings, pages 708–718, 2020. +3 + +[5] N. Shazeer and M. Stern. Adafactor: Adaptive learning rates with sublinear memory cost. In +International Conference on Machine Learning, pages 4596–4604. PMLR, 2018. +[6] N. Thakur, N. Reimers, A. Rücklé, A. Srivastava, and I. Gurevych. +Beir: A heteroge- +neous benchmark for zero-shot evaluation of information retrieval models. +arXiv preprint +arXiv:2104.08663, 2021. +[7] H. Wachsmuth, S. Syed, and B. Stein. Retrieval of the best counterargument without prior +topic knowledge. In Proceedings of the 56th Annual Meeting of the Association for Computa- +tional Linguistics (Volume 1: Long Papers), pages 241–251, Melbourne, Australia, July 2018. +Association for Computational Linguistics. +[8] B. Wang and A. Komatsuzaki. GPT-J-6B: A 6 Billion Parameter Autoregressive Language +Model. https://github.com/kingoflolz/mesh-transformer-jax, May 2021. +[9] J. Wei, M. Bosma, V. Y. Zhao, K. Guu, A. W. Yu, B. Lester, N. Du, A. M. Dai, and Q. V. Le. +Finetuned language models are zero-shot learners. arXiv preprint arXiv:2109.01652, 2021. +[10] H. Zhuang, Z. Qin, R. Jagerman, K. Hui, J. Ma, J. Lu, J. Ni, X. Wang, and M. Bendersky. +Rankt5: Fine-tuning t5 for text ranking with ranking losses. arXiv preprint arXiv:2210.10634, +2022. +4 + diff --git a/bNAzT4oBgHgl3EQf2v7I/content/tmp_files/load_file.txt b/bNAzT4oBgHgl3EQf2v7I/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..234cb11061c89b9d05baffd1824a55dd831d0f26 --- /dev/null +++ b/bNAzT4oBgHgl3EQf2v7I/content/tmp_files/load_file.txt @@ -0,0 +1,302 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf,len=301 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='01820v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='IR] 4 Jan 2023 InPars-v2: Large Language Models as Efficient Dataset Generators for Information Retrieval Vitor Jeronymo NeuralMind,' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Netherlands Abstract Recently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' InPars introduced a method to efficiently use large language models (LLMs) in information retrieval tasks: via few-shot examples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' an LLM is induced to generate relevant queries for documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' These synthetic query-document pairs can then be used to train a retriever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' However, InPars and, more re- cently, Promptagator, rely on proprietary LLMs such as GPT-3 and FLAN to generate such datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' In this work we introduce InPars-v2, a dataset genera- tor that uses open-source LLMs and existing powerful rerankers to select syn- thetic query-document pairs for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' A simple BM25 retrieval pipeline fol- lowed by a monoT5 reranker finetuned on InPars-v2 data achieves new state- of-the-art results on the BEIR benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' To allow researchers to further im- prove our method, we open source the code, synthetic data, and finetuned models: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='com/zetaalphavector/inPars/tree/master/tpu 1 Introduction and Background Data augmentation has been a reliable tool to improve the effectiveness of AI models in the face of the scarcity of high-quality in-domain training data, which is a common problem in practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Previous work by Bonifacio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' [1] and Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' [2] successfully leveraged the few-shot capabilities of LLMs to generate reliable synthetic training data for information retrieval models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' These training data helped their models achieve state-of-the-art (SOTA) results on the BEIR benchmark [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Bonifacio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' [1] propose InPars where they generate queries from documents in the corpus using LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Similarly to Bonifacio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' [1], the recently published Promptagator [2] model also feeds prompts to LLMs in order to generate alternative queries for a given document in an unsupervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' It differs primarily from InPars in that it uses dataset-specific prompts, a larger LLM to generate queries, and a fully trainable retrieval pipeline with smaller models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' This work extends the method of Bonifacio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' [1] by using a reranker as a filtering mechanism to select the best synthetically generated examples and further improving retrieval effectiveness on BEIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' We also use an open-source query generator as opposed to the proprietary one used by Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Bonifacio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' and provide the source code and data to reproduce our results on TPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' We refer to Bonifacio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' [1] model as Inpars-v1 and the model presented in this paper as Inpars-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' 2 Methodology In this section, we explain the experiments we performed and how they differ from InPars-v1 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' To generate synthetic queries, we use the open-source GPT-J [8] with 6B parameters to replace OpenAI’s curie model used in InPars-v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' For each dataset in the BEIR benchmark, we sample 100k documents from its corpus and generate one synthetic query per document using GPT-J prompted with 3 examples from MS MARCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' We use greedy decoding and the “gbq” prompt template from InPars-v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Some corpora in BEIR such as ArguAna [7] have less than 100k documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' In these cases, we generate as many synthetic queries as there are documents in the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' It takes on average 30 hours on an A100 GPU to generate 100k queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Once the synthetic queries are generated, we apply a filtering step to select query-document pairs that are more likely to be relevant to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' In InPars-v1, this filtering step consisted of selecting the top 10k query-document pairs with the highest log probabilities of generating a query given the 3-shot examples and the document as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' In InPars-v2, we use monoT5-3B [4] already finetuned on MS MARCO for one epoch1 to estimate a relevancy score for each of the 100k query-document pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Then, we keep only the top 10k pairs with the highest scores as our positive query-document pairs for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' It takes approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='5 hours to score 100k query-document pairs on a TPU v3-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' It should take twice as much on a A100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' To obtain negatives (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=', non-relevant) query-document pairs, we randomly sample one document from the top 1000 retrieved by BM25 when issued the synthetic query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Thus, our training set consists of 10k positive query-document pairs and 10k negative query-document pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' The rerankers are finetuned in the same manner as in InPars-v1: monoT5-3B is finetuned on MS MARCO for one epoch and then further finetuned for one epoch on the synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' We use the Adafactor optimizer [5] with a constant learning rate of 1e-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Each batch has 64 positive and 64 negative query-document pairs randomly sampled from the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' We finetune one model on each synthetic dataset from BEIR, that is, we end up with 18 different rerankers, one per dataset, which are then evaluated on the corresponding test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Finetuning on each synthetic dataset takes less around 10 minutes on a TPU v3-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Evaluation is performed using the following pipeline: first we use Pyserini’s [3] flat indexes2 to retrieve a thousand documents for each query using BM25 with default parameters (k1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='9, b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='4), for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Then we use the finetuned monoT5-3B models to rerank these documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' 3 Results Table 1 presents results for BM25 (2nd column), monoT5-3B finetuned on MS MARCO (3rd col- umn), monoT5-3b finetuned on MS MARCO and further finetuned on InPars-v1 (4th column), and monoT5-3B finetuned on MS MARCO and then finetuned on InPars-v2 data (5th column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Com- pared to InPars-v1, our approach is substantially better on TREC-News, Climate-FEVER, Robust and Touche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Additionally, we compare our method with Promptagator [2] and RankT5 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Taking into account the average of all BEIR datasets, these results represent a new state of the art on BEIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Promptagator and RankT5 strive on datasets that monoT5 and InPars-v2 cannot even surpass BM25, such as Touche and ArguAna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Note that these datasets focus on argument retrieval, which is slightly different from other datasets in the BEIR benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' As a result, they benefit from using cus- tom prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='3 Promptagator does this without using supervised data from MS MARCO and using smaller T5 models with 110M parameters for the retrieval and reranking steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' 1https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='co/castorini/monot5-3b-msmarco-10k 2As opposed to the multifield index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' 3In preliminary experiments, we also observed an improvement of more than 10 nDCG@10 points on ArguAna by using a dataset-specific prompt to generate synthetic queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' More details and results on the full BEIR benchmark will appear in an upcoming paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' 2 BM25 monoT5-3B PrGator RankT5 MARCO +InPars-v1 +InPars-v2 TREC-Covid 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='594 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='846 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='551 Avg PrGator 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='417 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='523 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='533 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content='536 Table 1: nDCG@10 on BEIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' “Avg PrGator” is the average of datasets reported by Promptagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Promptagator uses a proprietary model, FLAN [9], to generate synthetic queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' The RankT5 model is a modified version of the monoT5 reranker, but its checkpoint and code are not published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' In this work, we make the code, models, and data open-source and publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' 4 Conclusion In this work, we presented InPars-v2, an improved version of InPars [1] that uses a publicly available language model to generate queries and a better query-document pair selection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Our results show that we achieve effectiveness on par with the state of the art on BEIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' The synthetic data and finetuned models were publicly released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAzT4oBgHgl3EQf2v7I/content/2301.01820v1.pdf'} +page_content=' Acknowledgments This research was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) (project 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--- /dev/null +++ b/bNE0T4oBgHgl3EQfngFx/content/tmp_files/2301.02513v1.pdf.txt @@ -0,0 +1,3077 @@ +Information Carried by a Single Particle in Quantum Multiple-Access Channels +Xinan Chen,1, ∗ Yujie Zhang,2, ∗ Andreas Winter,3, 4, 5 Virginia O. Lorenz,2 and Eric Chitambar1 +1Department of Electrical and Computer Engineering, Coordinated Science Laboratory, +University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA +2Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA +3Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA), +Pg. +Llu´ıs Companys, 23, 08010 Barcelona, Spain +4Grup d’Informaci´o Qu`antica, Departament de F´ısica, +Universitat Aut`onoma de Barcelona, 08193 Bellaterra (Barcelona), Spain +5Institute for Advanced Study, Technische Universit¨at M¨unchen, +Lichtenbergstraße 2a, D-85748 Garching, Germany +(Dated: January 9, 2023) +Non-classical features of quantum systems have the potential to strengthen the way we currently +exchange information. In this paper, we explore this enhancement on the most basic level of single +particles. To be more precise, we compare how well multi-party information can be transmitted +to a single receiver using just one classical or quantum particle. +Our approach is based on a +multiple-access communication model in which messages can be encoded into a single particle that +is coherently distributed across multiple spatial modes. Theoretically, we derive lower bounds on the +accessible information in the quantum setting that strictly separate it from the classical scenario. +This separation is found whenever there is more than one sender, and also when there is just a single +sender who has a shared phase reference with the receiver. Experimentally, we demonstrate such +quantum advantage in single-particle communication by implementing a multi-port interferometer +with messages being encoded along the different trajectories. Specifically, we consider a two-sender +communication protocol built by a three-port optical interferometer. In this scenario, the rate sum +achievable with a classical particle is upper bounded by one bit, while we experimentally observe a +rate sum of 1.0152 ± 0.0034 bits in the quantum setup. +I. +INTRODUCTION +It is well known that a quantum particle exhibits +fundamentally different properties than its classical +counterpart. For instance, while a classical particle +has a definite trajectory in space, a quantum par- +ticle can be placed in a coherent superposition of +different paths as it moves from one point in space +to another. A natural practical question is whether +this superposition of trajectories can be utilized for +performing some communication task [1–6]. In this +paper, we focus on whether the path coherence of a +single particle can be used to enhance the commu- +nication of N spatially separated parties to a single +receiver. +Several previous papers have addressed similar +questions in this direction. Inspired by the famous +two-slit experiment, Massar first showed the advan- +tage of quantum particles in the bipartite finger- +printing task [7]. In such a task, Alice and Bob each +possesses one bit x, y ∈ {0, 1}, and they wish to let +a referee decide whether x = y by sending minimal +∗ X. Chen and Y. Zhang contributed equally to this paper. +amount of information to the referee. It is not dif- +ficult to see that one quantum particle in the state +1/ +√ +2(|0⟩A|1⟩B + |1⟩A|0⟩B) suffices for this objective, +while in the classical regime, the parties must send +both x and y for the referee to certify that x = y. +In Ref. [8], the authors reinterpreted this result as +two-way communication using only one single quan- +tum particle, which is forbidden if the information +medium is a classical particle. This idea was fur- +ther extended to the scenario where Alice and Bob +each have an n-bit string [9]. Using an n-level Mach- +Zehnder interferometer, one of Alice and Bob can re- +trieve the other’s full n-bit string, while only one bit +of information is revealed to the other party. Since +this can be done for an arbitrary n, this result sug- +gests, roughly speaking, that a single quantum par- +ticle can carry an arbitrarily large amount of infor- +mation in point-to-point communication. Comple- +menting the point-to-point communication results, +it was recently discovered via convex polytope analy- +sis that using a single quantum particle, one can gen- +erate multiple-access channels (MACs) that cannot +be constructed with a classical particle [6, 10]. How- +ever, these latter results pertain to the specific tran- +sition probabilities p(y|x1, · · · , xN) of the generated +N-party MACs. It has remained elusive whether the +arXiv:2301.02513v1 [quant-ph] 6 Jan 2023 + +discovered non-classical MACs actually have advan- +tages in terms of more practical figures of merit, such +as asymptotic communication rates. +In this paper, we provide a positive answer to this +question. Specifically, we utilize the framework of +single-particle multiple-access channels (MACs) de- +veloped in [10] to investigate the achievable rate re- +gions of distributed communication using a single +particle. While the communication rate sum of the +different senders is always upper bounded by 1 bit +if a single classical particle is used, in the quantum +setting a rate sum of at least 1.10 bits is achievable +for two senders. Even higher rates can be achieved if +there are more than two senders. Moreover, we ex- +perimentally demonstrate the quantum advantages +by implementing one of our designed protocols. In +particular, we achieve a quantum advantage within +five standard deviations using linear optics and a +single photon state. +This paper is organized as follows. +In Sec- +tion II, we introduce the operational framework of +single-particle MACs and review some information- +theoretic concepts such as the achievable rate re- +gions of MACs. +In Section III, we study in de- +tail the theoretical aspects of our work. In Section +IV, we give our experimental demonstration of the +two-sender coherent assisted communication proto- +col using linear optics and a heralded single pho- +ton state, where quantum-enhanced communication +is achieved by preparing a single photon in a super- +position of different trajectories. +II. +OPERATIONAL FRAMEWORK AND +INFORMATION THEORY PRELIMINARIES +A. +MACs constructed with one particle +To compare how much classical information can +be carried by a classical or quantum particle with +none of its internal degrees of freedom being accessi- +ble, we utilize the framework of single-particle MACs +developed in Ref. [10]. This framework, which we +now briefly describe, was inspired by previous work +[11, 12] that captured the resource-theoretic features +of quantum coherence in a multi-port interferome- +ter setup. +We denote the collection of senders as +A = (A1, A2, · · · , AN) and assume that each mes- +sage sent by each sender is finite. These N senders +will use a one-particle state to send information to +a single receiver B. Recall that the Fock space is de- +scribed by H = �∞ +i=0 Hi, where Hi is the i-particle +subspace of H. A one-particle state is represented +by a density operator ρA acting on the one-particle +subspace, which is +HA +1 := span {|ei⟩ : 1 ≤ i ≤ N} , +(1) +where |ei⟩ = |0⟩A1 · · · |1⟩Ai · · · |0⟩AN is the state of +the particle on path i, with |0⟩ being the vacuum +state. The senders then encode their messages using +completely positive trace-preserving (CPTP) maps. +For example if party Ai wishes to send message xi, +the CPTP map EAi +xi is locally applied. +The fully +encoded state for joint message x := (x1, · · · , xN) is +given by +σx := σx1···xN = EA1 +x1 ⊗ · · · ⊗ EAN +xN (ρA). +(2) +For the purposes of this investigation, we restrict +the allowed CPTP maps that the senders use to +encode. Specifically, since we are interested in the +information-carrying ability of a single particle, we +have to require that the encoding operations can- +not increase particle number. More specifically, we +model the encoding operations as CPTP maps with +a particle number-preserving unitary extension, that +is, +E(ρA) = TrE +� +U(ρA ⊗ |0⟩⟨0|E)U †� +, +(3) +where U preserves the overall particle number in +the system A and the environment E. This set of +operations was termed number-preserving extendible +(NPE) operations in Ref. [10] and was fully char- +acterized for an arbitrary number of particles. Here +we focus only on the case where there is at most one +particle. In this case, these operations are convex +combinations of channels with Kraus operators +K1 = +�1 +0 +0 eiφ1√1 − γ +� +, +K2 = +� +0 eiφ2√γ +0 +0 +� +. +(4) +Note they can be seen as generalized amplitude +damping channels with two additional relative phase +parameters. +In this work, we will rely heavily on +two particular NPE operations in the encoding: the +completely damping operation ρ �→ E(vac)(ρ) := +Tr(ρ)|0⟩⟨0| and the phase shift operation ρ +�→ +E(φ)(ρ) := e−iZφ/2ρeiZφ/2, where Z = +� 1 +0 +0 −1 +� +. Note +that E(vac) and E(φ) correspond to the choices of +γ = 1 and γ = 0 in Eq. +(4), respectively. +In +optical communication, these encoding operations +correspond to on-off keying (OOK) modulation and +phase-shift keying (PSK) modulation [13, 14]. +After the encoding operations, the state σx is sent +to the receiver, and the receiver tries to reconstruct +2 + +FIG. 1. +The general scheme for building a multiple- +access classical channel using a single particle. +the message using a positive operator-valued mea- +sure (POVM) {Πy}y. This process induces a classi- +cal channel by +p(y|x) := Tr(Πyσx). +(5) +A graphical representation of this framework is +shown in Fig. 1. Note that in our model we will +always assume that the receiver shares a phase ref- +erence with the particle source, and so |ei⟩ is defined +with the same overall phase for both the source and +detector [15]. +With this operational framework in mind, we can +define the set of N-sender MACs constructed from a +classical particle as MACs p(y|x1 · · · xN) of the form +p(y|x1 · · · xN) =Tr +� +Πy +� +EA1 +x1 ⊗ · · · ⊗ EAN +xN (ρA +cl) +�� +(6) +where ρA +cl = �N +i=1 pi|ei⟩⟨ei|. In other words, these +are MACs that can be constructed from a classi- +cal source state that has no coherence between any +two paths. On the other hand, in the truly quan- +tum case, no restriction is placed on the initial one- +particle density matrix. We say that the set of N- +sender MACs constructed from a quantum particle +are MACs that have the form +p(y|x1 · · · xN) =Tr +� +Πy +� +EA1 +x1 ⊗ · · · ⊗ EAN +xN (ρA) +�� +(7) +where ρA ∈ D(H1). +Here D(H1) denotes the set +of density operators on the one-particle subspace. +Throughout this work we assume that the message +xi of party Ai is chosen from alphabet set Xi, which +will always be a finite set of integers Xi = [mi] := +{0, · · · , mi − 1}. Similarly, we let Y denote the out- +put alphabet of the receiver B. For input and output +alphabet X := X1 ×X2 ×· · ·×XN and Y, we denote +the set of classical MACs by CN(X; Y) and quan- +tum MACs by QN(X; Y). We will use QN and CN +to denote general N-sender channels with arbitrary +input and output alphabets. +Despite the fact that both classical and quan- +tum MACs can be described using Born’s rule (i.e. +Eqs. (6) and (7), respectively), classical MACs ad- +mit a much simpler characterization. +The state +�N +i=1 pi|ei⟩⟨ei| can be understood simply as a clas- +sical particle that is sent along path i with proba- +bility pi. +A local NPE operation then reduces to +applying some local channel that lets the particle ei- +ther continue along its respective path or blocks it +from reaching the receiver B, i.e. either E(vac) or the +identity map is performed. With probability qi(0|xi) +the particle is blocked by party Ai for input choice +xi, and with probability qi(ei|xi) it is transmitted. +Hence if the input state is |ei⟩⟨ei|, then the state +received by B is +σxi = +� +j̸=i +|0⟩⟨0|Aj ⊗ EAi +xi (|1⟩⟨1|) += qi(ei|xi)|ei⟩⟨ei|A1···AN + qi(0|xi)|0⟩⟨0|A1···AN . +On the decoding end, party B examines each path +to see if it contains a particle. Output b is produced +with probability d(b|ei) when a particle is received +along path i and with probability d(b|0) when no +particle is received. Hence the channel obtained af- +ter averaging over all input states is +p(y|x1, · · · , xN) = +N +� +i=1 +pi[d(y|0)qi(0|xi) ++d(y|ei)qi(ei|xi)]. +(8) +The set CN(X1, · · · XN; Y) consists of MACs that can +be written in this form. +B. +Coherence-assisted communication +Thus far we have focused on scenarios where +the number of senders equals the number of paths +through which the particle source emits the particle. +We can generalize this model by allowing for extra +paths that are not acted upon by an sender (Fig. +2). We refer to these as coherence-assisted protocols, +with the extra paths being called called assistance +paths. Note that since the assistance paths are not +touched by any encoding operation, we can with- +out loss of generality combine amplitudes of multi- +ple assistance paths into one assistance path. +In- +tuitively, the assistance path can serve as a phase +reference for the other paths, which can help the +receiver better discriminate the encoded messages. +3 + +y +Particle +sourceFIG. 2. A coherence-assisted protocol allows for an un- +perturbed side channel through which the particle can +traverse coherently to the decoder. +One the other hand, as we will see in Theorem 2 +below, this assistance path cannot enhance commu- +nication rate when the source is a classical particle. +We let Qass +N (X; Y) denote the family of all coherence- +assisted channels built by N parties using a single +quantum particle and NPE operations. +An analogy can be drawn here to entanglement- +assisted communication [16, 17], in which entangle- +ment is shared between the senders and receiver. In +fact, one could imagine in Fig. 2 that the particle +is coherently distributed to the receiver along the +assisted path prior to the encoding of the senders. +Then the scenario becomes conceptually equivalent +to the entanglement-assisted setup except that the +shared resource between senders and receivers is co- +herence in single-particle spatial modes rather than +coherence in multi-particle states [18]. +C. +Communication rates of MACs +In this work we consider the achievable communi- +cation rates of the classical multiple-access channels +as constructed in the previous subsection. Roughly +speaking, a rate tuple (R1, · · · , RN) is achievable for +a given MAC if for every ϵ > 0 and n sufficiently +large, each sender i can send 2nRi possible messages +with average error no greater than ϵ (see Ref. [19] +for details). Remarkably the achievable rate region +of an N-sender MAC has a single-letter characteriza- +tion in terms of the conditional mutual information, +which, for random variables X1, X2, Y , is defined as +I(X1 : Y |X2) = I(X1X2 : Y ) − I(X2 : Y ). +Proposition +1 +([19–21]). +A +rate +tuple +(R1, · · · , RN) for MAC p(y|x) is achievable if +and only if it lies in the closure of the convex hull +of all rate tuples satisfying +RS ≤ I(XS : Y |XSC), +∀S ⊂ {1, · · · N} +(9) +for some product distribution p(x1) · · · p(xN) over +the input alphabet X. Here in a slight abuse of nota- +tion we denote XS := ×i∈SXi and RS := � +i∈S Ri. +In particular, for two parties, the achievable rate re- +gion is the convex hull of all rate pairs satisfying +R1 ≤ I(X1 : Y |X2) +R2 ≤ I(X2 : Y |X1) +R1 + R2 ≤ I(X1X2 : Y ), +(10) +for product distributions p(x1)p(x2). +For the purpose of this investigation, we will be +mainly interested in the largest amount of informa- +tion that can be jointly sent by the senders. In our +framework, this corresponds to the largest rate-sum +R := � +i∈{1,··· ,N} Ri that can be achieved using a +MAC constructed from a single particle. +D. +The accessible information and Holevo +information +As described in the previous sections, each com- +munication protocol using a single particle consists +of three elements: a choice of the initial one-particle +state ρ, an encoding strategy which specifies a fam- +ily of NPE encoding operations {EAi +xi }, and the de- +coding measurement {Πy}. We will be interested in +optimizing the joint achievable communication rate +under this framework, and to do so, we split the full +optimization into two parts. Every choice of initial +state, encoding strategy, and prior product distribu- +tion p(x) = p(x1) · · · p(xN) over the messages gives +rise to the classical-quantum (cq) state +σXA = +� +x +p(x)|x⟩⟨x|X ⊗ σA +x +(11) +where σA +x = EA1 +x1 ⊗ · · · ⊗ EAN +xN (ρ). For each such cq +state, when a POVM {Πy} is performed on systems +A, the resulting joint probability distribution can be +described by the classical-classical (cc) state +σXY = +� +x,y +p(x)p(y|x)|x⟩⟨x|X ⊗ |y⟩⟨y|Y. +(12) +where p(y|x) = Tr(Πyσx) is the constructed MAC in +QN(X; Y). If X denotes the random variable over +all N messages and Y denotes the output variable for +the receiver, then the information obtained by the +4 + +Particle +sourceFIG. 3. An illustration of our bounds on the one-sender coherence-assisted communication rate R(Qass +1 ) and two- +sender unassisted communication rate sum R(Q2). Each bound is established via the labeled proposition or theorem. +receiver about X is the mutual information I(X : +Y )σXY. Optimizing over all POVMs quantifies the +so-called accessible information of the cq state σXA, +Iacc(σXA) := max +{Πy} I(X : Y )σXY. +(13) +We then further optimize the accessible information +over all valid cq states (i.e. those having the form of +Eq. (11)), +R(QN) := max +σXA Iacc(σXA). +(14) +Thanks to Proposition 1, +R(QN) captures the +largest communication rate-sum that quantum me- +chanics allows when using a fixed encoding strategy +and decoding measurement on each particle. This is +the central quantity of interest in this paper. +Lower bounds of R(QN) are given by R(QN) ≥ +Iacc(σXA) ≥ I(X : Y )σXY, with I(X : Y )σXY aris- +ing from any explicit protocol. On the other hand, +the celebrated Holevo’s bound limits the accessible +information as +Iacc(σXA) ≤ χ(σXA) +(15) +where χ(σXA) := I(X : A)σXA = S +�� +x p(x)σA +x +� +− +� +x p(x)S +� +σA +x +� +is called the Holevo information [22]. +Therefore, a natural upper bound for R(QN) is +R(QN) = max +σXA Iacc(σXA) ≤ χ(QN) := max +σXA χ(σXA). +(16) +Again, the maximization is over cq state having the +form of Eq. (11). +In addition to providing an upper bound, the +Holevo information χ(σXA) admits an operational +interpretation [23, 24] within our one-particle com- +munication framework. Namely, it captures the sce- +nario in which the senders prepare independent and +identically distributed (i.i.d.) copies of σXA, yet the +receiver is allowed to perform joint decoding mea- +surement across all copies. In the asymptotic limit, +the largest amount of information that the receiver +can gain is exactly the Holevo information χ(σXA). +Therefore, the optimized Holevo information χ(QN) +represents the ultimate amount of information that +can be transmitted by N senders using a fixed single- +particle encoding scheme. We similarly let R(Qass +N ) +and χ(Qass +N ) be defined as in Eqns. (14) and (16), +respectively, except with the maximum now taken +over all cq states σXA built using a coherence assis- +tance path. +While R(QN) ≤ χ(QN) and R(Qass +N ) ≤ χ(Qass +N ), +for general N these bounds appear to be quite loose. +For example, we show below that χ(QN) ≥ log N +and χ(Qass +N ) ≥ log(N + 1). On the other hand, the +best lower bounds on R(QN) and R(Qass +N ) we obtain +do not even exceed 1.13. While this bound still ex- +ceeds the largest classical rate, which is the main fo- +cus of this paper, its divergence from the Holveo in- +formation reflects the strong communication degra- +dation that arises when restricting to single-copy +measurements. +5 + +Prop. 8 +Prop. 5 +Thm. 7 +Thm. 4 +Classical region R(Cn) +Prop. 2III. +THEORETICAL RESULTS +Having established our communication model, we +now probe the theoretical limits of single particle +communication in both the classical and quantum +settings. Our main goal is to place bounds on the +communication rates introduced in the previous sec- +tion. +For simplicity, we focus on multiple-access +channels with binary and ternary inputs/outputs. +In Section III A we compute the ultimate commu- +nication rates using a classical particle, which serve +as thresholds for our quantum protocols. In Section +III B we construct explicit quantum-enhanced com- +munication protocols. Lower bounds on R(Qass +1 ) and +R(Q2) are presented in Sections III B 1 and III B 2, +respectively; for N ≥ 2 lower bounds on R(QN) +and R(Qass +N ) are provided in Section III B 3; and fi- +nally in Section III B 4 we show that both χ(QN) +and χ(Qass +N ) grow as log N. +A. +Classical MACs +1. +N-party rate-sum +We begin by establishing the intuitive upper +bound of one bit for the N-party rate-sum using a +single classical particle. The following proposition +places a fundamental bound on N-party communi- +cation within our framework. +Proposition 2. R(CN) = 1 ∀ N. That is, we can +communicate at most 1 bit of information using a +classical particle. Furthermore, an assistance path +does not help in the classical setting. +Proof. We first show that R(CN) ≤ 1. According to +Eq. (8), any channel in CN admits the decomposition +p(y|x1 · · · xN) = +� +i +pi +� +m=0,ei +d(y|m)qi(m|xi) (17) +where d(y|m) and q(m|xi) are conditional probabil- +ity distributions associated with the decoder and the +encoder, respectively. By convexity of mutual infor- +mation I(X1 · · · XN : Y ) with respect to the under- +lying channel, we can conclude that the rate sum is +maximized by channels of the form p(y|x1 · · · xN) = +p(y|xi) = � +m=0,ei d(y|m)qi(m|xi). However, capac- +ities of these channels cannot exceed one bit since +d(y|m) is essentially a classical post-processing map, +and qi(m|xi) is a channel with binary outputs. +On the other hand, suppose the initial state is +|ei⟩⟨ei|, the i-th sender Ai encodes information by +either annihilating the particle or preserving the par- +ticle, and the receiver performs measurement in the +particle number basis. In this case, Ai can send 1 bit +of information, while other senders cannot send any +information. So, the total amount of transmitted in- +formation is one bit, and therefore R(CN) ≥ 1. To +see that an assistance path does not help, observe +that R(CN) = 1 holds for arbitrary N, and an as- +sistance path can be seen as a special case of CN+1 +where the (N + 1)-th party acts trivially. +2. +Classical canonical form +We next turn to the problem of identifying achiev- +able rate tuples using a single classical particle. This +task is simplified by recognizing that every canonical +MAC can be obtained from a canonical MAC com- +bined with stochastic encoders and a stochastic de- +coder. By the data processing inequality, stochastic +post-processing cannot increase the rate region, and +the same is true for stochastic pre-processing (Prob- +lem 14.5 in [25]). +Therefore, if (R1, · · · , RN) is a +rate tuple achievable by some single-particle classi- +cal MAC, then it is also achievable by a canonical +one defined below. +Proposition 3. For arbitrary input and output sets +X1×· · ·×XN and Y, every MAC in CN(X1, · · · XN; Y) +can be seen as arising from a canonical MAC that +has binary inputs for each sender and N +1 outputs +for the receiver. +Proof. For +a +given +classical +state +ρ += +�N +k=1 pk|ek⟩⟨ek| and induced MAC p(y|x1, · · · , xN) +having the form of Eq. +(8), define the canonical +MAC with transition probabilities +�p(k|j1, · · · , jN) = +� +pk +if jk = 1 +0 +if jk = 0 ; +(18a) +�p(0|j1, · · · , jN) = +� +k +such that jk = 0 +pk. +(18b) +This channel likewise has the form of Eq. (8) and +therefore belongs to C([2], · · · , [2]; [N + 1]). +Also, +define local pre-processing stochastic maps �qi +: +Xi → {0, 1} with �qi(0|xi) = qi(0|xi) and �qi(1|xi) = +qi(ei|xi), along with a post-processing stochastic +map �d : {0, 1 · · · , N} → Y by �d(y|k) = d(y|ek) +for k = 1, · · · , N and �d(y|0) = d(y|0). Then it is +6 + +FIG. 4. The shaded region is the union of all achiev- +able rate pairs as the weight λ of the source state +ρcl = λ|e1⟩⟨e1|+(1−λ)|e2⟩⟨e2| varies over interval [0, 1]. +straightforward to verify that +p(y|x1, · · · , xN) = +N +� +k=0 +1 +� +j1=0 +· · · +1 +� +jN=0 +{�d(y|k) +× �p(k|j1, · · · , jN)�q1(j1|x1) · · · �qN(jN|xN)}. (19) +3. +Two-sender classical rate regions +We now turn to the rate regions for two-sender +communication. +Consider the canonical MAC +p(y|x) that is generated by a classical particle ρcl = +λ|e1⟩⟨e1| + (1 − λ)|e2⟩⟨e2| and having the structure +of Eqns. (18a) and (18b). Since N = 2, the canon- +ical MAC is characterized by the single parameter +λ = p1, and the transition probabilities are given by +1 = p(00|00) +λ = p(01|01) = p(00|10) = p(01|11) +1 − λ = p(10|10) = p(00|01) = p(10|11). +(20) +For a fixed λ ∈ [0, 1], and prior p(x1)p(x2) the +achievable rate pairs (R1, R2) are determined by +Proposition 1, +which follows a pentangon con- +strained by Eq. 10. Combining all these regions with +with fixed λ ∈ [0, 1] but different priors p(x1)p(x2), +we could obtained the achievable rate region of a +specific MAC. +We are now interested in computing the union of +all achievable rate regions as λ is varied within the +interval [0, 1]. This will yield the total collection of +all asymptotic rate pairs (R1, R2) feasible by a MAC +built using a single classical particle. +Note that a rate pair (R1, R2) lies in the en- +closed region of Fig. 4 if and only if it is achiev- +able using many copies of the same source state +ρcl = λ|e1⟩⟨e1| + (1 − λ)|e2⟩⟨e2|, and the union of +these rate pairs evidently forms a non-convex set. +However, if we relax this i.i.d. constraint and allow +λ to vary across the multiple uses, then more rate +pairs are accessible by time-sharing. In this case, the +collection of achievable rate pairs is just the convex +hull of the region in Fig. 4, i.e. a triangle with outer +vertices (1, 0) and (0, 1). +B. +Quantum MACs +1. +Surpassing the classical bound with one sender and +coherence assistance +Given the classical communication bounds estab- +lished in the previous section, it is natural to con- +sider whether quantum mechanics can do better. +We begin by considering the special case of just +one sender, and the encoding scheme presented here +will generalize as more parties are added. +In the +one-sender scenario, if no coherence assistance is +used then the whole communication system is sim- +ply a two-dimensional space spanned by {|0⟩, |1⟩}. +By Holevo’s theorem, the communication rate is +bounded above by log 2 = 1, and therefore, quantum +mechanics offers no advantage over classical physics. +However, by leveraging coherence assistance in the +sense of Fig. 5, it is possible to communicate more +than one bit of information in the point-to-point sce- +nario. +To achieve a greater capacity using a single parti- +FIG. 5. +Coherence-assisted communication with one +sender. +7 + +R2 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +0.0 +R1 +0.2 +0.4 +0.6 +0.8 +1.0Particle +sourcecle, we construct a channel with ternary input sym- +bols. Suppose that the initial state distributed from +the particle source is |ψ⟩AR = cos θ|e1⟩ + sin θ|e2⟩ +with θ ∈ [0, π/2]. Note that this describes the most +general one-particle state since any relative phase +can be absorbed into the definition of |e1⟩, which +we assume is known to the receiver. For message +x ∈ {0, 1, 2}, let the sender A encode the state |ψ⟩ +according to the following NPE operations: +� +� +� +� +� +E0(ρ) = E(vac)(ρ) = Tr(ρ)|0⟩⟨0|; +E1(ρ) = ρ; +E2(ρ) = E(α)(ρ) = e−iαZ/2ρeiαZ/2. +(21) +Let +σx += +EA +x +⊗ idR(|ψ⟩⟨ψ|) +and +σXAR += +� +x p(x)|x⟩⟨x| ⊗ σx be an encoded cq state with +the prior distribution over messages have the form +p(0) = 1 − q and p(1) = p(2) = q/2. +As shown +in Section III B 4, the Holevo information χ(Qass +1 ) is +attained by this type of cq state. Hence, we are mo- +tivated to conjecture that the encoding scheme of +Eq. (21) is also optimal for the single-particle rate +R(Qass +1 ). Even if this conjecture fails to be true, the +accessible information of σXAR still provides a lower +bound on R(Qass +1 ). +In general, calculating the accessible information +of an arbitrary cq state is mathematically challeng- +ing. However, in our case, the encoded cq state en- +joys the following symmetries: (i) each σx is block- +diagonal in the particle number basis, and (ii) q/2·σ1 +and q/2 · σ2 are related by a reflection across the +line y = x tan(α/2) in the x − y plane of the Bloch +sphere. +Using similar arguments to those in Ref. +[26], we find (see the Supplemental Material) that +α = π provides an optimal encoding. Further anal- +ysis then shows that the accessible information is +maximized by a prior probability q and coherence +angle θ in the source state that together satisfy a pair +of transcendental equations. Solving these equations +numerically leads to the following theorem. +Theorem 4. There exists a one-sender coherence- +assisted communication protocol that sends approx- +imately 1.0931 bits of information, i.e., R(Qass +1 ) ≥ +1.0931. +The optimal (q, θ) that achieves this +are approximately (0.8701, arccos( +√ +0.4715)), and +the optimal measurement projects into the basis +{|00⟩, +1 +√ +2(|e1⟩ ± |e2⟩)}. +Note that the largest accessible information is not +attained using a state with uniform superposition +across both paths. Yet, the optimal decoding mea- +surement is a projection into uniform superposition +states +1 +√ +2(|e1⟩ ± |e2⟩). When using a source state +with uniform superposition across both paths (i.e. +θ = π/4), the largest communication rate is com- +puted to be 1.0875. +2. +Two-sender MACs +Let us now add a second sender to the communi- +cation picture. We first consider the scenario of two +senders with no coherence assistance. +Thus there +are only two paths connecting the source to the re- +ceiver, and we borrow ideas from the previous one- +sender coherence-assisted protocol, which also has +two paths. Let senders A1 and A2 share the state +|ψ⟩A1A2 = cos θ|e1⟩ + sin θ|e2⟩. +Consider first the +following binary encoding strategy: +� +EA1 +0 (ρ) = Tr(ρ)|0⟩⟨0|; +EA1 +1 (ρ) = ρ; +� +EA2 +0 (ρ) = ρ; +EA2 +1 (ρ) = e−iαZ/2ρeiαZ/2. +(22) +Observe that EA1 +0 ⊗EA2 +0 (|ψ⟩⟨ψ|) = EA1 +0 ⊗EA2 +1 (|ψ⟩⟨ψ|), +and so there only three distinct encoded states. In +fact, if A1 has prior probabilities {1 − q, q} over +messages {0, 1} and A2 has uniform prior probabil- +ities over the messages, then the resulting cq state +σXA1A2 is equivalent to the cq state σXAR constructed +in the one-sender assisted protocol. Therefore, by +Theorem 4, α = π is optimal, and the maximal +rate sum achievable with this protocol is 1.0931, +which is achieved by the source state +√ +0.4715|e1⟩ + +√ +0.5285|e2⟩ and encoding probability q ≈ 0.8701. +The full rate region can also be computed. For +each fixed θ ∈ [0, π/2], the initial state |ψ⟩A1A2 = +cos θ|e1⟩ + sin θ|e2⟩ induces a classical MAC [2] × +[2] → [3] when using the encoding of Eq. (22) and +the decoding measurement which projects into the +basis {|00⟩, +1 +√ +2(|e1⟩ ± |e2⟩)}. The specific transition +probabilities are found to be +p(0|00) = cos2 θ, +p(1|00) = p(2|00) = sin2 θ +2 +p(0|01) = cos2 θ, +p(1|01) = p(2|01) = sin2 θ +2 +p(1|10) = 1 +2 + cos θ sin θ, +p(2|10) = 1 +2 − cos θ sin θ +p(1|11) = 1 +2 − cos θ sin θ, +p(2|11) = 1 +2 + cos θ sin θ. +The rate region (R1, R2) is then found using Propo- +sition 1 (see Fig. 6). As we sweep θ over the interval +8 + +[0, π/2], the union of all achievable rate pairs using +encoding scheme (22) is identified in Fig. 7. The +solid line in this figure indicates the outer boundary +on achievable rates using a uniform superposition +input state |ψ⟩ = +1 +√ +2(|e1⟩ + |e2⟩). These values are +noteworthy since they are what we try to experimen- +tally replicate in Section IV. +We can enhance the rate sum even further if we al- +low one of the parties to have three inputs. Suppose +now that A2 encodes with the same ternary oper- +ation as in Eq. (21), and A1 again uses the on-off +keying encoding: +� +EA1 +0 (ρ) = Tr(ρ)|0⟩⟨0|; +EA1 +1 (ρ) = ρ; +� +� +� +� +� +EA2 +0 (ρ) = Tr(ρ)|0⟩⟨0|; +EA2 +1 (ρ) = ρ; +EA2 +2 (ρ) = e−iαZ/2ρeiαZ/2. +. +(23) +Suppose that Alice and Bob’s prior probability of +message 0 is q and q′, respectively. Then using the +same method of calculating the accessible informa- +tion of symmetric ensembles (see the Supplemental +Material), we again find that the optimal phase en- +coding is α = π. +This allows us to calculate the +accessible information of the encoded cq state for +any q, q′, and θ, which we then maximize. +FIG. 6. An example of rate region that is achievable by +using initial state |ψ⟩ = +� +1/3|e1⟩ + +� +2/3|e2⟩ and the +binary-input protocol. +FIG. 7. The union of all achievable rate regions using our +binary-input protocol (gray area). The solid line repre- +sents the boundary of rate region that is achievable us- +ing an equal superposition state +� +1/2(|e1⟩ + |e2⟩). The +dashed line represent the convex hull of all rate pairs +achievable using a classical particle. +Proposition 5. There exists a two-sender unas- +sisted communication protocol [2] × [3] → [3] that +sends 1.1014 bits of information per channel use, i.e., +R(Q2) ≥ 1.1014. The optimal (q, q′, θ) that achieves +this are approximately (0.9197, 0.9197, π/4), and the +optimal measurement is given by projecting on the +basis {|00⟩, +1 +√ +2(|e1⟩ ± |e2⟩)}. +Note that unlike in this case of binary encoding, +the optimal source state is a uniform superposition +across both paths (i.e. θ = π/4). +3. +A general encoding method for N ≥ 2 parties +without blocking +One drawback of the encoding schemes presented +in Eqns. (22) and (23) is that it requires one of the +parties to perform an on-off keying (i.e “blocking”) +operation. While intuitively simple, a reliable imple- +mentation of this encoding in an optical setup can +be quite demanding. +Here we show that through +the use of a coherence assistance path, a rate sum +strictly larger than one is always achievable using +simple 0, π phase encoding. The latter means that +9 + +1.0 +0.8 +0.6 +2 +R +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +R11.0 +Classicalparticle +Achievablewithequalsuperpositionstate +Achievablewithaguantumparticle +0.8 +0.6 +R +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +R1the sender either acts trivially on the particle or ap- +plies a rotation E(π)(ρ) = ZρZ. +Our protocol involves the idea of creating more +paths by successive uses of a beam splitter (see +Fig. +8). +Suppose that at the layer we start with +the uniform superposition state +1 +√ +2(|e1⟩ + |e2⟩). A +beam splitter is inserted along the second path yield- +ing the state +1 +√ +2|e1⟩ + 1 +2(|e2⟩ + |e3⟩). +This is re- +peated repeatedly until the initial state |ψ⟩A1···ANR = +�N +i=1 +1 +√ +2i |ei⟩+ +1 +√ +2N |eN+1⟩ is prepared for N senders +A1, · · · , AN and a coherence assistance path R. Each +sender encodes by applying a π phase shift +EAi +xi (ρ) = ZxiρZxi +(24) +for message xi +∈ +{0, 1} with prior probability +p(xi). +Upon receiving the encoded particle, the +receiver decodes using the projective measurement +{|bi⟩⟨bi| : i ∈ [N + 1]} with orthonormal vectors +|b0⟩ = +1 +√ +2|e1⟩ + +N +� +i=2 +1 +√ +2i |ei⟩ + +1 +√ +2N |eN+1⟩ +|b1⟩ = − 1 +√ +2|e1⟩ + +N +� +i=2 +1 +√ +2i |ei⟩ + +1 +√ +2N |eN+1⟩ +|b2⟩ = − 1 +√ +2|e2⟩ + +N +� +i=3 +1 +√ +2i−1 |ei⟩ + +1 +√ +2N−1 |eN+1⟩ +... +|bN⟩ = − 1 +√ +2|eN⟩ + 1 +√ +2|eN+1⟩. +This induces a classical channel p(y|x1, · · · , xN), and +for small N we can numerically compute their ca- +pacities using the generalized Blahut-Arimoto al- +gorithm adapted for MACs [27–29]. +The result +FIG. 8. A multi-path state with attenuated amplitudes +is generated by a successive application of beam splitters. +FIG. 9. Numerical calculation of rate sums achievable +with our N-sender protocol [Eq. (24)] for N up to 10. +is presented in Fig. +9. +Note that the general- +ized Blahut-Arimoto algorithm is not guaranteed +to converge to the optimal rate sum [30]. +How- +ever, let us consider p(y|x1, · · · , xN) as a single- +sender-single-receiver channel. The original Blahut- +Arimoto algorithm does in fact converge to its op- +timal point-to-point capacity. +This point-to-point +capacity serves as an upper bound for the rate sum +of p(y|x1, · · · , xN) since we are giving senders more +power to coordinate. +For N = 2, the problem allows for an analytic +solution, and we summarize the result in the propo- +sition below. +Proposition +6. +There +exists +a +two-sender +coherence-assisted +communication +protocol +that +does not require blocking operation or vacuum detec- +tion and sends log(17/8) ≈ 1.0875 bits per channel +use, i.e., R(Qass +2 ) ≥ 1.0875. +The optimal prior +probability that achieves this is p(x1 = 0) = 1/2 +and p(x2 = 0) = 15/17. +As N increases, we numerically find that the rate +sum does not increase significantly. +On the one +hand, this is not surprising since our encoding strat- +egy uses an initial state |ψ⟩ that places smaller and +smaller weight on the paths of additional parties. +However, on the other hand, we have not been able +to find any superior coding method, and in fact, +many coding schemes (such as the “fingerprinting” +protocol [6, 10]) have a rate sum that vanishes as +N grows large. A significant open problem is to find +upper bounds on the largest N-party rate sum using +a single quantum particle, which we conjecture will +10 + +Particle +Source1.125 +1.120 - +Achievable rate sum +1.115 +1.110 +N +Rate sum +N +Rate sum +N +Rate sum +1.105 +2 +1.0875 +5 +1.1220 +8 +1.1244 +1.100- +3 +1.1105 +6 +1.1235 +9 +1.1245 +4 +1.1187 +7 +1.1241 +10 +1.1246 +1.095 +x +B-A algorithm for MACs +1.090- +Point-to-point capacity +2 +3 +4 +5 +6 +7 +8 +9 +10 +Number of sendersnot be too far from the lower bound depicted in Fig. +9. +The coherence-assisted protocol just described +uses only phase encodings. +However, it can eas- +ily be converted to a coherence-unassisted commu- +nication protocol at the expense of needing block- +ing operations. To see the idea, consider the case +of N = 2. In the unassisted protocol, the encoded +states σA1A2R +x1x2 += |ψx1x2⟩⟨ψx1x2| have the form +|ψ00⟩ = +1 +√ +2|e1⟩ + 1 +2|e2⟩ + 1 +2|e3⟩ +|ψ01⟩ = +1 +√ +2|e1⟩ − 1 +2|e2⟩ + 1 +2|e3⟩ +|ψ10⟩ = − 1 +√ +2|e1⟩ + 1 +2|e2⟩ + 1 +2|e3⟩ +|ψ11⟩ = − 1 +√ +2|e1⟩ − 1 +2|e2⟩ + 1 +2|e3⟩. +Observe that these are made equivalent to the states +|ψ′ +00⟩ = +1 +√ +2|e1⟩ + 1 +√ +2|e2⟩ +|ψ′ +01⟩ = +1 +√ +2|e1⟩ + 1 +√ +2|e3⟩ +|ψ′ +10⟩ = − 1 +√ +2|e1⟩ + 1 +√ +2|e2⟩ +|ψ′ +11⟩ = − 1 +√ +2|e1⟩ + 1 +√ +2|e3⟩ +(25) +by a unitary operator that also transforms the mea- +surement vectors into +|b′ +0⟩ = +1 +√ +2|e1⟩ + 1 +√ +2|e2⟩ +|b′ +1⟩ = − 1 +√ +2|e1⟩ + 1 +√ +2|e2⟩ +|b′ +2⟩ = |e3⟩. +(26) +Hence, the states of Eq. (25) and measurement of +Eq. (26) will generate the same transition probabil- +ities as the original MAC. But since the |b′ +i⟩ have no +coherence between the {|e1⟩, |e2⟩} and {|e3⟩} sub- +spaces, we can first dephase the |ψ′ +x1x2⟩ across these +subspaces without altering the transition probabil- +ities. +Doing so and relabeling |0⟩ ≡ |e3⟩ leads to +states σx1x2 obtained by the unassisted encoding of +Eq. (22) (up to a swap A1 ↔ A2). This method +of converting a coherence-assisted protocol to an +unassisted-protocol generalizes for any N ≥ 2. +4. +The single-particle Holevo capacities +All of the communication rates computed thus far +assumes the receiver performs the same measure- +ment on each received quantum particle so as to +generate multiple uses of the same classical channel +p(y|x). While this leads to a definite communication +advantage compared to the use of a classical particle, +Fig. 9 suggests that this advantage is not that dra- +matic. On the other hand, if we enlarge the measure- +ment capabilities of the decoder and allow for collec- +tive measurements across multiple particle transmis- +sions, then the capacity can be enlarged significantly. +This quantity is the single-particle Holevo informa- +tion χ(QN) as defined in Eq. (16), with χ(Qass +N ) de- +noting its coherence-assisted form. +Our first result is the calculation of χ(Qass +N ) for +N = 1. +Theorem 7. +χ(Qass +1 ) = +max +q,cos2 θ∈[0,1] qh2(cos2 θ) + cos2 θh2(q) +≈ 1.2339, +where h(x) = −x log2 x − (1 − x) log2(1 − x) is the +binary entropy. +Note that since R(Qass +1 ) ≤ χ(Qass +1 ), this shows that +the encoding scheme of Theorem 4 is not too far from +optimal. The proof of this Theorem 7 is provided in +the Supplemental Material. As an intermediate step +in our proof, we show that the encoding strategy +of Eq. (21) maximizes the Holevo information for +each choice of initial state. +Then optimizing over +the initial state the maximum in Theorem 7 to be +obtained by the values (q, cos2 θ) ≈ (0.7035, 0.7035). +Turning to the N-sender case, +we find that +the single-particle Holevo information grows un- +bounded, in sharp contrast to the optimised accessi- +ble information, which we do not know exactly, but +seems to remain bounded for all N despite our best +efforts in searching for better protocols. +Proposition 8. log N ≤ χ(QN) ≤ log(N + 1) and +log(N + 1) ≤ χ(Qass +N ) ≤ log(N + 2). +To achieve the lower bounds, the parties use an +equal superposition state |ψ⟩ = � +i +1 +√ +N |ei⟩ and 0, π +phase encoding. If each local message xi has uniform +prior over {0, 1} then, the average encoded state is +� +x p(x)σA +x = +1 +N +�N +i=1 |ei⟩⟨ei|. +Hence, χ(QN) ≥ +log N. For the assisted case, a similar construction +yields χ(Qass +N ) ≥ log(N + 1). +The upper bounds +are simply dimensionality bounds based on the total +11 + +number of dimensions of the communication system. +The lower bound given here is in general not tight. +For instance, when N = 2, Proposition 5 shows that +χ(QN) ≥ 1.1014. +IV. +EXPERIMENTAL DEMONSTRATION +OF ENHANCED MULTIPLE-ACCESS +COMMUNICATION USING A SINGLE +PHOTON +We have applied our communication framework +to a multi-port optical interferometer experiment in +which each sender controls one path that the parti- +cle can traverse. A single photon is used as the mes- +sage carrier, prepared into the desired superposition +mode via the interferometric structure. +Messages +are coherently encoded by different senders onto the +photon along each optical path of the interferometer +and decoded by the single receiver, who collects the +photon at the output ports of the channel. +Not all the communication protocols described +above can be faithfully implemented using such a +setup, due to various unavoidable experimental im- +perfections, including finite transmission and de- +tection efficiencies, a non-ideal probabilistic single- +photon source with multi-photon pair generation, +and imperfect interference visibility of the optical +interferometer. +In particular, the photon loss in- +curred from the finite detection/transmission effi- +ciency prevents us from exploring the vacuum mode +as a valid decoding outcome. +Furthermore, the +quantum enhancement is extremely sensitive to in- +terferometric visibility, as we will explain in detail +later. Taking all these factors into consideration, the +most viable experiment to conduct is the two-sender +coherence-assisted communication protocol (Propo- +sition 6) presented in Section III B 3. The advantage +of this scenario is that quantum-enhanced communi- +cation can be achieved using only phase encoding by +each sender. However, as argued in Section III B 3, +the communication rates are the same as in a two- +sender unassisted protocol using path blocking and +phase encoding on the uniform superposition state +1 +√ +2(|e1⟩ + |e2⟩) (see the solid line in Fig. 7). +The experimental setup for this protocol is shown +in Fig. 10. A heralded single photon is created from +spontaneous parametric down conversion (SPDC), +and sent to a three-port interferometer with split- +ting ratio 1/2 : 1/4 : 1/4. The single-photon state is +filtered with a polarizer and spectral filter and cou- +pled into single-mode fiber (SMF), which allows us +to ignore all of its internal degrees of freedom and +write down the corresponding heralded state as a +superposition of different path basis states |ei⟩ = +|0⟩A1 · · · |1⟩Ai · · · |0⟩AN : +|ψ⟩ = +1 +√ +2|e1⟩ + 1 +2|e2⟩ + 1 +2|e3⟩, +(27) +where the third path is the assistance path, while +senders 1 and 2 each encode their input bits onto +the photon locally with tunable phase shifters in the +form of glass windows. The phase shifters are char- +acterized with respect to the angle of rotation of +the glass window and a phase-shift of 0 is set to +encode the bit “0” and π to encode the bit “1”. At +the output ports of the interferometric setup, single- +photon detectors are placed and information is de- +coded purely based on the which-port information. +A. +Experimental results +In order to claim the implementation of a com- +munication protocol with only one single particle +involved, we characterize the heralded second-order +cross-correlation function at zero delay g(2) +hcc(0) of our +photon-pair source. For an ideal source this num- +ber should be 0, which means exactly one photon is +produced in a heralded manner; however, without +a perfect photon-number-resolving detector, there +will always be a trade-off between having a higher +heralded-single-photon rate and lower g(2) +hcc(0). We +measure g(2) +hcc(0) = 0.0017 ± 0.001, which can basi- +cally rule out the possibility of having more than one +particle traveling into the communication setup after +heralding. This value is set to be an order of magni- +tude smaller than our expected quantum violation, +as we will elaborate on later. +Non-ideal single-photon source with small two +multi-photon contributions, encoding operations, +and/or decoding detections can all degrade the per- +formance of our quantum protocol to some extent. +Among them, most error in the setup is actually +caused by the non-unit interference visibility. Ide- +ally, when the three-port optical interferometer has +perfect interference visibility the following transition +probabilities can be achieved: +p(1|10) = 1, +p(2|11) = 1, +p(0|00) = 1 +2, +p(1|00) = 1 +4, +p(2|00) = 1 +4 +p(0|01) = 1 +2, +p(1|01) = 1 +4, +p(2|01) = 1 +4. +(28) +However, the communication protocol is extremely +sensitive to the interference visibility, as shown in +12 + +FIG. 10. Experimental setup: (a) Photon pairs are generated by pumping a PPKTP crystal with the second harmonic +of a pulsed laser (generated in BiBO); (b) The heralded single photons are sent to a three-port optical interferometer +consisting of an inner Sagnac loop and an outer Mach Zehnder (MZ) interferometer with information encoded by +auto-controlled phase plates; (c) The heralded single photons are coupled into single-mode fibers (SMF) and detected +by avalanche photodiodes (APD); different combinations of coincidence counts are processed by a time-to-digital +converter (TDC). +Fig. 11. +To obtain a greater quantum enhance- +ment with better interferometric visibility, we devise +a three-port optical interferometer comprised of a +passively stabilized Mach-Zehnder (MZ) interferom- +eter with an offset Sagnac interferometer embedded +within it. The visibility of the Sagnac interferometer +is achieved to above Vs = 99.5 ± 0.2% after tightly +filtering the single photon spectrally and spatially, +and the visibility for the outer MZ interferometer is +around Vz = 98.2 ± 0.4% averaged over 10 minutes. +Our experimental demonstration of quantum ad- +vantage comes in two forms. We first build a channel +having transition probabilities close to those of Eq. +(28). With this channel, it is, in principle, possible +to achieve asymptotic communication rates strictly +larger than what is possible using a classical par- +ticle. Second, we go one step further and actually +use the channel to establish correlated random vari- +ables between the senders and receivers whose mu- +tual information is above one, thereby exceeding the +accessible information of a classical particle. +FIG. 11. Expected enhanced two-sender communication +rate as a function of the interference visibilities of the +inner Sagnac and outer Mach-Zender (MZ) interferom- +eters. The maximal capacity rate of 1.0875 is achieved +when perfect interference visibility is obtained. +13 + +Time-to-digita +converter +Single photon +detector +SMFC +Lens +Spetral +filter +Tsunami +Dichroic +Modelocked +Mirror +BiBO +Laser +Beam splitter +PPKTP +Polarization +Phase +Beam splitter +Shifter +Mirror +DelayRate sum +100 +☆ Our experiment +1.08 +Visibility (%): Sagnac interferometer +99.8 +1.07 +1.06 +99.6 +1.05 +1.04 +99.4 +1.03 +99.2 +1.02 +1.01 +99 +100 +99.6 +99.2 +98.8 +98.4 +98 +Visibility (%): Mach-Zehnder interferometerFIG. 12. (a) Example of transition probability p(y|x) from direct characterization of the two-sender channel where +inputs x = (x1, x2) ∈ {0, 1}×2 and output y ∈ {0, 1, 2} (b) The union of achievable rate regions with the corresponding +channel, with experiment in blue and the ideal case in grey. The dashed line represents the bound of the rate region +achievable by a classical MAC. (c) Comparing our empirical joint distribution p(x, y)empirical to its theoretical value +for two-sender channel inputs x ∈ {0, 1}×2 and output y ∈ {0, 1, 2} where error bars are statistical uncertainty. +1. +Characterizing a two-sender assisted channel by +transition probabilities +To demonstrate quantum enhancement in the two- +sender communication protocol, we first character- +ize the transition probability of the channel p(y|x = +(x1, x2)), where xi is the bit encoded by sender i +corresponding to 0 (π) phase for xi = 0 (xi = 1), +while y is the trit decoded by the receiver based on +the “which-port” information of the output particle +measured. Given the low g(0) +hcc we set, we character- +ize each transition probability with different inputs +x = (x1, x2) by registering coincident counts over +a three minute period with around N = 105 events +registered (see Fig. 12 (a)). +Using +the +measured +transition +probabilities +p(y|x), the asymptotic rate region for the con- +structed channel can be computed. +In particular, +for the ideal channel of Eq. +(28), the mutual in- +formation between senders and receivers is found to +be maximized by a uniform prior distribution for +x1 ∈ {0, 1} and a biased distribution for x2 ∈ {0, 1} +with Pr{x2 = 0} ≈ 15 +17. With X = (X1, X2) denot- +ing input variables with these distributions, our con- +structed channel can thus achieve an input-output +mutual information of +I(X : Y )ch = +� +x,y +p(x)p(y|x) log p(y|x) +p(x) += 1.0152 ± 0.0034, +(29) +where the error is the standard deviation over 10 +runs of the experiment to take both statistical and +systematic error into consideration (the estimation +on the statistical error is given in the supplemen- +tary material). More generally, by varying the prior +p(x) = p(x1)p(x2), a different rate region is deter- +mined by the three mutual information quantities +{I(X1 : Y |X2), I(X2 : Y |X1), I(X1X2 : Y )} via +Eq. (10). The union of these regions is presented in +Fig. 12 (b). +2. +Characterizing a two-sender assisted channel by +mutual information +We take the demonstration further by generating +empirical random variables (X, Y ) that are corre- +lated using the single-particle channel we build. Ide- +ally we would like their mutual information I(X : Y ) +to be close to the maximum accessible information +R(Qass +2 ), but any value larger than one will already +yield a quantum advantage. To this end, we gen- +erate multiple series of random bits each of length +680 by independently sampling from the input set +{0, 1} with uniform probability p(0) = 1/2 for in- +put x1 and biased probability p(0) = 15 +17 for input +x2. Ideally, each sample would correspond to a spe- +cific choice of encoding in one run of the experiment. +However, in practice we can only change the encod- +ing map once per second. Hence, the ensemble we +generate has the form {p(x), σ⊗m +x +}⊗n rather than +(ideally) {p(x), σx}⊗mn, where both n = 680 and +m ≈ 600 to be the coincident count rates. Even if +we assume that the decoder does not try to exploit +this block structure (see the discussion on loopholes +below), there are still two sources of uncertainty in +this setup: (a) the generation of the random bit and +(b) the photon number fluctuation in each run of the +14 + +(a) +Theory +(b) +(c) +Theory +P(y|x=00) +P(y|x=10) + Experiment +1.00 +1.00 +1.0 +Experiment +Ideal +0.75 +0.75 +Experiment +0.5 +0.50 +0.50 +0.8 +0.4 +0.25 +0.25 +R2 +0.3 +0.00 1 +0.00 +y=o +y=1 +y=2 +y=o +y=1 +y=2 +0.2 +acity +P(y|x=01) +P(y|x=11) +0.1 +1.00 +1.00 +0.0 +0.75 +0.75 +x=11 +0.50 +0.2 +0.50 +x=10 +0.25 +0.25 +y=0 +x=01 +0.00 +0.00 +0.0. + +y=1 +y=o +y=1 +y=2 +y=o +y=1 +y=2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +X=00 +Capacity rate: Ri +y=2experiment. The result is a mutual information with +larger uncertainty and larger bias than I(X : Y )ch, +yet still above the classical threshold: +I(X : Y )empirical = +� +x,y +p(x, y) log p(x, y) +p(x)p(y) += 1.0117 ± 0.0047, +(30) +where again the error is the standard deviation over +10 runs of the experiment. Here, I(X : Y )empirical +is computed using the empirical joint distribution +p(x, y)empirical compiled from both the input and +output data. +B. +Experimental imperfections and loopholes +Similar to the problems encountered in most pho- +tonic Bell tests [31–33], our communication frame- +work suffers from several experimental loopholes. +While these can be fixed in principle, they make an +experimental demonstration of enhanced quantum +communication challenging to attain at the single- +particle level. +1. +Detection loophole +In optical experiments, the main difficulty in +demonstrating our theoretical protocols is the lim- +ited photon detection efficiency, which generates +many “no-click” events. +The single-photon detec- +tor we employ (APD, Excelitas SPCM-AQ4C) has +a photon detection efficiency around 40% at our +working wavelength of 810 nm. This ratio can be +improved up to 95% with superconducting single- +photon detectors. +Yet, even this relatively high +efficiency is insufficient to implement many single- +particle communication protocols. +The standard +way of demonstrating a detection-loophole free Bell +test is to classically relabel no-click events as some +other detection event. Unfortunately, this is not a +good strategy in any communication protocol that +uses blocking as an encoding operation since then +“no-click” events are intentionally used to transmit +information. To see this quantitatively, consider the +two-sender unassisted protocol in Section III B 2 that +uses blocking as an encoding operation. When start- +ing with a uniform superposition +1 +√ +2(|e1⟩+|e2⟩) and +following the encodings of Eq. (22), the resulting +channel without detection efficiency has the tran- +sition probabilities of Eq. (28). If we assume the +FIG. 13. +For a non-ideal single photon detector, no +quantum enhancement can be observed in the two-sender +unassisted scenario when the detector efficiency η drops +below roughly 97% (and all other apparatuses behave +flawlessly). +detection efficiency is a constant η for all detectors, +then the transition probabilities are replaced by +p(1|10) = η, +p(0|10) = 1 − η, +p(2|11) = η, +p(0|11) = 1 − η, +p(0|00) = 1 − 1 +2η, +p(1|00) = 1 +4η, +p(2|00) = 1 +4η, +p(0|01) = 1 − 1 +2η, +p(1|01) = 1 +4η, +p(2|01) = 1 +4η. +As shown in Fig. 15, the largest capacity rate sum +of this channel drops below one quickly. A similar +situation occurs if the transmission efficiency is low +(below 97% in the above case), which is almost in- +evitable in optical experiments. +This experimental imperfection leads to two con- +sequences. First, we cannot perform any protocol +with block operations using our current technologies. +Second, even for the case of using just phase en- +coding, our experiment does not close the detection +loophole but instead uses the assumption of “fair +sampling.” In other words, we assume that the ac- +cepted data in our experiment is representative of +the data that would have been recorded if the detec- +tors had unit efficiency [31]. +2. +Freedom-of-choice loophole +The freedom-of-choice loophole has recently been +proposed and fixed in Bell tests [32, 33]. This loop- +15 + +fixed prior +optimizedprior +1.05 +1.00 +0.95 +0.90 +0.85 +1 +0.99 +0.98 +0.97 +0.96 +0.95 +0.94 +0.93 +0.92 +0.91 +0.9 +Detection efficiency nhole refers to the possibility that “hidden variables” +may influence the choice of measurements in experi- +ments and thus enable cheating in acquiring the em- +pirical results. +A similar concern could also be raised in the exper- +imental demonstration presented in Section IV A 2. +As described, the time delay in our ability to switch +the encoding of each sender means that the same +channel input is selected in m = O(103) consecutive +experimental runs. This lack of input freedom for +each trial could be exploited in some classical proto- +col that is attempting to reproduce the same mutual +information I(X : Y )empirical > 1. The ultimate way +of fixing this problem is to independently choose an +input (x1, x2) and apply encoding EA1 +x1 ⊗EA2 +x2 for each +incoming photon. +This requires a phase encoding +operation as fast as 80 MHz in order to match our +laser repetition rate. This can be achieved poten- +tially with electro-optic devices or acousto-optical +devices; however, due to the demanding require- +ments of the overall interference visibility for the +interferometric setup, we could not easily introduce +such components into our setup. +3. +Locality loophole. +In standard Bell experiments, the locality con- +straint is set to prevent the two sites from commu- +nicating with each other [34]. Our experiment has +a similar loophole in that without sufficient separa- +tion between the senders, it is potentially possible +for them to communicate and perform some joint +(i.e. not independent) encoding on the particle. To +avoid this, at least we would need to design the ex- +periment so that the communication time between +senders is much longer than the time it takes the +photon to travel from one sender to the receiver. +In our case, the order of time differences should be +determinate by the coincident window we set, which +requires the spatial separation between senders to be +greater than 2ns × c = 0.6m. Closing this loophole +in our setup is challenging since the overall interfer- +ence visibility and stability is limited by the size of +the interferometer. +V. +CONCLUSION +In the present paper, we investigated how much +information +can +be +transmitted +from +multiple +senders to a single receiver by the use of a single +quantum or classical particle. To analyze this ques- +tion and to show an advantage of quantum over +classical particles, we have created a framework of +classical multiple-access channels constructed by lo- +cally modulating an initial superposition state of +different paths and afterwards detecting the parti- +cle with a general measurement. The classical case +is included when the initial state assigns a definite +path to the particle; on the other hand, if the ini- +tial state is a genuine quantum superposition, it +has the potential to induce channels not reachable +with a classical state. +Specifically, we found that +the communication rates of independent messages +of the separate users show a clear quantum advan- +tage. +Indeed, for a single classical particle, the +rate-sum for any number of senders is bounded by +one, while it exceeds one for two or more sender, +being monotonically increasing in the number of +senders. +The rate-sum can be even larger in the +model of coherence assistance, where there is an- +other path from the source directly to the decoder, +which allows a rate exceeding 1 even for the single- +sender model, to be precise 1.0931 bits per channel +use. We also experimentally demonstrated our pre- +dicted quantum advantage by implementing the two- +sender coherence-assisted protocol using an optical +interferometric setup. The constructed channel sup- +ports a communication rate-sum of 1.0152 ± 0.0034, +showing a four-standard-deviation quantum advan- +tage over the classical bound. +Alternatively, the +channel can be used to correlate random variables +whose mutual information we empirically found to +be I(X : Y )empirical = 1.012±0.005, again exceeding +the classical threshold of one. +We leave a number of open questions regarding +the basic theoretical understanding of the single- +particle MAC, starting with the actual maximum +value of the rate-sum for any number of senders and +the characterization of the full capacity region. Our +best upper bound on the rate-sum is the Holevo +quantity, and while we are just short of calculat- +ing that exactly, it scales as log N for large number +N of senders. +By contrast, we do not even know +if the achievable rate-sum via accessible informa- +tion diverges or not. It seems we would want bet- +ter outer bounds on the capacity region, but it is +perhaps much more exciting to search for improved +modulation and detection schemes. In another di- +rection, fixing the particular initial state, but opti- +mizing over modulations and detection, the achiev- +able rate region could give new quantifiers for the +amount of coherence in the state along the lines of +Refs. 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(21), the encoded cq state is +σXAR = (1 − q)|0⟩⟨0| ⊗ σ0 + q +2|1⟩⟨1| ⊗ σ1 + q +2|2⟩⟨2| ⊗ σ2 += (1 − q)|0⟩⟨0| ⊗ +� +cos2 θ|00⟩⟨00| + sin2 θ|e2⟩⟨e2| +� ++ q +2|1⟩⟨1| ⊗ (cos θ|e1⟩ + sin θ|e2⟩) (cos θ⟨e1| + sin θ⟨e2|) ++ q +2|2⟩⟨2| ⊗ +� +eiα cos θ|e1⟩ + sin θ|e2⟩ +� � +eiα cos θ⟨e1| + sin θ⟨e2| +� +. +To calculate its accessible information, we first note that the optimal POVM achieving the accessible in- +formation can be taken to be rank-1 projectors [37]. Additionally, as noted in the main text, the ensemble +has the following symmetries: (i) σXAR is diagonal in the number basis, and (ii) q/2 · σ1 and q/2 · σ2 are +related by a reflection across the line y = x tan(α/2) in the x − y plane of the Bloch sphere. Using the same +arguments in Ref. [26] (Proposition 1), we deduce that the optimal measurement attaining the accessible +information can be made to have the same symmetries. Therefore, the optimal POVM can be taken to be +{|00⟩⟨00|, wm|πm⟩⟨πm|, wm|π′ +m⟩⟨π′ +m|}, where +|πm⟩ = √σm|e1⟩ + √σmeiβm|e2⟩ +(A1) +|π′ +m⟩ = √σm|e1⟩ + √σme−i(α+βm)|e2⟩. +(A2) +Here σm = 1 − σm. Each m labels a pair of symmetric projectors specified by (wm, σm, βm). Now, since +� +m(Πm + Π′ +m) is the projector onto the |e1⟩, |e2⟩ subspace, we have +� +m +� +wm +� +σm +√σmσme−iβm +√σmσmeiβm +σm +� ++ wm +� +σm +√σmσmei(α+βm) +√σmσme−i(α+βm) +σm +�� += I, +(A3) +from which we can conclude that +� +m +wmσm = 1 +2, +� +m +wm = 1, +� +m +wm +√σmσm(eiβm + e−i(α+βm)) = 0. +(A4) +Denote the set of {(wm, σm, βm)}m satisfying all three constraints in Eq. (A4) as S. Following the same +approach laid out in [26], the accessible information of the ensemble (and hence the total communication +rate) is given by +Iacc = max +S +� +m +wmJ(σm, βm; q, θ, α), +(A5) +where +J(σ, β; q, θ, α) =q| +√ +σ cos θ + eiβ√σ sin θ|2 log | +√ +σ cos θ + eiβ√σ sin θ|2 ++ q| +√ +σ cos θ + ei(β−α)√σ sin θ|2 log | +√ +σ cos θ + ei(β−α)√σ sin θ|2 ++ 2(1 − q)σ sin2 θ log(σ sin2 θ) +− 2κ log κ − (1 − q) cos2 θ log(1 − q), +(A6) +in which κ = qσ cos2 θ + q [cos β + cos(β − α)] +√ +σσ cos θ sin θ + σ sin2 θ. +We can relax the restriction on wm, σm and βm by dropping the last condition in Eq. (A4), thus obtaining +an upper bound. Formally, let ˜S denote the set of {(wm, σm, βm)}m that satisfy only the first two conditions +in Eq. (A4), then +� +Iacc = max +˜ +S +� +m +wmJ(σm, βm; q, θ, α) ≥ Iacc. +(A7) +19 + +Note that dropping the third condition essentially allows us to optimize βm’s freely independent of any +other parameter. Our first goal is to find the optimal phase encoding α, denoted by α∗, that maximizes the +function J(σ, β; q, θ, α). +Lemma 9. For any σ, q, and θ, J(σ, β; q, θ, α) is maximized only if (α, β) = (0, 0), (0, π), (π, 0) or (π, π). +Proof. For J to attain a local maximum, it is necessary that the directional derivative D⃗uJ = 0 and the +second directional derivative D2 +⃗uJ ≤ 0 along any direction ⃗u on the α-β plane. Specifically, let us consider +two direction given by ⃗u1 = (1, 0)⊺ and ⃗u2 = (1, 1)⊺. Then we have: +D ⃗ +u1J = ∂J +∂α = 0, +D ⃗ +u2J = ∂J +∂α + ∂J +∂β = 0, +(A8) +D2 +⃗ +u1J = ∂2J +∂α2 ≤ 0, +D2 +⃗ +u2J = +� ∂ +∂α + ∂ +∂β +� �∂J +∂α + ∂J +∂β +� +≤ 0. +(A9) +Calculating the first derivatives gives: +D ⃗ +u1J = +1 +ln 2q +� +ln | +√ +σ cos θ + ei(β−α)√σ sin θ|2 + 1 +� +2 +√ +σσ cos θ sin θ sin(β − α) +− +1 +ln 22(ln κ + 1)q +√ +σσ cos θ sin θ sin(β − α) = 0, +(A10) +D ⃗ +u2J = − 1 +ln 2q +� +ln | +√ +σ cos θ + eiβ√σ sin θ|2 + 1 +� +2 +√ +σσ cos θ sin θ sin β ++ +1 +ln 22(ln κ + 1)q +√ +σσ cos θ sin θ sin β = 0. +(A11) +Assuming q +√ +σσ cos θ sin θ ̸= 0 (when one of q, cos θ, and sin θ is 0, the ensemble becomes trivial, and when +one of σ and σ is 0, then J reduces to −(1 − q) cos2 θ log(1 − q), which is independent of α and β), the two +equations simplify to: +� +sin(β − α) log | +√ +σ cos θ + ei(β−α)√σ sin θ|2 − sin(β − α) log κ = 0 +sin β log | +√ +σ cos θ + eiβ√σ sin θ|2 − sin β log κ = 0 +(A12) +This set of equations admits four possible conditions: +(i) sin(β − α) = 0, sin β = 0; +(A13) +(ii) | +√ +σ cos θ + ei(β−α)√σ sin θ|2 = κ, sin β = 0; +(A14) +(iii) sin(β − α) = 0, | +√ +σ cos θ + eiβ√σ sin θ|2 = κ; +(A15) +(iv) | +√ +σ cos θ + ei(β−α)√σ sin θ|2 = κ, | +√ +σ cos θ + eiβ√σ sin θ|2 = κ. +(A16) +Now, calculating the second derivatives gives us: +D2 +⃗ +u1J = 2q +√ +σσ cos θ sin θ +� +− cos(β − α) log +�| +√ +σ cos θ + ei(β−α)√σ sin θ|2 +κ +� ++ 1 +ln 2 sin(β − α) +� +2 +| +√ +σ cos θ + ei(β−α)√σ sin θ|2 − q +κ +� √ +σσ cos θ sin θ sin(β − α) +� +(A17) +D2 +⃗ +u2J = 2q +√ +σσ cos θ sin θ +� +− cos β log +�| +√ +σ cos θ + eiβ√σ sin θ|2 +κ +� ++ 1 +ln 2 sin β +� +2 +| +√ +σ cos θ + eiβ√σ sin θ|2 + q +κ +� √ +σσ cos θ sin θ sin β +� +(A18) +20 + +Recall that cos θ sin θ > 0 since θ can be taken to be in [0, π/2], and we assumed cos θ sin θ ̸= 0. Plugging +each of the four conditions into the two expressions above we find that D2 +⃗ +u1J > 0 for conditions (ii) and +(iv), while D2 +⃗ +u2J > 0 for conditions (iii) and (iv), unless sin(β − α) = sin β = 0 also holds. Therefore, points +satisfying (i), namely (α, β) = (0, 0), (0, π), (π, 0), or (π, π), are the only possible local maxima of J. +Since we have dropped some constraints on β and treated it as an independent variable when optimizing, +the optimal (α, β) may not actually be feasible. However, it is easy to check that (α, β) = (0, π), (π, 0), and +(π, π) satisfies all of the constraints in Eq. (A4), and therefore they correspond to physical POVMs. This +result tells us that the best phase encoding that the encoder can perform in our one-sender protocol is either +α = 0 or α = π. Additionally, note that (α, β) = (π, 0) and (α, β) = (π, π) are images of each other under +the reflection across y = x tan(α/2). So, they correspond to the same pair of projectors, and we can freely +choose either one. +Note that, if the encoder chooses α = 0, the encoded cq state σXAR effectively reduces to +σXAR = (1 − q)|0⟩⟨0| ⊗ +� +cos2 θ|00⟩⟨00| + sin2 θ|e2⟩⟨e2| +� ++ q|1⟩⟨1| ⊗ (cos θ|e1⟩ + sin θ|e2⟩) (cos θ⟨e1| + sin θ⟨e2|) . +The accessible information of this state is necessarily less than or equal to 1 bit, meaning that there is no +quantum advantage. In other words, for maximal quantum advantage, one should use π phase encoding. +This is summarized by the following proposition. +Proposition 10. In the one-sender coherence-assisted scenario, if the encoding maps is given by Eq. (21), +then for any initial state |ψ⟩AR and any measurement POVM for B, whenever there is a quantum advantage +in the communication rate (i.e., whenever the communication rate exceeds 1 bit), α = π is always the optimal +phase encoding that A can perform. +We can now prove the following theorem in the main text. +Theorem 4. There exists a one-sender coherence-assisted communication protocol that sends approximately +1.0931 bits of information, i.e., R(Q∗ +1) ≥ 1.0931. The optimal (q, θ) that achieves this are approximately +(0.8701, arccos( +√ +0.4715)), and the optimal measurement is the projective measurement {|00⟩, +1 +√ +2(|e1⟩ ± +|e2⟩)}. +Proof. Having established that α = π is the best encoding phase, in which case the best decoding phase is 0 +(or equivalently π) we will set α = β = π and obtain +J(σ, β = π; q, θ, α = π) ≡ ˜J(σ; q, θ) =q( +√ +σ cos θ + √σ sin θ)2 log( +√ +σ cos θ + √σ sin θ)2 ++ q( +√ +σ cos θ − √σ sin θ)2 log( +√ +σ cos θ − √σ sin θ)2 ++ 2(1 − q)σ sin2 θ log(σ sin2 θ) +− 2(qσ cos2 θ + σ sin2 θ) log(qσ cos2 θ + σ sin2 θ) +− (1 − q) cos2 θ log(1 − q) +(A19) +and +Iacc(q, θ) = +max +� +m wmσm=1/2 +� +m wm=1 +� +m +wm ˜J(σm; q, θ) +(A20) +Following the same argument presented in [26], which we briefly recapitulate here for completeness, we first +find that this maximization for the accessible information can be rewritten as a maximization with at most +two terms [38], that is, +Iacc(q, θ) = +max +σ1≤1/2≤σ2 +�σ2 − 1/2 +σ2 − σ1 +˜J(σ1; q, θ) + 1/2 − σ1 +σ2 − σ1 +˜J(σ2; q, θ) +� +(A21) +21 + +which can be again rewritten as +Iacc(q, θ) = +max +σ1≤1/2≤σ2 +� +˜J(σ1; q, θ) + 1/2 − σ1 +σ2 − σ1 +� +˜J(σ2; q, θ) − ˜J(σ1; q, θ) +�� +. +(A22) +The maxand can be understood as the value of the line through points (σ1, ˜J(σ1) and (σ2, ˜J(σ2) at 1 +2. For +each θ, we can find three different measurement regimes. When q is sufficiently small, the optimal (σ1, σ2) +is (0,1), corresponding to optimal measurement vectors |e1⟩ and |e2⟩. As q becomes larger, the optimal +(σ1, σ2) is 0 and some σ∗ ∈ [1/2, 1], corresponding to a POVM with |e1⟩ and a mirror-symmetric pair of +rank-one projectors. Finally, when q is sufficiently close to 1, the optimal (σ1, σ2) is (1/2,1/2), corresponding +to projective measurements { 1 +√ +2(|e1⟩ ± |e2⟩)}. +By straightforward calculation, we find that in region 1, the accessible information of the ensemble is +Iacc,1(q, θ) = −(1 − q) cos2 θ log(1 − q). +(A23) +In region 3, +Iacc,3(q, θ) =q − qh2 +�1 + sin 2θ +2 +� ++ (1 − q) sin2 θ log sin2 θ +− (q cos2 θ + sin2 θ) log(q cos2 θ + sin2 θ) − (1 − q) cos2 θ log(1 − q) +(A24) +where h2 is the binary entropy function. In region 2, the calculation is more involved, +Iacc,2(q, θ) = J(0) + 1 +2 +dJ +dσ (σ∗) +(A25) +where σ∗ is determined from the fact that the tangent line of J at σ∗ passes through (0, J(0)), in other +words, +J(0) + σ∗ ∂J +∂σ (σ∗) = J(σ), +(A26) +which after much algebra becomes +q cos2 θ log qσ∗ cos2 θ + σ∗ sin2 θ +σ∗ cos2 θ − σ∗ sin2 θ = (1 − q)σ∗ sin2 θ log σ∗ +(A27) +To plot the accessible information in the entire region of (q, θ), we note Iacc(q, θ) = maxi=1,2,3{Iacc,i(q, θ)}. +One can check by comparing the plot of Iacc,1(q, θ), Iacc,2(q, θ), and Iacc,3(q, θ) that the maximal accessible +information occurs in region 3. To compute its value, we take the derivative of Iacc,3 with respect to q and +θ and set both to 0. +∂Iacc,3 +∂θ += q cos 2θ log +�1 + sin 2θ +1 − sin 2θ +� ++ (1 − q) sin 2θ log +� (1 − q) sin2 θ +q cos2 θ + sin2 θ +� += 0 +(A28) +∂Iacc,3 +∂q += 1 − h2 +�1 + sin 2θ +2 +� +− log sin2 θ + cos2 θ log +� (1 − q) sin2 θ +q cos2 θ + sin2 θ +� += 0 +(A29) +There is no closed form solution for this system of transcendental equations. Solving these two equations +numerically gives sin θ∗ ≈ +√ +0.4715 and q∗ ≈ 0.8701. This optimal choice of θ and q corresponds to the +intial source state ≈ +√ +0.4715|e1⟩ + +√ +0.5285|e2⟩, prior probability p(x) ≈ (0.1299, 0.4351, 0.4351), and the +rate sum is approximately 1.0931. +Proposition 11. If the source state is the maximally coherent state +1 +√ +2|e1⟩ + +1 +√ +2|e2⟩, then the optimal rate +is log(17/8) ≈ 1.0875 > 1, and the optimal prior probabilities given by q∗ = 15/17 ≈ 0.8824 +Proof. Take θ = π/4 in Eq. (A24) and after simplification, we find that I(q, π/4) = 2q − 1 + h2 +� 1+q +2 +� +. The +maximizer q∗ can be found by setting the derivative with respect to q to 0, and we find that q∗ = 15/17, in +which case the mutual information is log(17/8). +22 + +(a) +(b) +FIG. 14. The accessible information of the ensemble (assuming α = π) in terms of q and θ. +Appendix B: Lower bound for Unassisted Two-Sender (R(Q2)) +In this section, we calculate the accessible information of cq states arising from the binary-ternary encoding +strategy given in Eqn. (23). Following the same steps laid out in Sect. A, the accessible information can be +expressed as +Iacc = max +S +� +m +wmJ(σm, βm; q, θ, α), +(B1) +where +J(σ, β; q, q′, θ, α) = (1 − q)q′(sin2 θ log sin2 θ + 2σ cos2 θ log σ cos2 θ) ++q(1 − q′)(cos2 θ log cos2 θ + 2σ sin2 θ log σ sin2 θ) ++qq′| +√ +σ cos θ + eiβ√σ sin θ|2 log | +√ +σ cos θ + eiβ√σ sin θ|2 ++qq′| +√ +σ cos θ + e−i(α+β)√σ sin θ|2 log | +√ +σ cos θ + e−i(α+β)√σ sin θ|2 +−ξ log ξ − 2η log η. +(B2) +in which +ξ = (1 − q)(1 − q′) + (1 − q)q′ sin2 θ + q(1 − q′) cos2 θ = 1 − q′ cos2 θ − q sin2 θ +(B3) +η = (1 − q)q′σ cos θ + q(1 − q′)σ sin2 θ + 1 +2qq′ � +| +√ +σ cos θ + eiβ√σ sin θ|2 + | +√ +σ cos θ + e−i(α+β)√σ sin θ|2� +(B4) +By the same argument as in Lemma 9, we can deduce that the local extrema of function J occurs only if α +and β are both multiples of π. And that α = π is the optimal phase encoding whenever there is a quantum +23 + +1.0 +0.8 +0.6 +0.4 +1.0 +0.2 +0.8 +0.6 +0.4 q +1.50 +1.251.000.750.500.250.00 +0.2 +0.01.0 +1.4 +1.2 +0.8 +1.0 +0.6 +@0.8 +0.6 +0.4 +Maximal accessibleinfo +0.4 +0.2 +0.2 +0.2 +0.4 +0.6 +0.8 +1.0 +badvantage. Taking α = β = π then, we have +J(σ, β = π; q, q′, θ, α = π) = (1 − q)q′(sin2 θ log sin2 θ + 2σ cos2 θ log σ cos2 θ) ++q(1 − q′)(cos2 θ log cos2 θ + 2σ sin2 θ log σ sin2 θ) ++qq′( +√ +σ cos θ − √σ sin θ)2 log( +√ +σ cos θ − √σ sin θ)2 ++qq′( +√ +σ cos θ + √σ sin θ)2 log( +√ +σ cos θ + √σ sin θ)2 +−ξ log ξ − 2η log η. +(B5) +where +ξ = 1 − q′ cos2 θ − q sin2 θ +(B6) +η = (1 − q)q′σ cos2 θ + q(1 − q′)σ sin2 θ + 1 +2qq′ � +( +√ +σ cos θ − √σ sin θ)2 + ( +√ +σ cos θ + √σ sin θ)2� += q′σ cos2 θ + qσ sin2 θ +(B7) +Following the same analysis, we find three regimes for the optimal measurement, and the accessible infor- +mation is maximized in the regime that corresponds to σ = 1/2. +Iacc(q, q′, θ) = J(σ = 1 +2, β = π; q, q′, θ, α = π) = (1 − q)q′(sin2 θ log sin2 θ + cos2 θ log 1 +2 cos2 θ) ++q(1 − q′)(cos2 θ log cos2 θ + sin2 θ log 1 +2 sin2 θ) +−qq′h2 +�1 + sin 2θ +2 +� +− ξ log ξ − 2η log η +(B8) +where +ξ = 1 − q′ cos2 θ − q sin2 θ +(B9) +η = 1 +2q′ cos2 θ + 1 +2q sin2 θ +(B10) +Again taking the derivative of Iacc with respect to q, q′, and θ, we obtain the following system of equations, +∂J +∂q = −q′(sin2 θ log sin2 θ + cos2 θ log 1 +2 cos2 θ) + (1 − q′)(cos2 θ log cos2 θ + sin2 θ log 1 +2 sin2 θ) +−q′h2 +�1 + sin 2θ +2 +� ++ sin2 θ log ξ +η +(B11) += (2q′ − 1)h2(sin2 θ) + q′ − sin2 θ − q′h2 +�1 + sin 2θ +2 +� ++ sin2 θ log ξ +η = 0 +(B12) +∂J +∂q′ = (1 − q)(sin2 θ log sin2 θ + cos2 θ log 1 +2 cos2 θ) − q(cos2 θ log cos2 θ + sin2 θ log 1 +2 sin2 θ) +−qh2 +�1 + sin 2θ +2 +� ++ cos2 θ log ξ +η +(B13) += (2q − 1)h2(sin2 θ) + q − cos2 θ − qh2 +�1 + sin 2θ +2 +� ++ cos2 θ log ξ +η = 0 +(B14) +∂J +∂θ = (q + q′ − 2qq′) log tan2 θ + qq′ cot 2θ log 1 + sin 2θ +1 − sin 2θ = 0 +(B15) +Numerically solving this system of equations, we obtain that the optimal q, q′, and θ is (0.9197, 0.9197, π/4), +and the optimal rate sum is 1.10138. +24 + +Appendix C: One-sender assisted Holevo information (χ(Q∗ +1)) - proof of Theorem 7 +Theorem 7. χ(Qass +1 ) = maxq,cos2 θ∈[0,1] qh2(cos2 θ) + cos2 θh2(q) ≈ 1.2339. +Proof. We first show that the encoding given by Eq. (21) is in fact the best encoding strategy. Consider the +most general encoding strategy using NPE operations. By convexity of the mutual information with respect +to the underlying channel, it is sufficient for us to consider pure initial state cos θ|e1⟩+sin θ|e2⟩ and encoding +strategies consisting of only extremal NPE operations (Eq. (4)). With this simplification, we only need to +optimize the Holevo information over cq states � +a pa|a⟩⟨a| ⊗ ρa where pa is the prior probability and +ρa = +� +� +γa cos2 θ +(1 − γa) cos2 θ +√1 − γa cos θ sin θeiφa +√1 − γa cos θ sin θe−iφa +sin2 θ +� +� . +(C1) +This means that +� +a +paρa = +� +� +� +a paγa cos2 θ +� +a pa(1 − γa) cos2 θ +� +a pa +√1 − γa cos θ sin θeiφa +� +a pa +√1 − γa cos θ sin θe−iφa +sin2 θ +� +� , +(C2) +and +� +a +paS(ρa) = +� +a +pah2(γa cos2 θ). +(C3) +Therefore, +χ +�� +a +pa|a⟩⟨a| ⊗ ρa +� += S +�� +a +paρa +� +− +� +a +paS(ρa) ≤ ˜χ(θ; pa, γa) +(C4) +where +˜χ(θ; pa, γa) := H +��� +a +paγa cos2 θ, +� +a +pa(1 − γa) cos2 θ, sin2 θ +�� +− +� +a +pah2(γa cos2 θ). +(C5) +Here H(·) is the Shannon entropy, and the inequality becomes equality when � +a pa +√1 − γaeiφa = 0. Taking +derivative of ˜χ with respect to γa, and after some algebra, we find +d˜χ +dγa += pa cos2 θ +� +log +� +a′ pa′(1 − γa′) +� +a′ pa′γa′ +− log 1 − γa cos2 θ +γa cos2 θ +� +(C6) +If there is a local max, then d˜χ +dγa += 0 for all a, which means γa = +� +a′ pa′γa′ +cos2 θ +∀a, i.e., they are all equal. +However, this then means γa = +γa +cos2 θ, which cannot be true unless cos2 θ = 1. Therefore, if cos2 θ ̸= 1, then +˜χ has no local extrema, and the maximum has to occur at the boundaries γa = 0 or γa = 1, corresponding +to phase shift or complete damping encoding operations. +Thus to maximize ˜χ(θ; pa, γa) with respect to γa, γa must be 0 or 1. In this case, let us define p := +� +a:γa=1 pa. Then we have +˜χ(θ; pa, γa) ≤ H +� +p cos2 θ, (1 − p) cos2 θ, sin2 θ +� +− ph2(cos2 θ) += cos2 θh2(p) + ph2(cos2 θ) +(C7) +25 + +which means that +χ(Qass +1 ) := max χ +�� +a +pa|a⟩⟨a| ⊗ ρa +� +≤ max +p,θ cos2 θh2(p) + ph2(cos2 θ) += +max +x,y∈[0,1] xh2(y) + yh2(x) +(C8) +Note that this upper bound can be achieved by precisely the encoding scheme given in Eq. (21). Solving +for the extrema of the last equation gives us (x∗, y∗) ≈ 0.7035, in other words, the optimal initial state is +|ψinit⟩ ≈ +√ +0.7035|e1⟩ + +√ +0.2965|e2⟩, and the optimal prior probabilities are p(a) ≈ (0.2965, 0.3518, 0.3518). +Additionally, χmax ≈ 1.2339. +Appendix D: Details of the experiment +Source preparation: The source of photon pairs is based on type II spontaneous parametric down- +conversion in a 2 mm periodically polled Potassium titanyl phosphate (PPKTP) crystal (with temperature +stabilizing oven). The crystal is pumped with frequency-doubled light pulses originating from a Tsunami +modelocked laser (a train of ∼100-fs pulses with center wavelength 810 nm and repetition rate 80 MHz), +doubled using a 0.5 mm Bismuth Borate (BiBO) crystal. +To prepare the photons in a single spectral, +polarization, and spatial mode, the heralding photons from the pair are filtered to ∼2 nm bandwidth at +full-width-at-half-maximum by a pair of tilted spectral filters, set to linear polarization by a polarizer, and +coupled into single-mode fiber. The existence of this idler photon is detected via a single-photon detector +(avalanche photodiode, Excelitas SPCM-AQ4C), while the other, heralded single photon is sent to a three- +port interferometer to prepare the desired state |ψ⟩ = +1 +√ +2|e1⟩ + 1 +2|e2⟩ + 1 +2|e3⟩, where we have ignored all the +internal degrees of freedom of the single particle and only represented it in a superposition of different path +basis states |ei⟩ = |0⟩A1 · · · |1⟩Ai · · · |0⟩AN . +To ensure the signal photon is close to a single photon source, we looked at the heralded signal photon +within a 2 ns coincident window after heralding the idler photon. We characterize the source by measuring +its second-order correlation g(2) +iss′, which can be calculated as: +g(2) +iss′ = Ciss′Ci +CisCis′ +(D1) +where Ciss′ are three-fold coincident counts between one idler photon and two single photons after splitting, +Cis(s′) represents two-fold coincident counts between idler photon and one signal photon, and Ci denotes +single counts for the idler. The power-dependence of the second-order correlation values is shown in Fig. 15 +(a), which indicates good agreement with the linear curve fitting and allows to set the pump power to +suppress the two-photon contribution from the source. +In our experiment, the heralded second-order correlation g(2) +iss′ has to be set extremely small, due to the +fact that large higher-order terms could in principle enable a higher capacity rate even in the classical case. +Taking the small violation we have estimated (1.02), we set g(2) +iss′(0) = 0.0017 ± 0.001 to be one order of +magnitude smaller than the violation to make sure the contribution from multiple-photon events can be +neglected. As a consequence, we have relatively low coincidence count rates around 600Hz +Interferometer design: The interference visibility of our three-port interferometer limits the performance +of our quantum-enhanced communication. To achieve a high enough visibility with free-space optics, we +design a three-port interferometer consisting of (1) an inner offset Sagnac interferometer, which is extremely +stable over a few hours with above 99.5% interference visibility; (2) an outer Mach–Zehnder interferom- +eter, which is passively stabilized thermally and vibrationally inside a small box and gives around 98.2% +interference visibility over 10 minutes. It was further actively adjusted by a piezo actuator implemented on +the translation stage in the delay line between different runs of measurement; (3) three 3-mm-thin glasses +26 + +FIG. 15. (a) Experimental result for heralded second-order correlation g(2) +iss′ under different SHG pump powers; (b) +Interference visibility of the Mach–Zehnder interferometer measured in 10 minutes. (c) Theoretical characterization +of our phase plate. +windows for controlling the phase independently; windows were chosen instead of other bulky electro-optical +devices, which could potentially degrade the interference visibility. +The whole setup can be maintained stable over ∼10 minutes with average interference visibilities around +99.5% and 98.2% for the inner and outer loops, respectively, while slight adjustment with the piezo actuator +helps to retrieve good interference visibility for the next round of the experiment. During the runs of our +experiment, we do not turn the active stabilization on so that the average stability remains the same over +10 minutes. +Encoding operation. As has been mentioned before, with the current type of single-photon detectors used +and the loss in our free-optics setup, performing general amplitude damping operations on the photons is +nontrivial. Instead, we devised our setup based on the coherent-assisted protocol where only phase encoding +is required. +One of the most commonly used phase shifters is electrically controlled liquid crystal, where the refractive +index along some axes of the crystal depends on the voltage applied to it and thus can be used to add +phase on single photons. However, the resolution of the applied phase (around 3◦) and the size and the +parallelism of most commercial liquid crystals prevent us from using them in our small-size, high-visibility +interferometer. Therefore, as a replacement, we create a phase shifter based on a d = 3 mm glass window +(with a reflective index around ng = 1.51) mounted on a rotation stage (with a resolution around 5 min-arc). +Starting from placing the glass plate perpendicular to the incoming beam, the phase added to the photon +after slightly tilting it with angle α can be computed as: +∆φ = 2πd +λ [( +� +n2g − sin(α)2 − cos(α)) − (ng − 1)], +(D2) +which is plotted in Fig. 15. The average resolution over 2π phase shift is around 2.3◦; however, due to its +nonlinear behavior, by carefully choosing the starting point, we can obtain a much finer resolution. +Error analysis: To estimate the experimental error, we note at first that we are limited mostly by the +interferometer stability. To ensure high interference visibility, we perform each run of our measurement for +∼10 minutes and re-optimize the setup between different runs. +In each run of the experiment, the statistical error can be calculated from standard error propagation. For +the case of characterizing channel transition probability p(y|x): +V [R1] = +1 +N 2 +� +xy +� +(p(x) log2 q(y) + p(x)H(q(y)))2 + (p(x) log2 p(y|x) + p(x)H(p(y|x)))2) +� +V [ny|x] +(D3) +where q(y) = � +a p(y|x)p(x) with fixed optimal prior p(x1 = 0) = 1/2 and p(x2 = 0) = 15/17. ny|x is the +total number of photons collected at port y conditional on input x and N is a total number of counts using +in characterizing the channel for every input x. The statistical error assuming the Poisson distribution is +given as V (ny|x) = Np(y|x)(1 − p(y|x)). With N ≈ 105, the statistical error is around +� +V (R1) ≈ 0.002. +27 + +100 +400 +0.014 +(b) +without active stabilization +(c) +with active stabilizatior +350 +86 +0.010 +cor +96 +Visility +0 200 +app +94 +150 +100 +92 +0.002 +50 +0.000 +90 + +20 +30 +40 +10 +50 +60 +70 +0 +2 +6 +8 +10 +1 +3 +6 +SHG power (mW) +min +rotation (degree)Similarly, for the case of measuring the joint distribution p(a, b), the error can be computed as: +V [R2] = +1 +N 2 +� +x,y +(log2 q(y) + log2 p(y)) − log2 p(x, y) + I(y : x))2 V [nxy], +(D4) +With extra uncertainty in p(x) from the generation of encoding random bits and V [nxy] = np(x)(1−p(x))× +m + n × mp(y|x)(1 − p(y|x)) where n = 680 (the number of random input bits) and m ≈ 600 (the number +of counts per second), we get a statistical error of estimating V [R2] to be around +� +V (R2) ≈ 0.011. +However, experimentally, besides the statistical error, the channels built from run to run are actually +slightly different since they are extremely sensitive to the overall interference visibility. +To take those +systematic errors into consideration and to show that our experimental result is repeatable, we calculate +the experimental result I(X : Y ) in 10 runs of experiments and average the capacity rate sum, which is what +we present in the main text. +28 + diff --git a/bNE0T4oBgHgl3EQfngFx/content/tmp_files/load_file.txt b/bNE0T4oBgHgl3EQfngFx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fab947dd8bbbb091f45ff7ccd5475ca36460f1c7 --- /dev/null +++ b/bNE0T4oBgHgl3EQfngFx/content/tmp_files/load_file.txt @@ -0,0 +1,1214 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf,len=1213 +page_content='Information Carried by a Single Particle in Quantum Multiple-Access Channels Xinan Chen,1, ∗ Yujie Zhang,2, ∗ Andreas Winter,3, 4, 5 Virginia O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Lorenz,2 and Eric Chitambar1 1Department of Electrical and Computer Engineering, Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA 2Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA 3Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA), Pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Llu´ıs Companys, 23, 08010 Barcelona, Spain 4Grup d’Informaci´o Qu`antica, Departament de F´ısica, Universitat Aut`onoma de Barcelona, 08193 Bellaterra (Barcelona), Spain 5Institute for Advanced Study, Technische Universit¨at M¨unchen, Lichtenbergstraße 2a, D-85748 Garching, Germany (Dated: January 9, 2023) Non-classical features of quantum systems have the potential to strengthen the way we currently exchange information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In this paper, we explore this enhancement on the most basic level of single particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' To be more precise, we compare how well multi-party information can be transmitted to a single receiver using just one classical or quantum particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Our approach is based on a multiple-access communication model in which messages can be encoded into a single particle that is coherently distributed across multiple spatial modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Theoretically, we derive lower bounds on the accessible information in the quantum setting that strictly separate it from the classical scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This separation is found whenever there is more than one sender, and also when there is just a single sender who has a shared phase reference with the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Experimentally, we demonstrate such quantum advantage in single-particle communication by implementing a multi-port interferometer with messages being encoded along the different trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Specifically, we consider a two-sender communication protocol built by a three-port optical interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In this scenario, the rate sum achievable with a classical particle is upper bounded by one bit, while we experimentally observe a rate sum of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0152 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0034 bits in the quantum setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' INTRODUCTION It is well known that a quantum particle exhibits fundamentally different properties than its classical counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For instance, while a classical particle has a definite trajectory in space, a quantum par- ticle can be placed in a coherent superposition of different paths as it moves from one point in space to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' A natural practical question is whether this superposition of trajectories can be utilized for performing some communication task [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In this paper, we focus on whether the path coherence of a single particle can be used to enhance the commu- nication of N spatially separated parties to a single receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Several previous papers have addressed similar questions in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Inspired by the famous two-slit experiment, Massar first showed the advan- tage of quantum particles in the bipartite finger- printing task [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In such a task, Alice and Bob each possesses one bit x, y ∈ {0, 1}, and they wish to let a referee decide whether x = y by sending minimal ∗ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Chen and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Zhang contributed equally to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' amount of information to the referee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' It is not dif- ficult to see that one quantum particle in the state 1/ √ 2(|0⟩A|1⟩B + |1⟩A|0⟩B) suffices for this objective, while in the classical regime, the parties must send both x and y for the referee to certify that x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' [8], the authors reinterpreted this result as two-way communication using only one single quan- tum particle, which is forbidden if the information medium is a classical particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This idea was fur- ther extended to the scenario where Alice and Bob each have an n-bit string [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Using an n-level Mach- Zehnder interferometer, one of Alice and Bob can re- trieve the other’s full n-bit string, while only one bit of information is revealed to the other party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Since this can be done for an arbitrary n, this result sug- gests, roughly speaking, that a single quantum par- ticle can carry an arbitrarily large amount of infor- mation in point-to-point communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Comple- menting the point-to-point communication results, it was recently discovered via convex polytope analy- sis that using a single quantum particle, one can gen- erate multiple-access channels (MACs) that cannot be constructed with a classical particle [6, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' How- ever, these latter results pertain to the specific tran- sition probabilities p(y|x1, · · · , xN) of the generated N-party MACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' It has remained elusive whether the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='02513v1 [quant-ph] 6 Jan 2023 discovered non-classical MACs actually have advan- tages in terms of more practical figures of merit, such as asymptotic communication rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In this paper, we provide a positive answer to this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Specifically, we utilize the framework of single-particle multiple-access channels (MACs) de- veloped in [10] to investigate the achievable rate re- gions of distributed communication using a single particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' While the communication rate sum of the different senders is always upper bounded by 1 bit if a single classical particle is used, in the quantum setting a rate sum of at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='10 bits is achievable for two senders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Even higher rates can be achieved if there are more than two senders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Moreover, we ex- perimentally demonstrate the quantum advantages by implementing one of our designed protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In particular, we achieve a quantum advantage within five standard deviations using linear optics and a single photon state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In Sec- tion II, we introduce the operational framework of single-particle MACs and review some information- theoretic concepts such as the achievable rate re- gions of MACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In Section III, we study in de- tail the theoretical aspects of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In Section IV, we give our experimental demonstration of the two-sender coherent assisted communication proto- col using linear optics and a heralded single pho- ton state, where quantum-enhanced communication is achieved by preparing a single photon in a super- position of different trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' OPERATIONAL FRAMEWORK AND INFORMATION THEORY PRELIMINARIES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' MACs constructed with one particle To compare how much classical information can be carried by a classical or quantum particle with none of its internal degrees of freedom being accessi- ble, we utilize the framework of single-particle MACs developed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This framework, which we now briefly describe, was inspired by previous work [11, 12] that captured the resource-theoretic features of quantum coherence in a multi-port interferome- ter setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We denote the collection of senders as A = (A1, A2, · · · , AN) and assume that each mes- sage sent by each sender is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' These N senders will use a one-particle state to send information to a single receiver B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Recall that the Fock space is de- scribed by H = �∞ i=0 Hi, where Hi is the i-particle subspace of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' A one-particle state is represented by a density operator ρA acting on the one-particle subspace, which is HA 1 := span {|ei⟩ : 1 ≤ i ≤ N} , (1) where |ei⟩ = |0⟩A1 · · · |1⟩Ai · · · |0⟩AN is the state of the particle on path i, with |0⟩ being the vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The senders then encode their messages using completely positive trace-preserving (CPTP) maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For example if party Ai wishes to send message xi, the CPTP map EAi xi is locally applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The fully encoded state for joint message x := (x1, · · · , xN) is given by σx := σx1···xN = EA1 x1 ⊗ · · · ⊗ EAN xN (ρA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (2) For the purposes of this investigation, we restrict the allowed CPTP maps that the senders use to encode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Specifically, since we are interested in the information-carrying ability of a single particle, we have to require that the encoding operations can- not increase particle number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' More specifically, we model the encoding operations as CPTP maps with a particle number-preserving unitary extension, that is, E(ρA) = TrE � U(ρA ⊗ |0⟩⟨0|E)U †� , (3) where U preserves the overall particle number in the system A and the environment E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This set of operations was termed number-preserving extendible (NPE) operations in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' [10] and was fully char- acterized for an arbitrary number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Here we focus only on the case where there is at most one particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In this case, these operations are convex combinations of channels with Kraus operators K1 = �1 0 0 eiφ1√1 − γ � , K2 = � 0 eiφ2√γ 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (4) Note they can be seen as generalized amplitude damping channels with two additional relative phase parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In this work, we will rely heavily on two particular NPE operations in the encoding: the completely damping operation ρ �→ E(vac)(ρ) := Tr(ρ)|0⟩⟨0| and the phase shift operation ρ �→ E(φ)(ρ) := e−iZφ/2ρeiZφ/2, where Z = � 1 0 0 −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Note that E(vac) and E(φ) correspond to the choices of γ = 1 and γ = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (4), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In optical communication, these encoding operations correspond to on-off keying (OOK) modulation and phase-shift keying (PSK) modulation [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' After the encoding operations, the state σx is sent to the receiver, and the receiver tries to reconstruct 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The general scheme for building a multiple- access classical channel using a single particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' the message using a positive operator-valued mea- sure (POVM) {Πy}y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This process induces a classi- cal channel by p(y|x) := Tr(Πyσx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (5) A graphical representation of this framework is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Note that in our model we will always assume that the receiver shares a phase ref- erence with the particle source, and so |ei⟩ is defined with the same overall phase for both the source and detector [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' With this operational framework in mind, we can define the set of N-sender MACs constructed from a classical particle as MACs p(y|x1 · · · xN) of the form p(y|x1 · · · xN) =Tr � Πy � EA1 x1 ⊗ · · · ⊗ EAN xN (ρA cl) �� (6) where ρA cl = �N i=1 pi|ei⟩⟨ei|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In other words, these are MACs that can be constructed from a classi- cal source state that has no coherence between any two paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' On the other hand, in the truly quan- tum case, no restriction is placed on the initial one- particle density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We say that the set of N- sender MACs constructed from a quantum particle are MACs that have the form p(y|x1 · · · xN) =Tr � Πy � EA1 x1 ⊗ · · · ⊗ EAN xN (ρA) �� (7) where ρA ∈ D(H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Here D(H1) denotes the set of density operators on the one-particle subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Throughout this work we assume that the message xi of party Ai is chosen from alphabet set Xi, which will always be a finite set of integers Xi = [mi] := {0, · · · , mi − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Similarly, we let Y denote the out- put alphabet of the receiver B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For input and output alphabet X := X1 ×X2 ×· · ·×XN and Y, we denote the set of classical MACs by CN(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Y) and quan- tum MACs by QN(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We will use QN and CN to denote general N-sender channels with arbitrary input and output alphabets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Despite the fact that both classical and quan- tum MACs can be described using Born’s rule (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (6) and (7), respectively), classical MACs ad- mit a much simpler characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The state �N i=1 pi|ei⟩⟨ei| can be understood simply as a clas- sical particle that is sent along path i with proba- bility pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' A local NPE operation then reduces to applying some local channel that lets the particle ei- ther continue along its respective path or blocks it from reaching the receiver B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' either E(vac) or the identity map is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' With probability qi(0|xi) the particle is blocked by party Ai for input choice xi, and with probability qi(ei|xi) it is transmitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Hence if the input state is |ei⟩⟨ei|, then the state received by B is σxi = � j̸=i |0⟩⟨0|Aj ⊗ EAi xi (|1⟩⟨1|) = qi(ei|xi)|ei⟩⟨ei|A1···AN + qi(0|xi)|0⟩⟨0|A1···AN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' On the decoding end, party B examines each path to see if it contains a particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Output b is produced with probability d(b|ei) when a particle is received along path i and with probability d(b|0) when no particle is received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Hence the channel obtained af- ter averaging over all input states is p(y|x1, · · · , xN) = N � i=1 pi[d(y|0)qi(0|xi) +d(y|ei)qi(ei|xi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (8) The set CN(X1, · · · XN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Y) consists of MACs that can be written in this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Coherence-assisted communication Thus far we have focused on scenarios where the number of senders equals the number of paths through which the particle source emits the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We can generalize this model by allowing for extra paths that are not acted upon by an sender (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We refer to these as coherence-assisted protocols, with the extra paths being called called assistance paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Note that since the assistance paths are not touched by any encoding operation, we can with- out loss of generality combine amplitudes of multi- ple assistance paths into one assistance path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In- tuitively, the assistance path can serve as a phase reference for the other paths, which can help the receiver better discriminate the encoded messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 3 y Particle sourceFIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' A coherence-assisted protocol allows for an un- perturbed side channel through which the particle can traverse coherently to the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' One the other hand, as we will see in Theorem 2 below, this assistance path cannot enhance commu- nication rate when the source is a classical particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We let Qass N (X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Y) denote the family of all coherence- assisted channels built by N parties using a single quantum particle and NPE operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' An analogy can be drawn here to entanglement- assisted communication [16, 17], in which entangle- ment is shared between the senders and receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In fact, one could imagine in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 2 that the particle is coherently distributed to the receiver along the assisted path prior to the encoding of the senders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Then the scenario becomes conceptually equivalent to the entanglement-assisted setup except that the shared resource between senders and receivers is co- herence in single-particle spatial modes rather than coherence in multi-particle states [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Communication rates of MACs In this work we consider the achievable communi- cation rates of the classical multiple-access channels as constructed in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Roughly speaking, a rate tuple (R1, · · · , RN) is achievable for a given MAC if for every ϵ > 0 and n sufficiently large, each sender i can send 2nRi possible messages with average error no greater than ϵ (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' [19] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Remarkably the achievable rate region of an N-sender MAC has a single-letter characteriza- tion in terms of the conditional mutual information, which, for random variables X1, X2, Y , is defined as I(X1 : Y |X2) = I(X1X2 : Y ) − I(X2 : Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Proposition 1 ([19–21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' A rate tuple (R1, · · · , RN) for MAC p(y|x) is achievable if and only if it lies in the closure of the convex hull of all rate tuples satisfying RS ≤ I(XS : Y |XSC), ∀S ⊂ {1, · · · N} (9) for some product distribution p(x1) · · · p(xN) over the input alphabet X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Here in a slight abuse of nota- tion we denote XS := ×i∈SXi and RS := � i∈S Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In particular, for two parties, the achievable rate re- gion is the convex hull of all rate pairs satisfying R1 ≤ I(X1 : Y |X2) R2 ≤ I(X2 : Y |X1) R1 + R2 ≤ I(X1X2 : Y ), (10) for product distributions p(x1)p(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For the purpose of this investigation, we will be mainly interested in the largest amount of informa- tion that can be jointly sent by the senders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In our framework, this corresponds to the largest rate-sum R := � i∈{1,··· ,N} Ri that can be achieved using a MAC constructed from a single particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The accessible information and Holevo information As described in the previous sections, each com- munication protocol using a single particle consists of three elements: a choice of the initial one-particle state ρ, an encoding strategy which specifies a fam- ily of NPE encoding operations {EAi xi }, and the de- coding measurement {Πy}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We will be interested in optimizing the joint achievable communication rate under this framework, and to do so, we split the full optimization into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Every choice of initial state, encoding strategy, and prior product distribu- tion p(x) = p(x1) · · · p(xN) over the messages gives rise to the classical-quantum (cq) state σXA = � x p(x)|x⟩⟨x|X ⊗ σA x (11) where σA x = EA1 x1 ⊗ · · · ⊗ EAN xN (ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For each such cq state, when a POVM {Πy} is performed on systems A, the resulting joint probability distribution can be described by the classical-classical (cc) state σXY = � x,y p(x)p(y|x)|x⟩⟨x|X ⊗ |y⟩⟨y|Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (12) where p(y|x) = Tr(Πyσx) is the constructed MAC in QN(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' If X denotes the random variable over all N messages and Y denotes the output variable for the receiver, then the information obtained by the 4 Particle sourceFIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' An illustration of our bounds on the one-sender coherence-assisted communication rate R(Qass 1 ) and two- sender unassisted communication rate sum R(Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Each bound is established via the labeled proposition or theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' receiver about X is the mutual information I(X : Y )σXY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Optimizing over all POVMs quantifies the so-called accessible information of the cq state σXA, Iacc(σXA) := max {Πy} I(X : Y )σXY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (13) We then further optimize the accessible information over all valid cq states (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' those having the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (11)), R(QN) := max σXA Iacc(σXA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (14) Thanks to Proposition 1, R(QN) captures the largest communication rate-sum that quantum me- chanics allows when using a fixed encoding strategy and decoding measurement on each particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This is the central quantity of interest in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Lower bounds of R(QN) are given by R(QN) ≥ Iacc(σXA) ≥ I(X : Y )σXY, with I(X : Y )σXY aris- ing from any explicit protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' On the other hand, the celebrated Holevo’s bound limits the accessible information as Iacc(σXA) ≤ χ(σXA) (15) where χ(σXA) := I(X : A)σXA = S �� x p(x)σA x � − � x p(x)S � σA x � is called the Holevo information [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Therefore, a natural upper bound for R(QN) is R(QN) = max σXA Iacc(σXA) ≤ χ(QN) := max σXA χ(σXA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (16) Again, the maximization is over cq state having the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In addition to providing an upper bound, the Holevo information χ(σXA) admits an operational interpretation [23, 24] within our one-particle com- munication framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Namely, it captures the sce- nario in which the senders prepare independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=') copies of σXA, yet the receiver is allowed to perform joint decoding mea- surement across all copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In the asymptotic limit, the largest amount of information that the receiver can gain is exactly the Holevo information χ(σXA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Therefore, the optimized Holevo information χ(QN) represents the ultimate amount of information that can be transmitted by N senders using a fixed single- particle encoding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We similarly let R(Qass N ) and χ(Qass N ) be defined as in Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (14) and (16), respectively, except with the maximum now taken over all cq states σXA built using a coherence assis- tance path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' While R(QN) ≤ χ(QN) and R(Qass N ) ≤ χ(Qass N ), for general N these bounds appear to be quite loose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For example, we show below that χ(QN) ≥ log N and χ(Qass N ) ≥ log(N + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' On the other hand, the best lower bounds on R(QN) and R(Qass N ) we obtain do not even exceed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' While this bound still ex- ceeds the largest classical rate, which is the main fo- cus of this paper, its divergence from the Holveo in- formation reflects the strong communication degra- dation that arises when restricting to single-copy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 5 Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 8 Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 5 Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 7 Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 4 Classical region R(Cn) Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 2III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' THEORETICAL RESULTS Having established our communication model, we now probe the theoretical limits of single particle communication in both the classical and quantum settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Our main goal is to place bounds on the communication rates introduced in the previous sec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For simplicity, we focus on multiple-access channels with binary and ternary inputs/outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In Section III A we compute the ultimate commu- nication rates using a classical particle, which serve as thresholds for our quantum protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In Section III B we construct explicit quantum-enhanced com- munication protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Lower bounds on R(Qass 1 ) and R(Q2) are presented in Sections III B 1 and III B 2, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' for N ≥ 2 lower bounds on R(QN) and R(Qass N ) are provided in Section III B 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' and fi- nally in Section III B 4 we show that both χ(QN) and χ(Qass N ) grow as log N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Classical MACs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' N-party rate-sum We begin by establishing the intuitive upper bound of one bit for the N-party rate-sum using a single classical particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The following proposition places a fundamental bound on N-party communi- cation within our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' R(CN) = 1 ∀ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' That is, we can communicate at most 1 bit of information using a classical particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Furthermore, an assistance path does not help in the classical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We first show that R(CN) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (8), any channel in CN admits the decomposition p(y|x1 · · · xN) = � i pi � m=0,ei d(y|m)qi(m|xi) (17) where d(y|m) and q(m|xi) are conditional probabil- ity distributions associated with the decoder and the encoder, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' By convexity of mutual infor- mation I(X1 · · · XN : Y ) with respect to the under- lying channel, we can conclude that the rate sum is maximized by channels of the form p(y|x1 · · · xN) = p(y|xi) = � m=0,ei d(y|m)qi(m|xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' However, capac- ities of these channels cannot exceed one bit since d(y|m) is essentially a classical post-processing map, and qi(m|xi) is a channel with binary outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' On the other hand, suppose the initial state is |ei⟩⟨ei|, the i-th sender Ai encodes information by either annihilating the particle or preserving the par- ticle, and the receiver performs measurement in the particle number basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In this case, Ai can send 1 bit of information, while other senders cannot send any information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' So, the total amount of transmitted in- formation is one bit, and therefore R(CN) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' To see that an assistance path does not help, observe that R(CN) = 1 holds for arbitrary N, and an as- sistance path can be seen as a special case of CN+1 where the (N + 1)-th party acts trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Classical canonical form We next turn to the problem of identifying achiev- able rate tuples using a single classical particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This task is simplified by recognizing that every canonical MAC can be obtained from a canonical MAC com- bined with stochastic encoders and a stochastic de- coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' By the data processing inequality, stochastic post-processing cannot increase the rate region, and the same is true for stochastic pre-processing (Prob- lem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='5 in [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Therefore, if (R1, · · · , RN) is a rate tuple achievable by some single-particle classi- cal MAC, then it is also achievable by a canonical one defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For arbitrary input and output sets X1×· · ·×XN and Y, every MAC in CN(X1, · · · XN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Y) can be seen as arising from a canonical MAC that has binary inputs for each sender and N +1 outputs for the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For a given classical state ρ = �N k=1 pk|ek⟩⟨ek| and induced MAC p(y|x1, · · · , xN) having the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (8), define the canonical MAC with transition probabilities �p(k|j1, · · · , jN) = � pk if jk = 1 0 if jk = 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (18a) �p(0|j1, · · · , jN) = � k such that jk = 0 pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (18b) This channel likewise has the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (8) and therefore belongs to C([2], · · · , [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' [N + 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Also, define local pre-processing stochastic maps �qi : Xi → {0, 1} with �qi(0|xi) = qi(0|xi) and �qi(1|xi) = qi(ei|xi), along with a post-processing stochastic map �d : {0, 1 · · · , N} → Y by �d(y|k) = d(y|ek) for k = 1, · · · , N and �d(y|0) = d(y|0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Then it is 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The shaded region is the union of all achiev- able rate pairs as the weight λ of the source state ρcl = λ|e1⟩⟨e1|+(1−λ)|e2⟩⟨e2| varies over interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' straightforward to verify that p(y|x1, · · · , xN) = N � k=0 1 � j1=0 · · 1 � jN=0 {�d(y|k) × �p(k|j1, · · · , jN)�q1(j1|x1) · · · �qN(jN|xN)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (19) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Two-sender classical rate regions We now turn to the rate regions for two-sender communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Consider the canonical MAC p(y|x) that is generated by a classical particle ρcl = λ|e1⟩⟨e1| + (1 − λ)|e2⟩⟨e2| and having the structure of Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (18a) and (18b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Since N = 2, the canon- ical MAC is characterized by the single parameter λ = p1, and the transition probabilities are given by 1 = p(00|00) λ = p(01|01) = p(00|10) = p(01|11) 1 − λ = p(10|10) = p(00|01) = p(10|11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (20) For a fixed λ ∈ [0, 1], and prior p(x1)p(x2) the achievable rate pairs (R1, R2) are determined by Proposition 1, which follows a pentangon con- strained by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Combining all these regions with with fixed λ ∈ [0, 1] but different priors p(x1)p(x2), we could obtained the achievable rate region of a specific MAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We are now interested in computing the union of all achievable rate regions as λ is varied within the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This will yield the total collection of all asymptotic rate pairs (R1, R2) feasible by a MAC built using a single classical particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Note that a rate pair (R1, R2) lies in the en- closed region of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 4 if and only if it is achiev- able using many copies of the same source state ρcl = λ|e1⟩⟨e1| + (1 − λ)|e2⟩⟨e2|, and the union of these rate pairs evidently forms a non-convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' However, if we relax this i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' constraint and allow λ to vary across the multiple uses, then more rate pairs are accessible by time-sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In this case, the collection of achievable rate pairs is just the convex hull of the region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' a triangle with outer vertices (1, 0) and (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Quantum MACs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Surpassing the classical bound with one sender and coherence assistance Given the classical communication bounds estab- lished in the previous section, it is natural to con- sider whether quantum mechanics can do better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We begin by considering the special case of just one sender, and the encoding scheme presented here will generalize as more parties are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In the one-sender scenario, if no coherence assistance is used then the whole communication system is sim- ply a two-dimensional space spanned by {|0⟩, |1⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' By Holevo’s theorem, the communication rate is bounded above by log 2 = 1, and therefore, quantum mechanics offers no advantage over classical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' However, by leveraging coherence assistance in the sense of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 5, it is possible to communicate more than one bit of information in the point-to-point sce- nario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' To achieve a greater capacity using a single parti- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Coherence-assisted communication with one sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 7 R2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 R1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0Particle sourcecle, we construct a channel with ternary input sym- bols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Suppose that the initial state distributed from the particle source is |ψ⟩AR = cos θ|e1⟩ + sin θ|e2⟩ with θ ∈ [0, π/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Note that this describes the most general one-particle state since any relative phase can be absorbed into the definition of |e1⟩, which we assume is known to the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For message x ∈ {0, 1, 2}, let the sender A encode the state |ψ⟩ according to the following NPE operations: � � � � � E0(ρ) = E(vac)(ρ) = Tr(ρ)|0⟩⟨0|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' E1(ρ) = ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' E2(ρ) = E(α)(ρ) = e−iαZ/2ρeiαZ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (21) Let σx = EA x ⊗ idR(|ψ⟩⟨ψ|) and σXAR = � x p(x)|x⟩⟨x| ⊗ σx be an encoded cq state with the prior distribution over messages have the form p(0) = 1 − q and p(1) = p(2) = q/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' As shown in Section III B 4, the Holevo information χ(Qass 1 ) is attained by this type of cq state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Hence, we are mo- tivated to conjecture that the encoding scheme of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (21) is also optimal for the single-particle rate R(Qass 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Even if this conjecture fails to be true, the accessible information of σXAR still provides a lower bound on R(Qass 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In general, calculating the accessible information of an arbitrary cq state is mathematically challeng- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' However, in our case, the encoded cq state en- joys the following symmetries: (i) each σx is block- diagonal in the particle number basis, and (ii) q/2·σ1 and q/2 · σ2 are related by a reflection across the line y = x tan(α/2) in the x − y plane of the Bloch sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Using similar arguments to those in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' [26], we find (see the Supplemental Material) that α = π provides an optimal encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Further anal- ysis then shows that the accessible information is maximized by a prior probability q and coherence angle θ in the source state that together satisfy a pair of transcendental equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Solving these equations numerically leads to the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' There exists a one-sender coherence- assisted communication protocol that sends approx- imately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0931 bits of information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=', R(Qass 1 ) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The optimal (q, θ) that achieves this are approximately (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8701, arccos( √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4715)), and the optimal measurement projects into the basis {|00⟩, 1 √ 2(|e1⟩ ± |e2⟩)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Note that the largest accessible information is not attained using a state with uniform superposition across both paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Yet, the optimal decoding mea- surement is a projection into uniform superposition states 1 √ 2(|e1⟩ ± |e2⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' When using a source state with uniform superposition across both paths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' θ = π/4), the largest communication rate is com- puted to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Two-sender MACs Let us now add a second sender to the communi- cation picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We first consider the scenario of two senders with no coherence assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Thus there are only two paths connecting the source to the re- ceiver, and we borrow ideas from the previous one- sender coherence-assisted protocol, which also has two paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Let senders A1 and A2 share the state |ψ⟩A1A2 = cos θ|e1⟩ + sin θ|e2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Consider first the following binary encoding strategy: � EA1 0 (ρ) = Tr(ρ)|0⟩⟨0|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' EA1 1 (ρ) = ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' � EA2 0 (ρ) = ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' EA2 1 (ρ) = e−iαZ/2ρeiαZ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (22) Observe that EA1 0 ⊗EA2 0 (|ψ⟩⟨ψ|) = EA1 0 ⊗EA2 1 (|ψ⟩⟨ψ|), and so there only three distinct encoded states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In fact, if A1 has prior probabilities {1 − q, q} over messages {0, 1} and A2 has uniform prior probabil- ities over the messages, then the resulting cq state σXA1A2 is equivalent to the cq state σXAR constructed in the one-sender assisted protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Therefore, by Theorem 4, α = π is optimal, and the maximal rate sum achievable with this protocol is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0931, which is achieved by the source state √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4715|e1⟩ + √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='5285|e2⟩ and encoding probability q ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The full rate region can also be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For each fixed θ ∈ [0, π/2], the initial state |ψ⟩A1A2 = cos θ|e1⟩ + sin θ|e2⟩ induces a classical MAC [2] × [2] → [3] when using the encoding of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (22) and the decoding measurement which projects into the basis {|00⟩, 1 √ 2(|e1⟩ ± |e2⟩)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The specific transition probabilities are found to be p(0|00) = cos2 θ, p(1|00) = p(2|00) = sin2 θ 2 p(0|01) = cos2 θ, p(1|01) = p(2|01) = sin2 θ 2 p(1|10) = 1 2 + cos θ sin θ, p(2|10) = 1 2 − cos θ sin θ p(1|11) = 1 2 − cos θ sin θ, p(2|11) = 1 2 + cos θ sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The rate region (R1, R2) is then found using Propo- sition 1 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' As we sweep θ over the interval 8 [0, π/2], the union of all achievable rate pairs using encoding scheme (22) is identified in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The solid line in this figure indicates the outer boundary on achievable rates using a uniform superposition input state |ψ⟩ = 1 √ 2(|e1⟩ + |e2⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' These values are noteworthy since they are what we try to experimen- tally replicate in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We can enhance the rate sum even further if we al- low one of the parties to have three inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Suppose now that A2 encodes with the same ternary oper- ation as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (21), and A1 again uses the on-off keying encoding: � EA1 0 (ρ) = Tr(ρ)|0⟩⟨0|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' EA1 1 (ρ) = ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' � � � � � EA2 0 (ρ) = Tr(ρ)|0⟩⟨0|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' EA2 1 (ρ) = ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' EA2 2 (ρ) = e−iαZ/2ρeiαZ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (23) Suppose that Alice and Bob’s prior probability of message 0 is q and q′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Then using the same method of calculating the accessible informa- tion of symmetric ensembles (see the Supplemental Material), we again find that the optimal phase en- coding is α = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This allows us to calculate the accessible information of the encoded cq state for any q, q′, and θ, which we then maximize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' An example of rate region that is achievable by using initial state |ψ⟩ = � 1/3|e1⟩ + � 2/3|e2⟩ and the binary-input protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The union of all achievable rate regions using our binary-input protocol (gray area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The solid line repre- sents the boundary of rate region that is achievable us- ing an equal superposition state � 1/2(|e1⟩ + |e2⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The dashed line represent the convex hull of all rate pairs achievable using a classical particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' There exists a two-sender unas- sisted communication protocol [2] × [3] → [3] that sends 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='1014 bits of information per channel use, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=', R(Q2) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='1014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The optimal (q, q′, θ) that achieves this are approximately (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='9197, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='9197, π/4), and the optimal measurement is given by projecting on the basis {|00⟩, 1 √ 2(|e1⟩ ± |e2⟩)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Note that unlike in this case of binary encoding, the optimal source state is a uniform superposition across both paths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' θ = π/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' A general encoding method for N ≥ 2 parties without blocking One drawback of the encoding schemes presented in Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (22) and (23) is that it requires one of the parties to perform an on-off keying (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='e “blocking”) operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' While intuitively simple, a reliable imple- mentation of this encoding in an optical setup can be quite demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Here we show that through the use of a coherence assistance path, a rate sum strictly larger than one is always achievable using simple 0, π phase encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The latter means that 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='6 2 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 R11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 Classicalparticle Achievablewithequalsuperpositionstate Achievablewithaguantumparticle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='6 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 R1the sender either acts trivially on the particle or ap- plies a rotation E(π)(ρ) = ZρZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Our protocol involves the idea of creating more paths by successive uses of a beam splitter (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Suppose that at the layer we start with the uniform superposition state 1 √ 2(|e1⟩ + |e2⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' A beam splitter is inserted along the second path yield- ing the state 1 √ 2|e1⟩ + 1 2(|e2⟩ + |e3⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This is re- peated repeatedly until the initial state |ψ⟩A1···ANR = �N i=1 1 √ 2i |ei⟩+ 1 √ 2N |eN+1⟩ is prepared for N senders A1, · · · , AN and a coherence assistance path R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Each sender encodes by applying a π phase shift EAi xi (ρ) = ZxiρZxi (24) for message xi ∈ {0, 1} with prior probability p(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Upon receiving the encoded particle, the receiver decodes using the projective measurement {|bi⟩⟨bi| : i ∈ [N + 1]} with orthonormal vectors |b0⟩ = 1 √ 2|e1⟩ + N � i=2 1 √ 2i |ei⟩ + 1 √ 2N |eN+1⟩ |b1⟩ = − 1 √ 2|e1⟩ + N � i=2 1 √ 2i |ei⟩ + 1 √ 2N |eN+1⟩ |b2⟩ = − 1 √ 2|e2⟩ + N � i=3 1 √ 2i−1 |ei⟩ + 1 √ 2N−1 |eN+1⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' |bN⟩ = − 1 √ 2|eN⟩ + 1 √ 2|eN+1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This induces a classical channel p(y|x1, · · · , xN), and for small N we can numerically compute their ca- pacities using the generalized Blahut-Arimoto al- gorithm adapted for MACs [27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The result FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' A multi-path state with attenuated amplitudes is generated by a successive application of beam splitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Numerical calculation of rate sums achievable with our N-sender protocol [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (24)] for N up to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Note that the general- ized Blahut-Arimoto algorithm is not guaranteed to converge to the optimal rate sum [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' How- ever, let us consider p(y|x1, · · · , xN) as a single- sender-single-receiver channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The original Blahut- Arimoto algorithm does in fact converge to its op- timal point-to-point capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This point-to-point capacity serves as an upper bound for the rate sum of p(y|x1, · · · , xN) since we are giving senders more power to coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For N = 2, the problem allows for an analytic solution, and we summarize the result in the propo- sition below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' There exists a two-sender coherence-assisted communication protocol that does not require blocking operation or vacuum detec- tion and sends log(17/8) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0875 bits per channel use, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=', R(Qass 2 ) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The optimal prior probability that achieves this is p(x1 = 0) = 1/2 and p(x2 = 0) = 15/17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' As N increases, we numerically find that the rate sum does not increase significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' On the one hand, this is not surprising since our encoding strat- egy uses an initial state |ψ⟩ that places smaller and smaller weight on the paths of additional parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' However, on the other hand, we have not been able to find any superior coding method, and in fact, many coding schemes (such as the “fingerprinting” protocol [6, 10]) have a rate sum that vanishes as N grows large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' A significant open problem is to find upper bounds on the largest N-party rate sum using a single quantum particle, which we conjecture will 10 Particle Source1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='125 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='120 - Achievable rate sum 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='115 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='110 N Rate sum N Rate sum N Rate sum 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='105 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0875 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='1220 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='1244 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='100- 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='1105 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='1235 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='1245 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='1187 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='1241 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='1246 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='095 x B-A algorithm for MACs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='090- Point-to-point capacity 2 3 4 5 6 7 8 9 10 Number of sendersnot be too far from the lower bound depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The coherence-assisted protocol just described uses only phase encodings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' However, it can eas- ily be converted to a coherence-unassisted commu- nication protocol at the expense of needing block- ing operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' To see the idea, consider the case of N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In the unassisted protocol, the encoded states σA1A2R x1x2 = |ψx1x2⟩⟨ψx1x2| have the form |ψ00⟩ = 1 √ 2|e1⟩ + 1 2|e2⟩ + 1 2|e3⟩ |ψ01⟩ = 1 √ 2|e1⟩ − 1 2|e2⟩ + 1 2|e3⟩ |ψ10⟩ = − 1 √ 2|e1⟩ + 1 2|e2⟩ + 1 2|e3⟩ |ψ11⟩ = − 1 √ 2|e1⟩ − 1 2|e2⟩ + 1 2|e3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Observe that these are made equivalent to the states |ψ′ 00⟩ = 1 √ 2|e1⟩ + 1 √ 2|e2⟩ |ψ′ 01⟩ = 1 √ 2|e1⟩ + 1 √ 2|e3⟩ |ψ′ 10⟩ = − 1 √ 2|e1⟩ + 1 √ 2|e2⟩ |ψ′ 11⟩ = − 1 √ 2|e1⟩ + 1 √ 2|e3⟩ (25) by a unitary operator that also transforms the mea- surement vectors into |b′ 0⟩ = 1 √ 2|e1⟩ + 1 √ 2|e2⟩ |b′ 1⟩ = − 1 √ 2|e1⟩ + 1 √ 2|e2⟩ |b′ 2⟩ = |e3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (26) Hence, the states of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (25) and measurement of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (26) will generate the same transition probabil- ities as the original MAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' But since the |b′ i⟩ have no coherence between the {|e1⟩, |e2⟩} and {|e3⟩} sub- spaces, we can first dephase the |ψ′ x1x2⟩ across these subspaces without altering the transition probabil- ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Doing so and relabeling |0⟩ ≡ |e3⟩ leads to states σx1x2 obtained by the unassisted encoding of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (22) (up to a swap A1 ↔ A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This method of converting a coherence-assisted protocol to an unassisted-protocol generalizes for any N ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The single-particle Holevo capacities All of the communication rates computed thus far assumes the receiver performs the same measure- ment on each received quantum particle so as to generate multiple uses of the same classical channel p(y|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' While this leads to a definite communication advantage compared to the use of a classical particle, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 9 suggests that this advantage is not that dra- matic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' On the other hand, if we enlarge the measure- ment capabilities of the decoder and allow for collec- tive measurements across multiple particle transmis- sions, then the capacity can be enlarged significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This quantity is the single-particle Holevo informa- tion χ(QN) as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (16), with χ(Qass N ) de- noting its coherence-assisted form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Our first result is the calculation of χ(Qass N ) for N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' χ(Qass 1 ) = max q,cos2 θ∈[0,1] qh2(cos2 θ) + cos2 θh2(q) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2339, where h(x) = −x log2 x − (1 − x) log2(1 − x) is the binary entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Note that since R(Qass 1 ) ≤ χ(Qass 1 ), this shows that the encoding scheme of Theorem 4 is not too far from optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The proof of this Theorem 7 is provided in the Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' As an intermediate step in our proof, we show that the encoding strategy of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (21) maximizes the Holevo information for each choice of initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Then optimizing over the initial state the maximum in Theorem 7 to be obtained by the values (q, cos2 θ) ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='7035, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='7035).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Turning to the N-sender case, we find that the single-particle Holevo information grows un- bounded, in sharp contrast to the optimised accessi- ble information, which we do not know exactly, but seems to remain bounded for all N despite our best efforts in searching for better protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' log N ≤ χ(QN) ≤ log(N + 1) and log(N + 1) ≤ χ(Qass N ) ≤ log(N + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' To achieve the lower bounds, the parties use an equal superposition state |ψ⟩ = � i 1 √ N |ei⟩ and 0, π phase encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' If each local message xi has uniform prior over {0, 1} then, the average encoded state is � x p(x)σA x = 1 N �N i=1 |ei⟩⟨ei|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Hence, χ(QN) ≥ log N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For the assisted case, a similar construction yields χ(Qass N ) ≥ log(N + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The upper bounds are simply dimensionality bounds based on the total 11 number of dimensions of the communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The lower bound given here is in general not tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For instance, when N = 2, Proposition 5 shows that χ(QN) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='1014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' EXPERIMENTAL DEMONSTRATION OF ENHANCED MULTIPLE-ACCESS COMMUNICATION USING A SINGLE PHOTON We have applied our communication framework to a multi-port optical interferometer experiment in which each sender controls one path that the parti- cle can traverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' A single photon is used as the mes- sage carrier, prepared into the desired superposition mode via the interferometric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Messages are coherently encoded by different senders onto the photon along each optical path of the interferometer and decoded by the single receiver, who collects the photon at the output ports of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Not all the communication protocols described above can be faithfully implemented using such a setup, due to various unavoidable experimental im- perfections, including finite transmission and de- tection efficiencies, a non-ideal probabilistic single- photon source with multi-photon pair generation, and imperfect interference visibility of the optical interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In particular, the photon loss in- curred from the finite detection/transmission effi- ciency prevents us from exploring the vacuum mode as a valid decoding outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Furthermore, the quantum enhancement is extremely sensitive to in- terferometric visibility, as we will explain in detail later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Taking all these factors into consideration, the most viable experiment to conduct is the two-sender coherence-assisted communication protocol (Propo- sition 6) presented in Section III B 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The advantage of this scenario is that quantum-enhanced communi- cation can be achieved using only phase encoding by each sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' However, as argued in Section III B 3, the communication rates are the same as in a two- sender unassisted protocol using path blocking and phase encoding on the uniform superposition state 1 √ 2(|e1⟩ + |e2⟩) (see the solid line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The experimental setup for this protocol is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' A heralded single photon is created from spontaneous parametric down conversion (SPDC), and sent to a three-port interferometer with split- ting ratio 1/2 : 1/4 : 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The single-photon state is filtered with a polarizer and spectral filter and cou- pled into single-mode fiber (SMF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' which allows us to ignore all of its internal degrees of freedom and write down the corresponding heralded state as a superposition of different path basis states |ei⟩ = |0⟩A1 · · · |1⟩Ai · · · |0⟩AN : |ψ⟩ = 1 √ 2|e1⟩ + 1 2|e2⟩ + 1 2|e3⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (27) where the third path is the assistance path,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' while senders 1 and 2 each encode their input bits onto the photon locally with tunable phase shifters in the form of glass windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The phase shifters are char- acterized with respect to the angle of rotation of the glass window and a phase-shift of 0 is set to encode the bit “0” and π to encode the bit “1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' At the output ports of the interferometric setup, single- photon detectors are placed and information is de- coded purely based on the which-port information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Experimental results In order to claim the implementation of a com- munication protocol with only one single particle involved, we characterize the heralded second-order cross-correlation function at zero delay g(2) hcc(0) of our photon-pair source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For an ideal source this num- ber should be 0, which means exactly one photon is produced in a heralded manner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' however, without a perfect photon-number-resolving detector, there will always be a trade-off between having a higher heralded-single-photon rate and lower g(2) hcc(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We measure g(2) hcc(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0017 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='001, which can basi- cally rule out the possibility of having more than one particle traveling into the communication setup after heralding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This value is set to be an order of magni- tude smaller than our expected quantum violation, as we will elaborate on later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Non-ideal single-photon source with small two multi-photon contributions, encoding operations, and/or decoding detections can all degrade the per- formance of our quantum protocol to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Among them, most error in the setup is actually caused by the non-unit interference visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Ide- ally, when the three-port optical interferometer has perfect interference visibility the following transition probabilities can be achieved: p(1|10) = 1, p(2|11) = 1, p(0|00) = 1 2, p(1|00) = 1 4, p(2|00) = 1 4 p(0|01) = 1 2, p(1|01) = 1 4, p(2|01) = 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (28) However, the communication protocol is extremely sensitive to the interference visibility, as shown in 12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Experimental setup: (a) Photon pairs are generated by pumping a PPKTP crystal with the second harmonic of a pulsed laser (generated in BiBO);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (b) The heralded single photons are sent to a three-port optical interferometer consisting of an inner Sagnac loop and an outer Mach Zehnder (MZ) interferometer with information encoded by auto-controlled phase plates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (c) The heralded single photons are coupled into single-mode fibers (SMF) and detected by avalanche photodiodes (APD);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' different combinations of coincidence counts are processed by a time-to-digital converter (TDC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' To obtain a greater quantum enhance- ment with better interferometric visibility, we devise a three-port optical interferometer comprised of a passively stabilized Mach-Zehnder (MZ) interferom- eter with an offset Sagnac interferometer embedded within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The visibility of the Sagnac interferometer is achieved to above Vs = 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2% after tightly filtering the single photon spectrally and spatially, and the visibility for the outer MZ interferometer is around Vz = 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4% averaged over 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Our experimental demonstration of quantum ad- vantage comes in two forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We first build a channel having transition probabilities close to those of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' With this channel, it is, in principle, possible to achieve asymptotic communication rates strictly larger than what is possible using a classical par- ticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Second, we go one step further and actually use the channel to establish correlated random vari- ables between the senders and receivers whose mu- tual information is above one, thereby exceeding the accessible information of a classical particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Expected enhanced two-sender communication rate as a function of the interference visibilities of the inner Sagnac and outer Mach-Zender (MZ) interferom- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The maximal capacity rate of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0875 is achieved when perfect interference visibility is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 13 Time-to-digita converter Single photon detector SMFC Lens Spetral filter Tsunami Dichroic Modelocked Mirror BiBO Laser Beam splitter PPKTP Polarization Phase Beam splitter Shifter Mirror DelayRate sum 100 ☆ Our experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='08 Visibility (%): Sagnac interferometer 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='06 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='04 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='03 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='01 99 100 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='6 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4 98 Visibility (%): Mach-Zehnder interferometerFIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (a) Example of transition probability p(y|x) from direct characterization of the two-sender channel where inputs x = (x1, x2) ∈ {0, 1}×2 and output y ∈ {0, 1, 2} (b) The union of achievable rate regions with the corresponding channel, with experiment in blue and the ideal case in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The dashed line represents the bound of the rate region achievable by a classical MAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (c) Comparing our empirical joint distribution p(x, y)empirical to its theoretical value for two-sender channel inputs x ∈ {0, 1}×2 and output y ∈ {0, 1, 2} where error bars are statistical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Characterizing a two-sender assisted channel by transition probabilities To demonstrate quantum enhancement in the two- sender communication protocol, we first character- ize the transition probability of the channel p(y|x = (x1, x2)), where xi is the bit encoded by sender i corresponding to 0 (π) phase for xi = 0 (xi = 1), while y is the trit decoded by the receiver based on the “which-port” information of the output particle measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Given the low g(0) hcc we set, we character- ize each transition probability with different inputs x = (x1, x2) by registering coincident counts over a three minute period with around N = 105 events registered (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 12 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Using the measured transition probabilities p(y|x), the asymptotic rate region for the con- structed channel can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In particular, for the ideal channel of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (28), the mutual in- formation between senders and receivers is found to be maximized by a uniform prior distribution for x1 ∈ {0, 1} and a biased distribution for x2 ∈ {0, 1} with Pr{x2 = 0} ≈ 15 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' With X = (X1, X2) denot- ing input variables with these distributions, our con- structed channel can thus achieve an input-output mutual information of I(X : Y )ch = � x,y p(x)p(y|x) log p(y|x) p(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0152 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0034, (29) where the error is the standard deviation over 10 runs of the experiment to take both statistical and systematic error into consideration (the estimation on the statistical error is given in the supplemen- tary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' More generally, by varying the prior p(x) = p(x1)p(x2), a different rate region is deter- mined by the three mutual information quantities {I(X1 : Y |X2), I(X2 : Y |X1), I(X1X2 : Y )} via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The union of these regions is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 12 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Characterizing a two-sender assisted channel by mutual information We take the demonstration further by generating empirical random variables (X, Y ) that are corre- lated using the single-particle channel we build.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Ide- ally we would like their mutual information I(X : Y ) to be close to the maximum accessible information R(Qass 2 ), but any value larger than one will already yield a quantum advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' To this end, we gen- erate multiple series of random bits each of length 680 by independently sampling from the input set {0, 1} with uniform probability p(0) = 1/2 for in- put x1 and biased probability p(0) = 15 17 for input x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Ideally, each sample would correspond to a spe- cific choice of encoding in one run of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' However, in practice we can only change the encod- ing map once per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Hence, the ensemble we generate has the form {p(x), σ⊗m x }⊗n rather than (ideally) {p(x), σx}⊗mn, where both n = 680 and m ≈ 600 to be the coincident count rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Even if we assume that the decoder does not try to exploit this block structure (see the discussion on loopholes below), there are still two sources of uncertainty in this setup: (a) the generation of the random bit and (b) the photon number fluctuation in each run of the 14 (a) Theory (b) (c) Theory P(y|x=00) P(y|x=10) Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 Experiment Ideal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='75 Experiment 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='25 R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='00 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='00 y=o y=1 y=2 y=o y=1 y=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 acity P(y|x=01) P(y|x=11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='75 x=11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='50 x=10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='25 y=0 x=01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' + y=1 y=o y=1 y=2 y=o y=1 y=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 X=00 Capacity rate: Ri y=2experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The result is a mutual information with larger uncertainty and larger bias than I(X : Y )ch, yet still above the classical threshold: I(X : Y )empirical = � x,y p(x, y) log p(x, y) p(x)p(y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0117 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0047, (30) where again the error is the standard deviation over 10 runs of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Here, I(X : Y )empirical is computed using the empirical joint distribution p(x, y)empirical compiled from both the input and output data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Experimental imperfections and loopholes Similar to the problems encountered in most pho- tonic Bell tests [31–33], our communication frame- work suffers from several experimental loopholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' While these can be fixed in principle, they make an experimental demonstration of enhanced quantum communication challenging to attain at the single- particle level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Detection loophole In optical experiments, the main difficulty in demonstrating our theoretical protocols is the lim- ited photon detection efficiency, which generates many “no-click” events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The single-photon detec- tor we employ (APD, Excelitas SPCM-AQ4C) has a photon detection efficiency around 40% at our working wavelength of 810 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This ratio can be improved up to 95% with superconducting single- photon detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Yet, even this relatively high efficiency is insufficient to implement many single- particle communication protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The standard way of demonstrating a detection-loophole free Bell test is to classically relabel no-click events as some other detection event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Unfortunately, this is not a good strategy in any communication protocol that uses blocking as an encoding operation since then “no-click” events are intentionally used to transmit information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' To see this quantitatively, consider the two-sender unassisted protocol in Section III B 2 that uses blocking as an encoding operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' When start- ing with a uniform superposition 1 √ 2(|e1⟩+|e2⟩) and following the encodings of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (22), the resulting channel without detection efficiency has the tran- sition probabilities of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' If we assume the FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For a non-ideal single photon detector, no quantum enhancement can be observed in the two-sender unassisted scenario when the detector efficiency η drops below roughly 97% (and all other apparatuses behave flawlessly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' detection efficiency is a constant η for all detectors, then the transition probabilities are replaced by p(1|10) = η, p(0|10) = 1 − η, p(2|11) = η, p(0|11) = 1 − η, p(0|00) = 1 − 1 2η, p(1|00) = 1 4η, p(2|00) = 1 4η, p(0|01) = 1 − 1 2η, p(1|01) = 1 4η, p(2|01) = 1 4η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 15, the largest capacity rate sum of this channel drops below one quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' A similar situation occurs if the transmission efficiency is low (below 97% in the above case), which is almost in- evitable in optical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This experimental imperfection leads to two con- sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' First, we cannot perform any protocol with block operations using our current technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Second, even for the case of using just phase en- coding, our experiment does not close the detection loophole but instead uses the assumption of “fair sampling.” In other words, we assume that the ac- cepted data in our experiment is representative of the data that would have been recorded if the detec- tors had unit efficiency [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Freedom-of-choice loophole The freedom-of-choice loophole has recently been proposed and fixed in Bell tests [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This loop- 15 fixed prior optimizedprior 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='85 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='9 Detection efficiency nhole refers to the possibility that “hidden variables” may influence the choice of measurements in experi- ments and thus enable cheating in acquiring the em- pirical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' A similar concern could also be raised in the exper- imental demonstration presented in Section IV A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' As described, the time delay in our ability to switch the encoding of each sender means that the same channel input is selected in m = O(103) consecutive experimental runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This lack of input freedom for each trial could be exploited in some classical proto- col that is attempting to reproduce the same mutual information I(X : Y )empirical > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The ultimate way of fixing this problem is to independently choose an input (x1, x2) and apply encoding EA1 x1 ⊗EA2 x2 for each incoming photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This requires a phase encoding operation as fast as 80 MHz in order to match our laser repetition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This can be achieved poten- tially with electro-optic devices or acousto-optical devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' however, due to the demanding require- ments of the overall interference visibility for the interferometric setup, we could not easily introduce such components into our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Locality loophole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In standard Bell experiments, the locality con- straint is set to prevent the two sites from commu- nicating with each other [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Our experiment has a similar loophole in that without sufficient separa- tion between the senders, it is potentially possible for them to communicate and perform some joint (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' not independent) encoding on the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' To avoid this, at least we would need to design the ex- periment so that the communication time between senders is much longer than the time it takes the photon to travel from one sender to the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In our case, the order of time differences should be determinate by the coincident window we set, which requires the spatial separation between senders to be greater than 2ns × c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='6m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Closing this loophole in our setup is challenging since the overall interfer- ence visibility and stability is limited by the size of the interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' CONCLUSION In the present paper, we investigated how much information can be transmitted from multiple senders to a single receiver by the use of a single quantum or classical particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' To analyze this ques- tion and to show an advantage of quantum over classical particles, we have created a framework of classical multiple-access channels constructed by lo- cally modulating an initial superposition state of different paths and afterwards detecting the parti- cle with a general measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The classical case is included when the initial state assigns a definite path to the particle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' on the other hand, if the ini- tial state is a genuine quantum superposition, it has the potential to induce channels not reachable with a classical state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Specifically, we found that the communication rates of independent messages of the separate users show a clear quantum advan- tage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Indeed, for a single classical particle, the rate-sum for any number of senders is bounded by one, while it exceeds one for two or more sender, being monotonically increasing in the number of senders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The rate-sum can be even larger in the model of coherence assistance, where there is an- other path from the source directly to the decoder, which allows a rate exceeding 1 even for the single- sender model, to be precise 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0931 bits per channel use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We also experimentally demonstrated our pre- dicted quantum advantage by implementing the two- sender coherence-assisted protocol using an optical interferometric setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The constructed channel sup- ports a communication rate-sum of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0152 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0034, showing a four-standard-deviation quantum advan- tage over the classical bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Alternatively, the channel can be used to correlate random variables whose mutual information we empirically found to be I(X : Y )empirical = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='012±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='005, again exceeding the classical threshold of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We leave a number of open questions regarding the basic theoretical understanding of the single- particle MAC, starting with the actual maximum value of the rate-sum for any number of senders and the characterization of the full capacity region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Our best upper bound on the rate-sum is the Holevo quantity, and while we are just short of calculat- ing that exactly, it scales as log N for large number N of senders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' By contrast, we do not even know if the achievable rate-sum via accessible informa- tion diverges or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' It seems we would want bet- ter outer bounds on the capacity region, but it is perhaps much more exciting to search for improved modulation and detection schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In another di- rection, fixing the particular initial state, but opti- mizing over modulations and detection, the achiev- able rate region could give new quantifiers for the amount of coherence in the state along the lines of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' [12, 35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 16 ACKNOWLEDGEMENTS This work was supported by the National Sci- ence Foundation Award Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 1839177 and 2112890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' AW is supported by the European Commission QuantERA grant ExTRaQT (Spanish MICINN project PCI2022-132965), by the Spanish MINECO (project PID2019-107609GB-I00) with the support of FEDER funds, the Generalitat de Catalunya (project 2017-SGR-1127), by the Spanish MCIN with funding from European Union NextGener- ationEU (PRTR-C17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='I1) and the Generalitat de Catalunya, and by the Alexander von Humboldt Foundation, as well as the Institute of Advanced Study of the Technical University Munich.' metadata={'source': 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+page_content=' The extrema of the expected value of a function of independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The Annals of Mathematical Statistics, 26(2):268– 275, 1955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 18 Appendix A: Lower bound for one-sender (R(Q∗ 1)) Using the encoding operations given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (21), the encoded cq state is σXAR = (1 − q)|0⟩⟨0| ⊗ σ0 + q 2|1⟩⟨1| ⊗ σ1 + q 2|2⟩⟨2| ⊗ σ2 = (1 − q)|0⟩⟨0| ⊗ � cos2 θ|00⟩⟨00| + sin2 θ|e2⟩⟨e2| � + q 2|1⟩⟨1| ⊗ (cos θ|e1⟩ + sin θ|e2⟩) (cos θ⟨e1| + sin θ⟨e2|) + q 2|2⟩⟨2| ⊗ � eiα cos θ|e1⟩ + sin θ|e2⟩ � � eiα cos θ⟨e1| + sin θ⟨e2| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' To calculate its accessible information, we first note that the optimal POVM achieving the accessible in- formation can be taken to be rank-1 projectors [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Additionally, as noted in the main text, the ensemble has the following symmetries: (i) σXAR is diagonal in the number basis, and (ii) q/2 · σ1 and q/2 · σ2 are related by a reflection across the line y = x tan(α/2) in the x − y plane of the Bloch sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Using the same arguments in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' [26] (Proposition 1), we deduce that the optimal measurement attaining the accessible information can be made to have the same symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Therefore, the optimal POVM can be taken to be {|00⟩⟨00|, wm|πm⟩⟨πm|, wm|π′ m⟩⟨π′ m|}, where |πm⟩ = √σm|e1⟩ + √σmeiβm|e2⟩ (A1) |π′ m⟩ = √σm|e1⟩ + √σme−i(α+βm)|e2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (A2) Here σm = 1 − σm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Each m labels a pair of symmetric projectors specified by (wm, σm, βm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Now, since � m(Πm + Π′ m) is the projector onto the |e1⟩, |e2⟩ subspace, we have � m � wm � σm √σmσme−iβm √σmσmeiβm σm � + wm � σm √σmσmei(α+βm) √σmσme−i(α+βm) σm �� = I, (A3) from which we can conclude that � m wmσm = 1 2, � m wm = 1, � m wm √σmσm(eiβm + e−i(α+βm)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (A4) Denote the set of {(wm, σm, βm)}m satisfying all three constraints in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (A4) as S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Following the same approach laid out in [26], the accessible information of the ensemble (and hence the total communication rate) is given by Iacc = max S � m wmJ(σm, βm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q, θ, α), (A5) where J(σ, β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q, θ, α) =q| √ σ cos θ + eiβ√σ sin θ|2 log | √ σ cos θ + eiβ√σ sin θ|2 + q| √ σ cos θ + ei(β−α)√σ sin θ|2 log | √ σ cos θ + ei(β−α)√σ sin θ|2 + 2(1 − q)σ sin2 θ log(σ sin2 θ) − 2κ log κ − (1 − q) cos2 θ log(1 − q), (A6) in which κ = qσ cos2 θ + q [cos β + cos(β − α)] √ σσ cos θ sin θ + σ sin2 θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We can relax the restriction on wm, σm and βm by dropping the last condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (A4), thus obtaining an upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Formally, let ˜S denote the set of {(wm, σm, βm)}m that satisfy only the first two conditions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (A4), then � Iacc = max ˜ S � m wmJ(σm, βm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q, θ, α) ≥ Iacc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (A7) 19 Note that dropping the third condition essentially allows us to optimize βm’s freely independent of any other parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Our first goal is to find the optimal phase encoding α, denoted by α∗, that maximizes the function J(σ, β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q, θ, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For any σ, q, and θ, J(σ, β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q, θ, α) is maximized only if (α, β) = (0, 0), (0, π), (π, 0) or (π, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For J to attain a local maximum, it is necessary that the directional derivative D⃗uJ = 0 and the second directional derivative D2 ⃗uJ ≤ 0 along any direction ⃗u on the α-β plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Specifically, let us consider two direction given by ⃗u1 = (1, 0)⊺ and ⃗u2 = (1, 1)⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Then we have: D ⃗ u1J = ∂J ∂α = 0, D ⃗ u2J = ∂J ∂α + ∂J ∂β = 0, (A8) D2 ⃗ u1J = ∂2J ∂α2 ≤ 0, D2 ⃗ u2J = � ∂ ∂α + ∂ ∂β � �∂J ∂α + ∂J ∂β � ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (A9) Calculating the first derivatives gives: D ⃗ u1J = 1 ln 2q � ln | √ σ cos θ + ei(β−α)√σ sin θ|2 + 1 � 2 √ σσ cos θ sin θ sin(β − α) − 1 ln 22(ln κ + 1)q √ σσ cos θ sin θ sin(β − α) = 0, (A10) D ⃗ u2J = − 1 ln 2q � ln | √ σ cos θ + eiβ√σ sin θ|2 + 1 � 2 √ σσ cos θ sin θ sin β + 1 ln 22(ln κ + 1)q √ σσ cos θ sin θ sin β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (A11) Assuming q √ σσ cos θ sin θ ̸= 0 (when one of q, cos θ, and sin θ is 0, the ensemble becomes trivial, and when one of σ and σ is 0, then J reduces to −(1 − q) cos2 θ log(1 − q), which is independent of α and β), the two equations simplify to: � sin(β − α) log | √ σ cos θ + ei(β−α)√σ sin θ|2 − sin(β − α) log κ = 0 sin β log | √ σ cos θ + eiβ√σ sin θ|2 − sin β log κ = 0 (A12) This set of equations admits four possible conditions: (i) sin(β − α) = 0, sin β = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (A13) (ii) | √ σ cos θ + ei(β−α)√σ sin θ|2 = κ, sin β = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (A14) (iii) sin(β − α) = 0, | √ σ cos θ + eiβ√σ sin θ|2 = κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (A15) (iv) | √ σ cos θ + ei(β−α)√σ sin θ|2 = κ, | √ σ cos θ + eiβ√σ sin θ|2 = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (A16) Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' calculating the second derivatives gives us: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='⃗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='u1J = 2q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='σσ cos θ sin θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='− cos(β − α) log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='�| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='σ cos θ + ei(β−α)√σ sin θ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='ln 2 sin(β − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='σ cos θ + ei(β−α)√σ sin θ|2 − q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='� √ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='σσ cos θ sin θ sin(β − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='(A17) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='⃗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='u2J = 2q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='σσ cos θ sin θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='− cos β log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='�| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='σ cos θ + eiβ√σ sin θ|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='ln 2 sin β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='σ cos θ + eiβ√σ sin θ|2 + q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='� √ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='σσ cos θ sin θ sin β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='(A18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='Recall that cos θ sin θ > 0 since θ can be taken to be in [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' π/2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' and we assumed cos θ sin θ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Plugging each of the four conditions into the two expressions above we find that D2 ⃗ u1J > 0 for conditions (ii) and (iv), while D2 ⃗ u2J > 0 for conditions (iii) and (iv), unless sin(β − α) = sin β = 0 also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Therefore, points satisfying (i), namely (α, β) = (0, 0), (0, π), (π, 0), or (π, π), are the only possible local maxima of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Since we have dropped some constraints on β and treated it as an independent variable when optimizing, the optimal (α, β) may not actually be feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' However, it is easy to check that (α, β) = (0, π), (π, 0), and (π, π) satisfies all of the constraints in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (A4), and therefore they correspond to physical POVMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This result tells us that the best phase encoding that the encoder can perform in our one-sender protocol is either α = 0 or α = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Additionally, note that (α, β) = (π, 0) and (α, β) = (π, π) are images of each other under the reflection across y = x tan(α/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' So, they correspond to the same pair of projectors, and we can freely choose either one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Note that, if the encoder chooses α = 0, the encoded cq state σXAR effectively reduces to σXAR = (1 − q)|0⟩⟨0| ⊗ � cos2 θ|00⟩⟨00| + sin2 θ|e2⟩⟨e2| � + q|1⟩⟨1| ⊗ (cos θ|e1⟩ + sin θ|e2⟩) (cos θ⟨e1| + sin θ⟨e2|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The accessible information of this state is necessarily less than or equal to 1 bit, meaning that there is no quantum advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In other words, for maximal quantum advantage, one should use π phase encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This is summarized by the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In the one-sender coherence-assisted scenario, if the encoding maps is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (21), then for any initial state |ψ⟩AR and any measurement POVM for B, whenever there is a quantum advantage in the communication rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=', whenever the communication rate exceeds 1 bit), α = π is always the optimal phase encoding that A can perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We can now prove the following theorem in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' There exists a one-sender coherence-assisted communication protocol that sends approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0931 bits of information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=', R(Q∗ 1) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The optimal (q, θ) that achieves this are approximately (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8701, arccos( √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4715)), and the optimal measurement is the projective measurement {|00⟩, 1 √ 2(|e1⟩ ± |e2⟩)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Having established that α = π is the best encoding phase, in which case the best decoding phase is 0 (or equivalently π) we will set α = β = π and obtain J(σ, β = π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q, θ, α = π) ≡ ˜J(σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q, θ) =q( √ σ cos θ + √σ sin θ)2 log( √ σ cos θ + √σ sin θ)2 + q( √ σ cos θ − √σ sin θ)2 log( √ σ cos θ − √σ sin θ)2 + 2(1 − q)σ sin2 θ log(σ sin2 θ) − 2(qσ cos2 θ + σ sin2 θ) log(qσ cos2 θ + σ sin2 θ) − (1 − q) cos2 θ log(1 − q) (A19) and Iacc(q, θ) = max � m wmσm=1/2 � m wm=1 � m wm ˜J(σm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q, θ) (A20) Following the same argument presented in [26], which we briefly recapitulate here for completeness, we first find that this maximization for the accessible information can be rewritten as a maximization with at most two terms [38], that is, Iacc(q, θ) = max σ1≤1/2≤σ2 �σ2 − 1/2 σ2 − σ1 ˜J(σ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q, θ) + 1/2 − σ1 σ2 − σ1 ˜J(σ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q, θ) � (A21) 21 which can be again rewritten as Iacc(q, θ) = max σ1≤1/2≤σ2 � ˜J(σ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q, θ) + 1/2 − σ1 σ2 − σ1 � ˜J(σ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q, θ) − ˜J(σ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q, θ) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (A22) The maxand can be understood as the value of the line through points (σ1, ˜J(σ1) and (σ2, ˜J(σ2) at 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For each θ, we can find three different measurement regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' When q is sufficiently small, the optimal (σ1, σ2) is (0,1), corresponding to optimal measurement vectors |e1⟩ and |e2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' As q becomes larger, the optimal (σ1, σ2) is 0 and some σ∗ ∈ [1/2, 1], corresponding to a POVM with |e1⟩ and a mirror-symmetric pair of rank-one projectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Finally, when q is sufficiently close to 1, the optimal (σ1, σ2) is (1/2,1/2), corresponding to projective measurements { 1 √ 2(|e1⟩ ± |e2⟩)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' By straightforward calculation, we find that in region 1, the accessible information of the ensemble is Iacc,1(q, θ) = −(1 − q) cos2 θ log(1 − q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (A23) In region 3, Iacc,3(q, θ) =q − qh2 �1 + sin 2θ 2 � + (1 − q) sin2 θ log sin2 θ − (q cos2 θ + sin2 θ) log(q cos2 θ + sin2 θ) − (1 − q) cos2 θ log(1 − q) (A24) where h2 is the binary entropy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In region 2, the calculation is more involved, Iacc,2(q, θ) = J(0) + 1 2 dJ dσ (σ∗) (A25) where σ∗ is determined from the fact that the tangent line of J at σ∗ passes through (0, J(0)), in other words, J(0) + σ∗ ∂J ∂σ (σ∗) = J(σ), (A26) which after much algebra becomes q cos2 θ log qσ∗ cos2 θ + σ∗ sin2 θ σ∗ cos2 θ − σ∗ sin2 θ = (1 − q)σ∗ sin2 θ log σ∗ (A27) To plot the accessible information in the entire region of (q, θ), we note Iacc(q, θ) = maxi=1,2,3{Iacc,i(q, θ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' One can check by comparing the plot of Iacc,1(q, θ), Iacc,2(q, θ), and Iacc,3(q, θ) that the maximal accessible information occurs in region 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' To compute its value, we take the derivative of Iacc,3 with respect to q and θ and set both to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' ∂Iacc,3 ∂θ = q cos 2θ log �1 + sin 2θ 1 − sin 2θ � + (1 − q) sin 2θ log � (1 − q) sin2 θ q cos2 θ + sin2 θ � = 0 (A28) ∂Iacc,3 ∂q = 1 − h2 �1 + sin 2θ 2 � − log sin2 θ + cos2 θ log � (1 − q) sin2 θ q cos2 θ + sin2 θ � = 0 (A29) There is no closed form solution for this system of transcendental equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Solving these two equations numerically gives sin θ∗ ≈ √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4715 and q∗ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' This optimal choice of θ and q corresponds to the intial source state ≈ √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4715|e1⟩ + √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='5285|e2⟩, prior probability p(x) ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='1299, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4351, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4351), and the rate sum is approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' If the source state is the maximally coherent state 1 √ 2|e1⟩ + 1 √ 2|e2⟩, then the optimal rate is log(17/8) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0875 > 1, and the optimal prior probabilities given by q∗ = 15/17 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8824 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Take θ = π/4 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (A24) and after simplification, we find that I(q, π/4) = 2q − 1 + h2 � 1+q 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The maximizer q∗ can be found by setting the derivative with respect to q to 0, and we find that q∗ = 15/17, in which case the mutual information is log(17/8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 22 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The accessible information of the ensemble (assuming α = π) in terms of q and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Appendix B: Lower bound for Unassisted Two-Sender (R(Q2)) In this section, we calculate the accessible information of cq states arising from the binary-ternary encoding strategy given in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Following the same steps laid out in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' A, the accessible information can be expressed as Iacc = max S � m wmJ(σm, βm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q, θ, α), (B1) where J(σ, β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q, q′, θ, α) = (1 − q)q′(sin2 θ log sin2 θ + 2σ cos2 θ log σ cos2 θ) +q(1 − q′)(cos2 θ log cos2 θ + 2σ sin2 θ log σ sin2 θ) +qq′| √ σ cos θ + eiβ√σ sin θ|2 log | √ σ cos θ + eiβ√σ sin θ|2 +qq′| √ σ cos θ + e−i(α+β)√σ sin θ|2 log | √ σ cos θ + e−i(α+β)√σ sin θ|2 −ξ log ξ − 2η log η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (B2) in which ξ = (1 − q)(1 − q′) + (1 − q)q′ sin2 θ + q(1 − q′) cos2 θ = 1 − q′ cos2 θ − q sin2 θ (B3) η = (1 − q)q′σ cos θ + q(1 − q′)σ sin2 θ + 1 2qq′ � | √ σ cos θ + eiβ√σ sin θ|2 + | √ σ cos θ + e−i(α+β)√σ sin θ|2� (B4) By the same argument as in Lemma 9, we can deduce that the local extrema of function J occurs only if α and β are both multiples of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' And that α = π is the optimal phase encoding whenever there is a quantum 23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4 q 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='6 @0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4 Maximal accessibleinfo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0 badvantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Taking α = β = π then, we have J(σ, β = π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q, q′, θ, α = π) = (1 − q)q′(sin2 θ log sin2 θ + 2σ cos2 θ log σ cos2 θ) +q(1 − q′)(cos2 θ log cos2 θ + 2σ sin2 θ log σ sin2 θ) +qq′( √ σ cos θ − √σ sin θ)2 log( √ σ cos θ − √σ sin θ)2 +qq′( √ σ cos θ + √σ sin θ)2 log( √ σ cos θ + √σ sin θ)2 −ξ log ξ − 2η log η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (B5) where ξ = 1 − q′ cos2 θ − q sin2 θ (B6) η = (1 − q)q′σ cos2 θ + q(1 − q′)σ sin2 θ + 1 2qq′ � ( √ σ cos θ − √σ sin θ)2 + ( √ σ cos θ + √σ sin θ)2� = q′σ cos2 θ + qσ sin2 θ (B7) Following the same analysis, we find three regimes for the optimal measurement, and the accessible infor- mation is maximized in the regime that corresponds to σ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Iacc(q, q′, θ) = J(σ = 1 2, β = π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' α = π) = (1 − q)q′(sin2 θ log sin2 θ + cos2 θ log 1 2 cos2 θ) +q(1 − q′)(cos2 θ log cos2 θ + sin2 θ log 1 2 sin2 θ) −qq′h2 �1 + sin 2θ 2 � − ξ log ξ − 2η log η (B8) where ξ = 1 − q′ cos2 θ − q sin2 θ (B9) η = 1 2q′ cos2 θ + 1 2q sin2 θ (B10) Again taking the derivative of Iacc with respect to q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' and θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' we obtain the following system of equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='∂J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='∂q = −q′(sin2 θ log sin2 θ + cos2 θ log 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 cos2 θ) + (1 − q′)(cos2 θ log cos2 θ + sin2 θ log 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 sin2 θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='−q′h2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='�1 + sin 2θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='+ sin2 θ log ξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='(B11) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='= (2q′ − 1)h2(sin2 θ) + q′ − sin2 θ − q′h2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='�1 + sin 2θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='+ sin2 θ log ξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='η = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='(B12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='∂J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='∂q′ = (1 − q)(sin2 θ log sin2 θ + cos2 θ log 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 cos2 θ) − q(cos2 θ log cos2 θ + sin2 θ log 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 sin2 θ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='−qh2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='�1 + sin 2θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='+ cos2 θ log ξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='(B13) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='= (2q − 1)h2(sin2 θ) + q − cos2 θ − qh2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='�1 + sin 2θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='+ cos2 θ log ξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='η = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='(B14) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='∂J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='∂θ = (q + q′ − 2qq′) log tan2 θ + qq′ cot 2θ log 1 + sin 2θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='1 − sin 2θ = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='(B15) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='Numerically solving this system of equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' we obtain that the optimal q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' and θ is (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='9197, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='9197, π/4), and the optimal rate sum is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='10138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 24 Appendix C: One-sender assisted Holevo information (χ(Q∗ 1)) - proof of Theorem 7 Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' χ(Qass 1 ) = maxq,cos2 θ∈[0,1] qh2(cos2 θ) + cos2 θh2(q) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We first show that the encoding given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (21) is in fact the best encoding strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Consider the most general encoding strategy using NPE operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' By convexity of the mutual information with respect to the underlying channel, it is sufficient for us to consider pure initial state cos θ|e1⟩+sin θ|e2⟩ and encoding strategies consisting of only extremal NPE operations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' With this simplification, we only need to optimize the Holevo information over cq states � a pa|a⟩⟨a| ⊗ ρa where pa is the prior probability and ρa = � � γa cos2 θ (1 − γa) cos2 θ √1 − γa cos θ sin θeiφa √1 − γa cos θ sin θe−iφa sin2 θ � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (C1) This means that � a paρa = � � � a paγa cos2 θ � a pa(1 − γa) cos2 θ � a pa √1 − γa cos θ sin θeiφa � a pa √1 − γa cos θ sin θe−iφa sin2 θ � � , (C2) and � a paS(ρa) = � a pah2(γa cos2 θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (C3) Therefore, χ �� a pa|a⟩⟨a| ⊗ ρa � = S �� a paρa � − � a paS(ρa) ≤ ˜χ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' pa, γa) (C4) where ˜χ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' pa, γa) := H ��� a paγa cos2 θ, � a pa(1 − γa) cos2 θ, sin2 θ �� − � a pah2(γa cos2 θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (C5) Here H(·) is the Shannon entropy, and the inequality becomes equality when � a pa √1 − γaeiφa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Taking derivative of ˜χ with respect to γa, and after some algebra, we find d˜χ dγa = pa cos2 θ � log � a′ pa′(1 − γa′) � a′ pa′γa′ − log 1 − γa cos2 θ γa cos2 θ � (C6) If there is a local max, then d˜χ dγa = 0 for all a, which means γa = � a′ pa′γa′ cos2 θ ∀a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=', they are all equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' However, this then means γa = γa cos2 θ, which cannot be true unless cos2 θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Therefore, if cos2 θ ̸= 1, then ˜χ has no local extrema, and the maximum has to occur at the boundaries γa = 0 or γa = 1, corresponding to phase shift or complete damping encoding operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Thus to maximize ˜χ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' pa, γa) with respect to γa, γa must be 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In this case, let us define p := � a:γa=1 pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Then we have ˜χ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' pa, γa) ≤ H � p cos2 θ, (1 − p) cos2 θ, sin2 θ � − ph2(cos2 θ) = cos2 θh2(p) + ph2(cos2 θ) (C7) 25 which means that χ(Qass 1 ) := max χ �� a pa|a⟩⟨a| ⊗ ρa � ≤ max p,θ cos2 θh2(p) + ph2(cos2 θ) = max x,y∈[0,1] xh2(y) + yh2(x) (C8) Note that this upper bound can be achieved by precisely the encoding scheme given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Solving for the extrema of the last equation gives us (x∗, y∗) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='7035, in other words, the optimal initial state is |ψinit⟩ ≈ √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='7035|e1⟩ + √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2965|e2⟩, and the optimal prior probabilities are p(a) ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2965, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='3518, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='3518).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Additionally, χmax ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Appendix D: Details of the experiment Source preparation: The source of photon pairs is based on type II spontaneous parametric down- conversion in a 2 mm periodically polled Potassium titanyl phosphate (PPKTP) crystal (with temperature stabilizing oven).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The crystal is pumped with frequency-doubled light pulses originating from a Tsunami modelocked laser (a train of ∼100-fs pulses with center wavelength 810 nm and repetition rate 80 MHz), doubled using a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='5 mm Bismuth Borate (BiBO) crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' To prepare the photons in a single spectral, polarization, and spatial mode, the heralding photons from the pair are filtered to ∼2 nm bandwidth at full-width-at-half-maximum by a pair of tilted spectral filters, set to linear polarization by a polarizer, and coupled into single-mode fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The existence of this idler photon is detected via a single-photon detector (avalanche photodiode, Excelitas SPCM-AQ4C), while the other, heralded single photon is sent to a three- port interferometer to prepare the desired state |ψ⟩ = 1 √ 2|e1⟩ + 1 2|e2⟩ + 1 2|e3⟩, where we have ignored all the internal degrees of freedom of the single particle and only represented it in a superposition of different path basis states |ei⟩ = |0⟩A1 · · · |1⟩Ai · · · |0⟩AN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' To ensure the signal photon is close to a single photon source, we looked at the heralded signal photon within a 2 ns coincident window after heralding the idler photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' We characterize the source by measuring its second-order correlation g(2) iss′, which can be calculated as: g(2) iss′ = Ciss′Ci CisCis′ (D1) where Ciss′ are three-fold coincident counts between one idler photon and two single photons after splitting, Cis(s′) represents two-fold coincident counts between idler photon and one signal photon, and Ci denotes single counts for the idler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The power-dependence of the second-order correlation values is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 15 (a), which indicates good agreement with the linear curve fitting and allows to set the pump power to suppress the two-photon contribution from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In our experiment, the heralded second-order correlation g(2) iss′ has to be set extremely small, due to the fact that large higher-order terms could in principle enable a higher capacity rate even in the classical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Taking the small violation we have estimated (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='02), we set g(2) iss′(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='0017 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='001 to be one order of magnitude smaller than the violation to make sure the contribution from multiple-photon events can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' As a consequence, we have relatively low coincidence count rates around 600Hz Interferometer design: The interference visibility of our three-port interferometer limits the performance of our quantum-enhanced communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' To achieve a high enough visibility with free-space optics, we design a three-port interferometer consisting of (1) an inner offset Sagnac interferometer, which is extremely stable over a few hours with above 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='5% interference visibility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (2) an outer Mach–Zehnder interferom- eter, which is passively stabilized thermally and vibrationally inside a small box and gives around 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2% interference visibility over 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' It was further actively adjusted by a piezo actuator implemented on the translation stage in the delay line between different runs of measurement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (3) three 3-mm-thin glasses 26 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (a) Experimental result for heralded second-order correlation g(2) iss′ under different SHG pump powers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (b) Interference visibility of the Mach–Zehnder interferometer measured in 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (c) Theoretical characterization of our phase plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' windows for controlling the phase independently;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' windows were chosen instead of other bulky electro-optical devices, which could potentially degrade the interference visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The whole setup can be maintained stable over ∼10 minutes with average interference visibilities around 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='5% and 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='2% for the inner and outer loops, respectively, while slight adjustment with the piezo actuator helps to retrieve good interference visibility for the next round of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' During the runs of our experiment, we do not turn the active stabilization on so that the average stability remains the same over 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Encoding operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' As has been mentioned before, with the current type of single-photon detectors used and the loss in our free-optics setup, performing general amplitude damping operations on the photons is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Instead, we devised our setup based on the coherent-assisted protocol where only phase encoding is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' One of the most commonly used phase shifters is electrically controlled liquid crystal, where the refractive index along some axes of the crystal depends on the voltage applied to it and thus can be used to add phase on single photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' However, the resolution of the applied phase (around 3◦) and the size and the parallelism of most commercial liquid crystals prevent us from using them in our small-size, high-visibility interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Therefore, as a replacement, we create a phase shifter based on a d = 3 mm glass window (with a reflective index around ng = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='51) mounted on a rotation stage (with a resolution around 5 min-arc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Starting from placing the glass plate perpendicular to the incoming beam, the phase added to the photon after slightly tilting it with angle α can be computed as: ∆φ = 2πd λ [( � n2g − sin(α)2 − cos(α)) − (ng − 1)], (D2) which is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The average resolution over 2π phase shift is around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='3◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' however, due to its nonlinear behavior, by carefully choosing the starting point, we can obtain a much finer resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' Error analysis: To estimate the experimental error, we note at first that we are limited mostly by the interferometer stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' To ensure high interference visibility, we perform each run of our measurement for ∼10 minutes and re-optimize the setup between different runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' In each run of the experiment, the statistical error can be calculated from standard error propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' For the case of characterizing channel transition probability p(y|x): V [R1] = 1 N 2 � xy � (p(x) log2 q(y) + p(x)H(q(y)))2 + (p(x) log2 p(y|x) + p(x)H(p(y|x)))2) � V [ny|x] (D3) where q(y) = � a p(y|x)p(x) with fixed optimal prior p(x1 = 0) = 1/2 and p(x2 = 0) = 15/17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' ny|x is the total number of photons collected at port y conditional on input x and N is a total number of counts using in characterizing the channel for every input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' The statistical error assuming the Poisson distribution is given as V (ny|x) = Np(y|x)(1 − p(y|x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' With N ≈ 105, the statistical error is around � V (R1) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 27 100 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='014 (b) without active stabilization (c) with active stabilizatior 350 86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='010 cor 96 Visility 0 200 app 94 150 100 92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='002 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='000 90 + 20 30 40 10 50 60 70 0 2 6 8 10 1 3 6 SHG power (mW) min rotation (degree)Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' for the case of measuring the joint distribution p(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' the error can be computed as: V [R2] = 1 N 2 � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='y (log2 q(y) + log2 p(y)) − log2 p(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' y) + I(y : x))2 V [nxy],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' (D4) With extra uncertainty in p(x) from the generation of encoding random bits and V [nxy] = np(x)(1−p(x))× m + n × mp(y|x)(1 − p(y|x)) where n = 680 (the number of random input bits) and m ≈ 600 (the number of counts per second),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' we get a statistical error of estimating V [R2] to be around � V (R2) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content='011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' However, experimentally, besides the statistical error, the channels built from run to run are actually slightly different since they are extremely sensitive to the overall interference visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' To take those systematic errors into consideration and to show that our experimental result is repeatable, we calculate the experimental result I(X : Y ) in 10 runs of experiments and average the capacity rate sum, which is what we present in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} +page_content=' 28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE0T4oBgHgl3EQfngFx/content/2301.02513v1.pdf'} diff --git a/btFST4oBgHgl3EQfCziU/content/tmp_files/2301.13708v1.pdf.txt b/btFST4oBgHgl3EQfCziU/content/tmp_files/2301.13708v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dce8880cfe592ccbc306e98d858c9183dfecfb17 --- /dev/null +++ b/btFST4oBgHgl3EQfCziU/content/tmp_files/2301.13708v1.pdf.txt @@ -0,0 +1,2829 @@ +Draft version February 1, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +YunMa: Enabling Spectral Retrievals of Exoplanetary Clouds +Sushuang Ma +,1 Yuichi Ito +,2, 1 Ahmed Faris Al-Refaie +,1 Quentin Changeat +,3, 1 Billy Edwards +,4, 5, 1 and +Giovanna Tinetti +1 +1Department of Physics and Astronomy, University College London +Gower Street, WC1E 6BT London, United Kingdom +2National Astronomical Observatory of Japan +2 Chome-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan +3European Space Agency (ESA) +ESA Office, Space Telescope Science Institute (STScI), 3700 San Martin Drive, Baltimore MD 21218, United States of America +4SRON, Netherlands Institute for Space Research, Niels Bohrweg 4, NL-2333 CA, Leiden, The Netherlands +5AIM, CEA, CNRS, Universit´e Paris-Saclay, Universit´e de Paris, F-91191 Gif-sur-Yvette, France +ABSTRACT +In this paper, we present YunMa, a model which enables the study of cloud microphysics and radiative +properties in exoplanetary atmospheres. YunMa simulates the vertical distribution and sizes of cloud +particles and their corresponding scattering signature in transit spectra. We validated YunMa against +results from the literature. +When coupled to the TauREx 3 platform, an open Bayesian framework for spectral retrievals, YunMa +enables the retrieval of the cloud properties and parameters from transit spectra of exoplanets. The +sedimentation efficiency (fsed), which controls the cloud microphysics, is set as a free parameter in +retrievals. We assess the retrieval performances of YunMa through 28 instances of a cloudy super- +Earth’s atmosphere. This work also highlights the need for cloud radiative transfer and microphysics +modelling to retrieve next-generation data of exoplanets. +Keywords: Exoplanets(498) — Exoplanet atmospheres(487) — Transmission spectroscopy(2133) — +Atmospheric clouds(2180) +1. INTRODUCTION +Thousands of exoplanets have been detected since the +late 20th century. During the past decade, transit spec- +troscopy has become one of the most powerful tech- +niques for studying exoplanets’ atmospheres in-depth +(e.g., reviews by Tinetti et al. 2013; Burrows 2014; Mad- +husudhan 2019). +Data recorded from space-borne in- +struments (e.g., Hubble, Spitzer and James Webb Space +Telescopes) or from the ground have revealed important +information about exoplanet atmospheric chemistry and +dynamics (e.g., Sing et al. 2016; Tsiaras et al. 2018; Wel- +banks et al. 2019; Changeat et al. 2022; Edwards et al. +2022; JWST Transiting Exoplanet Community Early +Release Science Team 2022; Venot et al. 2020; Roudier +et al. 2021) and may provide insight into planetary in- +Corresponding author: Sushuang Ma +sushuang.ma.20@ucl.ac.uk +terior composition and formation (Madhusudhan et al. +2020; Yu et al. 2021; Tsai et al. 2021; Charnay et al. +2022). +A number of spectral retrieval models have been de- +veloped by different teams to interpret the atmospheric +data and quantify their information content; these in- +clude e.g., Madhusudhan & Seager (2009), Lee et al. +(2012), TauREx 3 (Al-Refaie et al. 2021), NEMESIS +(Irwin et al. 2008), CHIMERA (Line et al. 2013), AR- +CiS (Min et al. 2020; Ormel & Min 2019), PICASO +(Robbins-Blanch et al. 2022; Batalha et al. 2019), BART +(Harrington et al. 2022), petitRADTRANS (Molli`ere, +P. et al. 2020), HELIOS (Kitzmann et al. 2020), PO- +SEIDON (MacDonald & Madhusudhan 2017), HyDRA +(Gandhi & Madhusudhan 2018), SCARLET (Benneke +2015), PLATON II (Zhang et al. 2020), and Pyrat-Bay +(Cubillos & Blecic 2021). Up to date, most of the re- +trieval studies of exoplanetary atmospheres are highly +parameterised. +This approach has been very sensible +arXiv:2301.13708v1 [astro-ph.EP] 31 Jan 2023 + +ID2 +Ma et al. +given the relatively poor information content of cur- +rent atmospheric data. However, a number of papers in +the literature (e.g., Caldas et al. 2019; Changeat et al. +2021a; Changeat et al. 2022) have cautioned against +this approach when applied to data recorded with next- +generation facilities. +Clouds are omnipresent in planetary, exoplanetary +and brown dwarf atmospheres (see e.g., review by +Helling 2022) and have often been detected in exoplanet +atmospheric data (Kreidberg et al. 2014; Sing et al. 2016; +Stevenson 2016; Tsiaras et al. 2018). +Their presence +imposes additional complexity and uncertainties in the +interpretation of exoplanet atmospheric spectra (e.g., +Changeat et al. 2021b; Tsiaras et al. 2019; Mai & Line +2019). +Models simulating the formation and radiative prop- +erties of clouds and hazes have been published in the lit- +erature, e.g., Exo-REM (Baudino et al. 2015; Charnay +et al. 2018), Gao et al. (2020), Windsor et al. (2022) and +Kawashima & Ikoma (2018). +Due to the – currently limited – observational con- +straints and computational resources available to simu- +late the complexity of clouds, retrieval studies of cloudy +atmospheres are still in their infancy (see e.g., Fortney +et al. 2021). For instance, many studies have adopted +wavelength-independent opaque clouds, where all the ra- +diation beneath the cloud top is blocked from reaching +the telescope, and retrieve the vertical location of clouds +(Boucher et al. 2021; Brogi & Line 2019). +Wakeford +et al. (2018) used a grey, uniform cloud in the ATMO +Retrieval Code (ARC) (Goyal et al. 2017; Drummond, +B. et al. 2016; Tremblin et al. 2015). +Other models +constrain from radiative transfer the uniform cloud par- +ticle sizes without being estimated through cloud mi- +crophysics models. For instance, Benneke et al. (2019 +a) have initially estimated the particle sizes in the at- +mosphere of GJ 3470 b using Mie-scattering theory. +Extended from this highly parametric approach, cloud +scattering parameters and inhomogeneous coverage were +also retrieved: NEMESIS was used by Barstow (2020) +and Wang et al. (2022) to retrieve the cloud’s opacity, +scattering index, top and base pressures, particle sizes +and shape factor. Pinhas et al. (2019) run POSEIDON +to constrain the cloud’s top pressure and coverage frac- +tion. Wang et al. (2022) adopted PICASO to extract the +cloud’s base pressure, optical thickness, single scatter- +ing albedo, scattering asymmetry and coverage. Lueber +et al. (2022) extended the use of Helios-r2 to retrieve +non-grey clouds, with extinction efficiencies estimated +from Mie theory calculations. The model Aurora (Wel- +banks & Madhusudhan 2021) presents inhomogeneities +in cloud and haze distributions by separating the atmo- +sphere horizontally into four distinct areas. +The data provided by the next-generation telescopes +will be greatly superior in quality and quantity, allow- +ing us to obtain more stringent constraints to our under- +standing of clouds in exoplanetary atmospheres. Transit +spectra of exoplanets recorded from space by the James +Webb Space Telescope (JWST, 0.6–28.3 µm, Bean et al. +2018; Greene et al. 2016; Gardner et al. 2006), Ariel +(0.5–7.8 µm, Tinetti et al. 2018; Tinetti et al. 2021) +and Twinkle (0.5–4.5 µm, Edwards et al. 2019) at rela- +tively high spectral resolution and/or broad wavelength +coverage will open the possibility of integrating self- +consistent, cloud microphysics approaches into atmo- +spheric retrieval codes. A good example of such models +is ARCiS (ARtful modelling Code for exoplanet Science, +Min et al. 2020; Ormel & Min 2019), which simulates +cloud formation from diffusion processes and parametric +coagulation. ARCiS also generates cloudy transit spec- +tra from Mie theory (Fleck & Canfield 1984) and Dis- +tribution of Hollow Spheres (DHS, Min, M. et al. 2005; +Molli`ere, P. et al. 2019), and can be used to retrieve the +cloud diffusivity and nuclei injection from transit spec- +tra. +In this work, we present a new optimised model to +study cloud microphysical processes directly integrated +into a spectral retrieval framework. We consider clouds +as a thermochemical product, i.e. +the aggregation of +condensates in the atmosphere, while hazes form pho- +tochemically (Kawashima & Ikoma 2018). +The cloud +distribution depends on the atmospheric conditions. Be- +ing generated thermochemically, clouds form and diffuse +depending on the atmospheric thermal structure and, in +return, contribute to it. They also depend on the mix- +ing profiles of the condensable gases in the atmosphere. +Clouds act as absorbers and/or scatterers and therefore +may dampen the atomic and molecular spectroscopic +features and change the continuum. +Based on studies of the Earth and Solar System’s plan- +etary atmospheres, Lewis (1969) published a 1-D cloud +model optimised to describe tropospheric clouds in gi- +ant planets. This model assumes that the fall speeds +of all condensates are equivalent to the updraft veloc- +ities, and only vapour is transported upward. Lunine +et al. (1989) included a correlation between cloud par- +ticle sizes, downward sedimentation and upward turbu- +lent mixing. Based on previous models by Lewis (1969), +Carlson et al. (1988), Lunine et al. (1989) and Marley +et al. (1999), Ackerman & Marley (2001) proposed a new +method to estimate the mixing ratio and vertical size +distribution of cloud particles (A-M model hereafter). +In the A-M model, the sedimentation timescale is esti- + +YunMa: Enabling Spectral Retrievals of Exoplanetary Clouds +3 +mated through cloud microphysics, taking into account +the atmospheric gas kinetics and dynamical viscosity. +The model assumes an equilibrium between upward tur- +bulent mixing and sedimentation, where the turbulent +mixing is derived from the eddy diffusion in the atmo- +sphere. The key assumptions of the A-M model are as +follows: +1. Clouds are distributed uniformly in the horizontal +direction. +2. Condensable +particles +rain +out +at +(su- +per)saturation while maintaining a balance of +the upward and downward drafts. +3. It does not consider the cloud cover variations +caused by precipitation or the microphysics be- +tween different types of clouds. +The A-M model was originally proposed for giant exo- +planets and brown dwarfs and was tested on Jupiter’s +ammonia clouds, demonstrating that this approach is +applicable to a broad range of temperatures and plane- +tary types. +Another popular 1-D cloud microphysics model is the +Community Aerosol and Radiation Model for Atmo- +spheres (CARMA), initially developed for the Earth’s +stratospheric sulfate aerosols (Turco et al. 1979; Toon +et al. 1979). +CARMA is a time-dependent cloud mi- +crophysics model which solves the discretised continuity +equations for aerosol particles starting from nucleation. +Gao et al. (2018) extended the use of CARMA to sim- +ulate clouds on giant exoplanets and brown dwarfs by +including additional condensates predicted to form in +hot atmospheres and compared the results with the A- +M model. The A-M model, while able to provide the +cloud particle sizes and number density distributions, +is of intermediate numerical complexity and, therefore, +potentially adaptable to be included in retrieval codes. +In addition to the original implementation by Ackerman +& Marley (2001), Virga (Rooney et al. 2022) simulates +the cloud’s particle size distribution from the A-M ap- +proach and estimates separately the sedimentation effi- +ciency. PICASO (Robbins-Blanch et al. 2022; Batalha +et al. 2019) adopts Virga to simulate cloudy exoplan- +etary atmospheres. Adams et al. (2022) couples MIT +GCM and Virga to include clouds in 3-D models. Xuan +et al. (2022) present high-resolution spectroscopic sim- +ulations with petitTRANS (Molli`ere, P. et al. 2020). +There, clouds are modelled from the A-M theory and +spectra are simulated with Mie theory and DHS. None +of these models, however, are currently available for re- +trieval simulations. +To simulate inhomogeneities for cloud formation in +the horizontal direction, we would need to consider +global circulation atmospheric effects, such as those +modelled in Cho et al. (2021). An example of 3-D at- +mospheric model with clouds is Aura-3D (Welbanks & +Madhusudhan 2021; Nixon & Madhusudhan 2022). The +retrieval part for Aura-3D is highly parametrised, both +for the atmospheric and cloud parameters. Helling et al. +(2022, 2019) have simulated global cloud distributions +by generating inputs to their kinetic cloud model from +pre-calculated 3D Global Circulation Models (GCMs). +Unfortunately, these complex models require excessive +computing time. In addition, the data expected to be +observed in the near future are unlikely to constrain +the large number of parameters needed in a 3-D model. +Therefore, while theoretical studies with 3-D models +are very important to progress in our understanding of +clouds in exoplanetary atmospheres and as benchmarks, +they are currently less useful for interpreting available +data. +In this paper, we present a new cloud retrieval model, +YunMa, optimised for transit spectroscopy. In YunMa, +we built the cloud model based on Ackerman & Marley +(2001) and simulated the cloud contribution to transit +spectra using extinction coefficients as calculated by the +open-source B-H Mie code (Bohren & Huffman 2008). +YunMa is fully integrated into the TauREx 3 retrieval +platform (Al-Refaie et al. 2022; Al-Refaie et al. 2021; +Al-Refaie et al. 2019) and, for the first time, provides +cloud microphysics capabilities into a retrieval model. +We describe the model in Section 2. In Section 3, we +detail the experimental setups. In Section 4, we validate +particle size distributions and spectroscopic simulations +against previous studies published in the literature. Af- +ter validation, we show new spectral and retrieval sim- +ulations obtained with YunMa. +In Section 5, we dis- +cuss our results and assumptions and identify possible +improvements to the model to be considered in future +developments. +2. MODEL DESCRIPTION +YunMa estimates the vertical distribution of the cloud +particle sizes (VDCP hereafter, see, e.g. Fig. B1 b and +B4 a, b in Appendix B) based on A-M model and their +contribution to the radiative transfer calculations. The +YunMa module has been integrated into the TauREx +3 retrieval platform: +the combined YunMa-TauREx +model is able to constrain the VDCP from observed at- +mospheric spectra, as described in detail below. +2.1. Modelling the cloud particle size distribution +YunMa model contains a numerical realisation of the +A-M microphysical approach to simulate the VDCP. We + +4 +Ma et al. +show in Fig. +1 a pictorial representation of the A-M +approach: it assumes that clouds form with different +VDCP to maintain the balance between the upward tur- +bulent mixing and downward sedimentation of the con- +densable species. +Depending on the atmospheric T-p +profile, multiple cloud layers may form. +Figure 1. +Sketch of the A-M micophysical approach +adopted in YunMa. Cloud particles may form when the mix- +ing ratio of the condensable gas exceeds the saturation mix- +ing ratio, which is derived from its saturation vapour pressure +(SVP). The vertical distribution of the cloud particle sizes +(VDCP) is derived from the balance between the sedimen- +tation of the cloud particles and the atmospheric turbulent +mixing. +2.1.1. Cloud mixing profile +Cloud particles start forming when the partial pres- +sure of a certain gas exceeds the saturation vapour pres- +sure (SVP): the formation strongly depends on the at- +mospheric thermal structure. The condensation process, +occurring when the partial pressure exceeds the SVP, is +estimated by comparing the molecular mixing ratio of +the gas phase with its saturation vapour mixing ratio: +qc(z) = max[0, qt(z) − (Sc + 1) qs(z)], +(1) +where qc is the mixing ratio of the condensed species, +z is the altitude, qs is the mixing ratio where the con- +densable gas saturates, qt is the total mixing ratio of a +condensable chemical species, including both the con- +densate and gas phases, and Sc is the supersaturation +factor which persists after condensation. qs can be es- +timated from the ratio between the SVP of a certain +chemical species and the atmospheric pressure at the +same altitude. Note that in this paper, the mixing ratio +refers to the volume fraction of a chemical species in the +atmosphere. +In the A-M approach, the turbulent mixing of the con- +densate and vapour is assumed to be in equilibrium with +the sedimentation of the condensate: +−K(z) ∂qt(z) +∂z +− fsed w∗(z) qc(z) = 0, +(2) +where K (m2 s−1) represents the vertical eddy diffusion +coefficient, and w∗ (m s−1) is the convective velocity. +fsed is the ratio between the mass-weighted droplet sed- +imentation velocity and w∗, defined as: +fsed = +� ∞ +0 vf dm +dr dr +ερaw∗qc +; +(3) +here ρa is the atmospheric mass density, which can be +estimated through the Ideal Gas Law, ρp is the mass +density of a condensed particle, ε is the ratio between +the molecular weights of the condensates and the at- +mosphere, and vf is the sedimentation velocity which +will be explained later. The first term in equation (2) +describes the upward vertical draft derived from the +macroscopic eddy diffusion equation. The second term +describes the downward sedimentation, which is in equi- +librium with the first term. +The eddy diffusion coefficient (K) is one of the key +parameters affecting cloud formation. In free convection +(Gierasch & Conrath 1985), it can be estimated as: +K = H +3 +� L +H +� 4 +3 � RF +µρacp +� 1 +3 +, +(4) +where H, µ and cp are, respectively, the atmospheric +scale height, mean molecular weight and specific heat +capacity. F = σT 4 +eff is the approximated radiative flux. +The turbulent mixing length (L) is the scale height of +the local stability in eddy diffusion, as opposed to the +atmospheric scale height (H). In YunMa, we use the +value of K as estimated by the atmospheric chemistry +models embedded in TauREx 3 (Al-Refaie et al. 2022; +Al-Refaie et al. 2021) and derive L, accordingly, from +equation (4). The convective velocity scale (w⋆) men- +tioned above can also be estimated as a ratio between +K and L. +The sedimentation velocity, denoted by vf, is the +speed at which a cloud particle settles within a hetero- +geneous mixture due to the force of gravity. vf can be + +Space/Anothercloud layer +Horizontally homogeneous +Clouddeck +Turbulent mixing +Cloud· +Partialpres. +Vapour... +Particles +VSSVP +Sedimentation +Cloudbase +Clearatmosphere +Surface/AnothercloudlayerYunMa: Enabling Spectral Retrievals of Exoplanetary Clouds +5 +estimated through viscous fluid physics: +vf = 2 +9 +βgr2∆ρ +η +, +(5) +where ∆ρ is the difference between ρp and ρa, β is the +Cunningham slip factor, and η is the atmospheric dy- +namical viscosity (See Appendix A for more details of +SVP, β and η). +2.1.2. Particle size and number density +Following the A-M approach, we assume spherical +cloud particles with radii r. The particle radius at w⋆, +denoted as rw, can be obtained using these relationships +between vf and w⋆: +vf(rw) = w⋆, +(6) +and +vf = w⋆( r +rw +)α, +(7) +where α corresponds to the sedimentation velocity de- +crease in viscous flows. In A-M, the particle size distri- +bution was constrained by in-situ measurements of Cal- +ifornian stratocumulus clouds, which followed a broad +lognormal distribution. +The assumption of lognormal +distribution allows estimating the geometric mean ra- +dius (rg), the effective radius (reff) and the total cloud +particle number density (N), using the detailed defini- +tions and derivations listed in Appendix A. +2.2. Cloud contribution in transit spectra +To estimate the wavelength-dependent cloud contri- +bution to transit spectra, YunMa adopts the scatter- +ing theory and absorption cross sections as described in +Bohren & Huffman (2008, BH-Mie hereafter), assuming +spherical cloud particles. The cross-section of the cloud +particles (kλ) at each wavelength (λ) and particle size +are estimated through the extinction coefficient (Qext) +of the corresponding wavelength and particle size, de- +rived by BH-Mie from the refractive indices of the cloud +particles: +kλ = Qext πr2. +(8) +We used the water ice refractive indices reported in War- +ren & Brandt (2008) for our simulations of the tem- +perate super-Earth. +We show some examples of at- +mospheres with water ice clouds in Section 4. +Post- +experimental tests were conducted to avoid the contam- +ination of liquid water particles. +We simulate the cloud optical depths from the parti- +cle sizes and number densities along each optical path, +which passes the terminator at altitude zter, with a path +length szter of each atmospheric layer: +τλ = +� ztop +zter +� ∞ +0 +kλ +dn +dr dr dszter +dz +dz, +(9) +where n is the accumulated number density of particles +with a radius smaller than r and ztop is the altitude +at the top of the atmosphere. The contribution of the +clouds to the transit spectra, ∆Fc, can be estimated as: +∆Fc = +2 +� ztop +zbottom(Rp + z)(1 − e−τλ) dz +R2s +, +(10) +where zbottom is the altitude at Rp. While YunMa has +the capability to include any customised cloud particle +size distribution in the spectral simulations, in this pa- +per, we aim at model testing and, for simplicity, we use a +single radius bin, i.e. uniform cloud particle size rc = rg +(see equation A5 in Appendix A) for each atmospheric +layer in the radiative-transfer simulation. +2.3. YunMa-TauREx: retrieval of cloudy atmospheres +We integrate the YunMa VDCP and τλ simulations +in the Tau Retrieval of Exoplanets framework (Tau- +REx 3, Al-Refaie et al. 2022; Al-Refaie et al. 2021; Al- +Refaie et al. 2019), which allows atmospheric retrieval +simulations. TauREx 3 combined to YunMa allow us +to perform retrievals which include cloud microphysical +processes and cloud scattering properties. Parameters +estimated by TauREx 3 include atmospheric T-p and +chemical profiles, planetary (e.g., mass and radius) and +stellar (e.g., temperature and metallicity) parameters. +The radiative transfer calculations executed by TauREx +3 consider molecular and atomic absorptions, Rayleigh +scattering and collisionally induced absorptions (CIA) +of H2-H2 and H2-He pairs from Cox (2015). +YunMa uses as initial condition the gas mixing ratio +profiles provided by TauREx 3 chemistry models (qt, +equation 1, 2). In this paper, for simplicity, we assume +the baseline chemical abundances are constant with al- +titude instead of a more complex chemical structure. +YunMa then adjusts the gas phase mixing ratios, atmo- +spheric mean molecular weight and atmospheric density +in the TauREx 3 chemistry models as a result of the for- +mation of clouds. To simulate transit spectra and per- +form retrievals, we use the atmospheric grids and optical +paths defined in TauREx 3 and add the cloud opaci- +ties as estimated by YunMa B-H Mie to the absorptions +caused by the chemical species, using the methods ex- +plained in Section 2.2. The retrievals were tested on 80 +Intel(R) Xeon(R) Gold 6248 CPU @ 2.50GHz. +3. METHODOLOGY +In this paper, we use YunMa to perform retrieval sim- +ulations of small temperate planets, where we expect a +considerable amount of H2O to be present in the at- +mosphere. For simplicity, we consider only water clouds + +6 +Ma et al. +Table 1. +Priors for spectral retrieval exper- +iments using YunMa of all the cases listed in +Table 3. +Parameter +Unit +Ground +Mode +Priors +Rp +RN +Je +0.20 +factor +0.75 – 1.25 +fsed +· · · +Table 3 +log +10−3 – 102 +XH2O +· · · +Table 3 +log +10−12 – 1 +pc +bar +Table 3 +log +10−4 – 1 +Tc +K +200 +linear +0 – 500 +Tsurf +K +1000 +linear +500 – 2000 +XN2 +· · · +Table 3 +linear +10−12 – 1 +Note—XH2O represents the water vapour mixing ratio +forming in the atmosphere and we do not consider super- +saturation cases. The planetary parameters are inspired +by K2-18 b (Tsiaras et al. 2019; Charnay et al. 2021 b; +Yu et al. 2021), which is a suitable candidate for cloud +model testing. We list all the priors of our experiment +in Table 1. In this work, we set K as a constant and +estimate η using the approximation proposed by Ros- +ner (2012, equation A4). +We include both scattering +and absorption due to water clouds based on Bohren +& Huffman (2008), Rayleigh scattering of all the gas +species and CIA of H2-H2 and H2-He pairs, which are +enabled by TauREx 3. We fine-tune the abundance of +N2 to adjust the scale heights of the atmosphere. H2 and +He act as the filling gases. We use the POKAZATEL +dataset for 1H16 +2 O (Polyansky et al. 2018) from the Exo- +Mol database1 (Tennyson & Yurchenko 2012; Tennyson +& Yurchenko 2021; Chubb et al. 2021) to estimate the +water vapour absorption and Rayleigh scattering. The +CIA data is from HITRAN2 (Karman et al. 2019). We +use the PHOENIX library (Husser et al. 2013) to simu- +late the stellar atmospheres spectra. +For the numerical parameter settings, after a num- +ber of tests, we decided to use the explicit Runge-Kutta +method of order 8 (DOP853, Hairer et al. 1988) with +relative tolerance (rtol) of 1 × 10−13 and absolute tol- +erance (atol) of 1 × 10−16 to solve the partial differ- +ential equation (2) for all the experiments presented in +Section 4. We have opted for a logarithm sampling to +retrieve most of the atmospheric parameters, e.g., fsed, +XH2O and pc. We have used a linear sampling, instead, +for N2 to obtain a better numerical performance. The +priors are sufficiently unconstrained to avoid biases gen- +erated by excessive pre-knowledge, as discussed, e.g. in +1 https://exomol.com +2 https://hitran.org +Changeat et al. (2021a). After a number of tests, we +have chosen to use 400 live points for 3-dimensional re- +trievals and 1000 for more dimensions. +To begin with, we run a sensitivity study with YunMa +about the planetary and instrumental parameters. We +set the planetary radius (Rp), fsed and XH2O as free pa- +rameters in our 3-dimensional retrieval tests. We list in +Table 1 the planetary parameters adopted in the simu- +lations, the prior ranges and the sampling modes. The +simulations are conducted with 80 atmospheric layers, +from 10 bar to 10−6 bar, which encompass the typical +observable atmospheric range for super-Earths. We se- +lect Case 2 in Table 3 as the nominal case, and test the +model sensitivity to the key parameters in the retrievals. +In Case 2, clouds dampen the gas spectroscopic features +but do not obscure them entirely (see Fig. 3). The nom- +inal fsed refers to the value adopted in A-M. The water +SVP (both liquid and ice) used are taken from Appendix +A in A-M. +SENSITIVITY STUDIES TO KEY ATMOSPHERIC +PARAMETERS +We have performed sensitivity studies to test how the +model behaves when changing some of the key atmo- +spheric parameters, including pc, fsed, XH2O, XN2 (Case +1–9). Isothermal T-p profiles, as commonly used in tran- +sit retrieval studies, are too simplistic for cloud studies. +We first assume T-p profiles with a dry adiabatic lapse +rate in the troposphere, a moist adiabatic lapse rate in +the cloud-forming region and a colder isothermal pro- +file above the tropopause. To be compatible with the +computing requirements in retrieval, we simplify it to +a ”two-point” profile (e.g., Fig. 5 in Section 4) in Tau- +REx while keeping the lapse rate at the cloud deck, since +the cloud formation is more sensitive to it than the T-p +profile in the deeper atmosphere from our preliminary +sensitivity tests. We define as Tc and pc the temperature +and pressure at the tropopause and Tsurf the tempera- +ture at 10 bar. By tuning pc, the altitude at which the +cloud forms and the cloud’s optical thickness will be al- +tered. Here we test pc from 10−3 to 10−1 bar (Case 1–5), +which is a much broader range than the one considered +in previous literature about K2-18 b. In Case 6, we set +the sedimentation efficiency fsed to 0.01, i.e. the down- +ward sedimentation of the cloud particles is relatively +slow compared to the net upward molecular mixing of +the condensable species. By contrast, in Case 7 where +fsed = 10, we have a larger downward draft velocity scale +compared to the upward one. In Case 8 and 9, we mod- +ulate the amount of condensable gas, here represented +by the water vapour mixing ratio (XH2O). + +YunMa: Enabling Spectral Retrievals of Exoplanetary Clouds +7 +SENSITIVITY STUDIES TO DATA QUALITY +The new generation of space-based facilities, such as +JWST and Ariel, will deliver unprecedentedly high- +quality data in terms of wavelength coverage, signal-to- +noise ratio and spectral resolution. We select as nominal +case transit spectra covering 0.4–14 µm, at a spectral +resolution of 100, with 10 ppm uncertainty across wave- +lengths. We chose 0.4 µm as the blue cut-off to maximise +the information content about Rayleigh scattering and +14 µm as the red cut-off to maximise the information +content about water vapour and atmospheric tempera- +ture for the type of planets considered here. We then +test how YunMa’s performances degrade when we com- +promise with any of those parameters, for instance: un- +certainties of 30 ppm in Case 11 and spectral resolution +of 10 in Case 12, which should have similar effects. We +move the blue cut-off at longer wavelengths in Case 13. +RETRIEVALS OF ATMOSPHERIC THERMAL +PROFILES +We have run retrieval experiments to test how YunMa +performs with increasingly complex model assumptions +and which parameters may be problematic in these re- +trievals. +YunMa’s ability to retrieve pc, Tc and Tsurf +were studied in Cases 14–17. In Case 15, we use a higher +quality spectrum as input and compare the results with +Case 14. +ADDITION OF N2 IN RETRIEVALS +Inactive, featureless gases, such as N2, inject much +uncertainty in the retrieval. In Cases 18–20, we include +different amounts of the inert and featureless gas N2 +in the atmosphere. N2 may exist in super-Earths’ at- +mospheres, as it happens for Solar System planets at +similar temperatures. Being N2 heavier than H2O, we +adjust the mean molecular weight and the scale heights +accordingly by modulating XN2. Heavier atmospheres +have smaller scale heights: the spectral features are less +prominent and harder to detect. We first try to retrieve +only Rp, fsed, XH2O and XN2. +DEGENERACY BETWEEN CLOUDS AND HEAVY +ATMOSPHERE +In Cases 14–20, N2 and the T-p profile were retrieved +separately. Here we attempt to retrieve the cloud mi- +crophysics parameters (fsed) together with XN2, the T-p +profile and other atmospheric parameters. Case 21 re- +trieves these parameters except XN2 fixed to 50% for +comparison with Case 22 to investigate the degeneracy +imposed by the uncertainty of XN2. In Case 22, we per- +form the full retrieval including both XN2 and the T-p +profile as free parameters. Similar to N2, decreases the +scale height and compresses the transit spectrum, the +existence of clouds mitigates the spectral features, and +the difference between these two scenarios may be dif- +ficult to distinguish from the current data quality. One +hypothesis is that if we do not include N2 among the +priors, the model will add clouds to compensate for the +missing N2. In Case 23, we test this hypothesis by forc- +ing XN2 to zero and then monitor the cloud parameters +in the posteriors obtained. +COMPARISON OF CLOUD RETRIEVAL MODELS +Here forward spectra are simulated with YunMa cloud +microphysics and then retrieved with another simplified +cloud retrieval framework. In Case 24, clouds are de- +scribed as an opaque cloud deck across wavelengths in +TauREx 3, which is commonly used to retrieve data from +the last decades with narrow wavelength coverage, e.g., +HST/WFC3. +In Case 25, we deliberately omit cloud +parameters in the retrieval priors and learn if and how +other parameters can compensate for those missing. +RETRIEVALS OF FEATURELESS SPECTRA +In Cases 26–28, we retrieve the atmospheric parame- +ters from spectra with minimal spectroscopic features, +where spectral standard deviation (σspec) is less than 1 +ppm. The spectra are featureless due to heavy atmo- +spheres and cloud contribution. In Case 28, we unreal- +istically decrease the uncertainty to 1 ppm at resolving +power 100 to evaluate YunMa’s performance with ide- +alised data quality. +4. RESULTS +4.1. Model validation +We validated our model against Ackerman & Mar- +ley (2001) by comparing the condensate mixing ratios +of Jovian ammonia clouds (qc) with different values of +fsed, as shown in Fig. 2. The two sets of results are +consistent and the small differences in qc translate into +∼ 10−2 ppm in transit depth, which is completely neg- +ligible compared to typical observational noise. +We further validated our model against the results +from Gao et al. (2018) by comparing the KCl cloud +molecular mixing ratio and particle-size profile (see Fig. +B1 in Appendix B). Gao et al. (2018) used CARMA +to simulate cloud microphysics in exoplanets and brown +dwarfs with Teff = 400 K and log g = 3.25, 4.25 and +5.25 (in cgs units), corresponding to planetary masses of +0.72 MJ, 8.47 MJ, and 44.54 MJ. In the best fit between +CARMA to A-M model, the fsed = 0.125, 0.093 and +0.025 for the cases with K = 104, 103 and 102 m2 s−1, +respectively. +The mixing length is derived from con- +stant eddy diffusion, as described in equation (4). The + +8 +Ma et al. +Figure 2. +Validation of the YunMa cloud microphysics +model against the Jovian ammonia clouds in Ackerman & +Marley (2001). qc corresponding to different sedimentation +efficiencies (fsed) are shown. Solid lines: results from YunMa. +Dashed lines: results from Fig. +1 in Ackerman & Marley +(2001). Dotted line: T-p profile. +results agree with each other within 8 %, i.e. ∼ 10−3 +ppm difference in the transit depth. +We compared simulations from YunMa with the re- +sults from Charnay et al. (2021 b, see Fig. B2 in Ap- +pendix B), which include the horizontal effects generated +by global circulation. Being YunMa a 1D model, we can- +not reproduce the exact same results, but we can have an +approximated comparison with their simulations at the +substellar point, where the horizontal winds are weak. +We assume 100 × solar composition, with fsed = 3 and +K = 106 cm2s−1. Both models predict clouds forming +in the region between 3 × 10−2 and 1 × 10−2 bar and a +cloud molecular mixing ratio ∼ 10−4. +We have validated the BH-Mie module in YunMa +against PyMieScatt, an open-source model simulating +atmospheric particle scattering properties (Sumlin et al. +2018), as shown in Fig. B3. Cloud particle radii were +selected in the range of 0.1-100 µm. The largest discrep- +ancy in Qext is within ±0.002, which corresponds to an +average of 0.01 ppm in the planetary transit depth of +the nominal scenario in our experiments. +4.2. Simulated transit spectra with YunMa +We present here the transmission spectra generated +with YunMa of a cloudy super-Earth. Fig. 3 shows five +examples at resolving power of 100 which corresponds +to the ground truths of some of the retrieval cases in +Table 3: pc = 2 × 10−3 bar with XN2 = 0 (blue, Case +2, 10, 11, 14 and 15) and = 0.5 (purple, Case 20–25); +pc = 1 × 10−2 bar with XN2 = 0 (green, Case 3 and +16) and = 0.5 (yellow, Case 27 and 28) and one case +of opaque cloud with pc = 1 × 10−2 bar in the H2/He +dominated atmosphere (red, Case 4 and 17). The plan- +etary and atmospheric parameters are listed in Table 1 +and 3. All the simulations contain baseline 10% H2O +abundance across the atmosphere, which is then altered +by the cloud formation. The rest of the atmosphere is +N2 and H2/He. We select fsed = 3 for all the scenarios. +The simulation results are summarised in Table 2. +In the experiment without N2 and pc set to 2 × 10−3 +bar, clouds form at high altitude, where the atmospheric +density (ρa) is low compared to the cases where pc = 5 +× 10−3 bar and = 1 × 10−2 bar. Here the sedimen- +tation velocity (vf) is small with small cloud particle +radii and number density. The cloud contribution (blue +dash-dotted line) has a mean transit depth of 2660 ppm +and σspec of 17 ppm. It is an optically thin cloud which +does not completely block the spectral features shaped +by water vapour absorption (blue dotted line). When pc += 1 × 10−2 bar, clouds form at relatively low altitudes, +where the atmospheric density (ρa) is high. Here vf is +large and the cloud particles have relatively large radii +and number density, which increase the opacity. +The +cloud contribution (red dash-dotted line) has a mean +flux depth of 2727 ppm and σspec of 0.62 ppm. Since the +clouds are optically thick, they contribute significantly +to the mean transit depth and obscure the spectral fea- +tures of water vapour (red dotted line). However, the +water vapour features are still able to show due to the +low altitude of the clouds. Still, the spectral deviation is +only 12.16 ppm, where the spectroscopic features have +a high chance of being hidden by the observational un- +certainty. pc = 5 × 10−3 bar is an intermediate case +regarding the simulated cloud altitude and opacity. The +simulation suggests that the intermediate combination +of these two cloud properties does not result in more +significant atmospheric features than in other cases. +In atmospheres with relatively high mean molecular +weight – and therefore small scale height – for the same +value of pc, the transit depth is smaller, as expected. In +the case with XN2 = 0.5 and pc = 2 × 10−2 and = 5 +× 10−2 bar, the mean value of the transit depths are +∼ 200 ppm smaller than them in a H2/He dominated +atmosphere. Here the cloud particles form at higher ρa +and therefore have larger particle size and larger number +density compared to those formed in the H2/He domi- +nated atmospheres. The spectrum has σspec of 2.58 and +0.62 ppm, which are negligible compared to the obser- +vational uncertainty. +Besides pc and XN2, we also have tested different fsed +to understand how this parameter controls the cloud mi- +crophysics. The particle radii, rc, number density and +transit spectra across all the cloud pressure levels and + +Temperature (K) +110 +120 +130 +140 +150 +160 +10-1 +Pressure (bar) +3 × 10-1 +4 × 10-1 +6 × 10-1 +100 +10-8 +10-7 +10-6 +10-5 +10-4 +Cloud molecular mixing ratio (qc) +fsed = O (YunMa) +fsed = 1 (A-M 01) +fsed = 10 (YunMa) +fsed = 0 (A-M 01) +fsed = 3 (YunMa) +fsed = 10 (A-M 01) +fsed = 1 (YunMa) +fsed = 3 (A-M 01) +TYunMa: Enabling Spectral Retrievals of Exoplanetary Clouds +9 +0.5 +1 +2 +5 +10 +Wavelength ( m) +2500 +2550 +2600 +2650 +2700 +2750 +2800 +Transit depth (ppm) +pc = 2 × 10 +3 bar all contribution +pc = 2 × 10 +3 bar gas phase H2O absorption +pc = 2 × 10 +3 bar absorption and scattering of H2O cloud +pc = 2 × 10 +3 bar with XN2 = 0.5 all contribution +pc = 2 × 10 +3 bar with XN2 = 0.5 gas phase H2O absorption +pc = 2 × 10 +3 bar with XN2 = 0.5 absorption and scattering of H2O cloud +pc = 5 × 10 +3 bar all contribution +pc = 5 × 10 +3 bar gas phase H2O absorption +pc = 5 × 10 +3 bar absorption and scattering of H2O cloud +pc = 5 × 10 +3 bar with XN2 = 0.5 all contribution +pc = 5 × 10 +3 bar with XN2 = 0.5 gas phase H2O absorption +pc = 5 × 10 +3 bar with XN2 = 0.5 absorption and scattering of H2O cloud +pc = 1 × 10 +2 bar all contribution +pc = 1 × 10 +2 bar gas phase H2O absorption +pc = 1 × 10 +2 bar absorption and scattering of H2O cloud +Figure 3. Simulated transit spectra of cloudy super-Earths using YunMa. Solid line: total transit depth with all contributions +included; dot-dashed line: water ice clouds; faint-dotted line: water vapour. Blue, green and red lines: H2/He dominated, +cloudy atmospheres with different pc (see legend). Purple and yellow lines: heavier cloudy atmospheres with 50% N2. +. +obtained with different fsed are shown in Fig. B4. Here +we note that, from the results, the cloud particle sizes in- +crease with fsed while the number densities at each layer +behave reversely. Also, which is easy to understand, the +more atmospheric layers with clouds, the larger the op- +tical depth. +4.3. Retrieval results +We show in this section how YunMa performs with dif- +ferent model assumptions and ground truth parameters +(GTPs), following the approach described in Section 3. +GTPs and priors are listed in Table 1. The posterior +median values and their standard deviations obtained +for all the simulated cases are summarised in Table 3. +Cases 1–5 test the effects of different pc to the tran- +sit spectra and retrievals. The transit spectra of Cases +2 and 4 are shown in Fig. 3 (blue and red solid lines +respectively). The larger is pc, the more opaque mean- +while, the lower altitude becomes the clouds. The sig- +nificance of spectroscopic features owns to both factors, +and generally speaking, the more significant features are, +the easier it is to retrieve the atmospheric parameters. +We choose pc = 2 × 10−3 bar as nominal case and +show the corresponding posterior distributions in Fig. +4. Cases 2, 6 and 7 test the impact on the transit spec- +Rp = 0.20+0.00 +0.00 +1.6 +1.2 +0.8 +0.4 +log(H2O) +log(H2O) = +0.99+0.17 +0.18 +0.40 +0.45 +0.50 +0.55 +0.60 +log(fsed) +log(fsed) = 0.48+0.03 +0.02 +0.192 +0.196 +0.200 +0.204 +0.208 +Rp +1.5 +3.0 +4.5 +6.0 +7.5 + (derived) +1.6 +1.2 +0.8 +0.4 +log(H2O) +0.40 +0.45 +0.50 +0.55 +0.60 +log(fsed) +1.5 +3.0 +4.5 +6.0 +7.5 + (derived) + (derived) = 3.90+0.77 +0.54 +Figure 4. +Example of cloud retrieval using YunMa inte- +grated in TauREx 3 (Case 2 in Table 3). The sedimenta- +tion efficiency (fsed), which is the main parameter control- +ling the cloud microphysics, is well recovered together with +the vapour water mixing ratio. +tra and retrievals of the sedimentation efficiency (fsed), +which controls the cloud microphysics in our model. In +all three cases, GTPs for fsed, XH2O and Rp are well + +10 +Ma et al. +Table 2. Atmospheric and cloud parameters included in YunMa simulations. The corresponding transit depths are also +reported. +pc +2 × 10−3 (bar) +5 × 10−3 (bar) +1 × 10−2 (bar) +2 × 10−3 (bar) +5 × 10−3 (bar) +XN2 +· · · +· · · +· · · +0.5 +0.5 +Parameter +Unit +All contribution, mean +ppm +2770.17 +2750 +2729.82 +2558.65 +2553.71 +All contribution, std (σspec) +ppm +15.72 +11.59 +12.16 +2.58 +0.62 +Cloud contribution, mean +ppm +2659.65 +2729.60 +2726.53 +2534.78 +2542.78 +Cloud contribution, std +ppm +17.21 +4.39 +0.62 +2.15 +0.85 +MMW (bottom of the atmosphere) +g mol−1 +3.88 +3.88 +3.88 +16.73 +16.73 +Atmospheric pressure (cloud base) +bar +2.34 × 10 −3 +6.40 × 10 −3 +1.43 × 10 −2 +2.34 × 10 −3 +6.40 × 10 −3 +Atmospheric pressure (cloud deck) +bar +8.54 × 10 −4 +2.34 × 10 −3 +4.28 × 10 −3 +8.54 × 10 −4 +2.33 × 10 −3 +Cloud MMR (cloud base) +· · · +6.35 × 10 −4 +9.51 × 10 −5 +2.00 × 10 −4 +2.92 × 10 −4 +4.35 × 10 −4 +Cloud MMR (cloud deck) +· · · +7.56 × 10 −8 +2.92 × 10 −7 +7.28 × 10 −8 +3.49 × 10 −7 +1.33 × 10 −6 +vf +m s−1 +6.54 – 7.53 +6.07 - 6.67 +4.43 – 6.41 +7.14 – 7.97 +4.37 – 7.28 +rc +µm +9.03 – 6.39 +9.79 – 8.88 +12.13 – 9.35 +13.99 – 11.09 +25.53 – 13.66 +N (cloud base) +m−3 +1.03 × 104 +3.19 × 104 +7.47 × 104 +1.28 × 104 +8.22 × 10 3 +N (cloud deck) +m−3 +1.45 × 101 +5.72 × 101 +2.24 × 101 +1.28 × 101 +7.16 × 10 1 +Note—MMR and MMW: molecular mixing ratio and mean molecular weight. The pressure at the bottom of the atmosphere is assumed to +be 10 bar. +within the retrieved parameter space, with more or less +peaked and symmetric posterior distributions. +When +fsed = 0.01 and 3, the posterior distributions of fsed are +closer to normal distribution centred at the retrieved +solution. When fsed = 10, the upper limit is not well +constrained in the posterior distribution of fsed. This +effect is not unexpected, as high fsed scenarios tend to +have negligible impact on the planetary transit depth +due to thinner cloud layers and smaller number density +(N) compares to a low fsed scenario, e.g., the fsed = +10 and 100 cases in Fig. B4. From the experiments on +the baseline water abundance (XH2O), in atmospheres +where water can condense, when the XH2O is less than 1 +× 10−3, clouds cannot form. When XH2O = 0.01 (Case +8) a thin cloud with low opacity may form, the water +vapour spectral features are well visible and the retrieval +performs well. By contrast, a higher mixing ratio of the +condensable gas increases the partial pressure and con- +tributes to the condensation process. +This is, for in- +stance, the case of XH2O = 0.5 (Case 9) where the cloud +is thick and largely blocks the spectral features, making +the retrieval of the atmospheric parameters difficult. +From the experiments on observational data qual- +ity, Case 10’s Bayesian evidence (4568.52) compared to +the ones calculated for Case 2 (3755.80) and Case 11 +(3368.29) showcases how the retrieval performance im- +proves when the observational uncertainties are small. +Case 12 performs better than Case 2, as the spectral +resolution of the transit spectrum used as input to the +retrieval is higher. In Case 13, we omitted the informa- +tion contained in the optical wavelengths, which means +that we have less information about the cloud scatter- +ing properties. The retrieval performances are degraded +compared to Case 2, which includes the optical wave- +lengths. +In Case 14–17, we retrieved the T-p profiles as free pa- +rameters for different pc. Our results show that both the +T-p profiles and cloud parameters can be constrained, +although the retrieved gas phase mixing ratio and fsed +have large standard deviations. Case 18 compared to +Case 2 showcases how the addition of the spectrally in- +active N2 injects uncertainty in the retrieval. +Differ- +ent amounts of N2 (XN2), are considered in Cases 18– +20. Despite the minimal spectral features of the cloudy +heavy atmospheres due to N2 injection, it is still able to +retrieve the atmospheric parameters in simple cases. +In Cases 21–25 we retrieve both the T-p profile and +XN2. The corresponding transit spectra are shown in +Fig. +3 (purple lines). +Case 21 shows how the atmo- +spheric parameters, with the exception of XN2, can be +retrieved in a heavy atmosphere with XN2 = 0.5. When +we include the uncertainty of XN2 (Case 22), the GTPs +for Rp, XH2O, fsed, Tsurf and Ttop (see the retrieved +T-p profile in Fig. 5) still fall into the 2σ confidence +range. fsed is significantly less constrained in Case 22 +than Case 21 due to the uncertainty of XN2. +Cases +22–24 are compared with each other in Fig. +6. +Po- +tential degeneracy could happen according to Case 23: +when we omit N2 among the priors, the retrieval tries +to compensate for the missing radiative-inactive gas by + +YunMa: Enabling Spectral Retrievals of Exoplanetary Clouds +11 +decreasing Rp and Tsurf, while increasing XH2O, and the +Bayesian evidence of Case 23 (3757.08) is close to Case +22 (3757.40). In the experiment of model comparison, if +we use the simple opaque cloud retrieval model in Tau- +REx 3 (Case 24) instead of YunMa for the not opaque +case, the cloud and other atmospheric parameters are +significantly more degenerated and less constrained than +using YunMa. Without the cloud in the prior (Case 25), +the results show a lack of constraints on the T-p profile +while comparable performance on other parameters with +Case 22. +0 +250 +500 +750 +1000 +1250 +1500 +1750 +Temperature (K) +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +Pressure (bar) +Fitted profile +Figure 5. Retrieved T-p profile of Case 22 in Table 3. The +isothermal temperature (Tc), surface temperature (Tsurf) and +the pressure where the isothermal profile starts (pc) are re- +trieved using YunMa. The shaded area indicates the stan- +dard deviation of the posterior distribution. +Cases 26–28 focus on retrieval of plain spectra (pc = +5 × 10−2 bar in a heavy atmosphere). The transit spec- +trum for Case 27 and 28 is shown in Fig. +3, yellow +line: the spectral signal is very small compared to the +observational uncertainty (10 ppm). As expected, the +retrieval performance is limited unless the observational +uncertainty is unrealistically decreased to 1 ppm (Fig. +7). + +12 +Ma et al. +Figure 6. Retrieval posteriors for Case 22 (purple), 23 (orange) and 24 (green) in Table 3. Blue crosses indicate the ground truth +parameters and the vertical dashed lines in histograms indicate the 1σ and 2σ confidence ranges of the posterior distribution. +These three retrievals use the same transit spectrum as input (thin cloud with XN2 = 0.5 in Fig. 3) but different retrieval +assumptions. Case 22 (purple): H2O and N2 are included as priors and the cloud formation is simulated by YunMa. Case 23 +(orange): same as Case 22 except that N2 is not included among the priors. Case 24 (green): same as Case 22 except that the +cloud is simulated by a simpler model from TauREx, in which the atmosphere becomes opaque below the cloud deck. The only +retrieved cloud parameter for the simpler model is the cloud deck pressure, which is not shown here for simplicity. + +YunMa cloud with XN, as free parameter in retrieval +YunMa cloud with fixed X, = O in retrieval + Flat-line cloud with Xn, as free parameter in retrieval +Juns +T = 111.55+%99 +L +20 +log(Pc) = 3.23±8:3 +log(Pc) +log(H20) = -0.82±8.44 +log(H20) +2.4 +N2 = 0.68±8:31 +0.6 +0.2 +fsed = 0.60±0:91 +seo +0.8 +2.4 +-1.6 +Tsurf +Tc +log(Pc) +log(H20) +fsed +N2YunMa: Enabling Spectral Retrievals of Exoplanetary Clouds +13 +Figure 7. YunMa retrieval experiments with thick clouds and XN2 = 0.5 (orange line in Fig. 3). Observational uncertainty = +10 ppm (blue line, Case 27 in Table 3) and = 1 ppm (brown line, Case 28 in Table 3). Red lines indicate GTPs. + +Rp = 0.20±0:00 +Tsurf = 976.57+41:46 +pc = 5 × 10-2 bar, Error = 10 ppm +Te= 187.18±15:86 +pc = 5 × 10-2 bar, Error = 1 ppm +T +180 +120 +6 +log(Pc) = 2.69+8:18 +4 +log(Pc) +log(H20) = -1.20±8:18 +0% +? +(02H)60) +1? +log(fsed)= 0.46±0.19 +log(fsed) +0.0 +1.6 +-2.4 +N2 = 0.90±8.82 +1.0 +to +0.0 +6 +10 +% +. +1 +Tsurf +Te +log(Pc) +log(H20) +log(fsed) +N214 +Ma et al. +Table 3. YunMa retrieval experimental results. The ground truth parameters (GTPs) assumed in the simulations are listed in the left columns of the table and the +retrieved posteriors on the right. The retrieval priors are listed in Table 1. In each case, we test the model sensitivity of the atmospheric parameters of fsed, pc, XH2O, +XN2, and the observational parameters of the error, wavelength coverage (λ) and the spectral resolution. We choose the nominal value of pc as 2 × 10−3 bar, where the +cloud is not either too thin to be detected or too thick to block the spectroscopic features. +Case +GTPs +Posteriors +log(fsed) +log(pc) +log(XH2O) +XN2 +Error +λ +Res. +Rp +log(fsed) +log(XH2O) +XN2 +log(pc) +Tc +Tsurf +log(bar) +(ppm) +(µm) +RN +Je +log(bar) +(K) +(K) +1 +0.48 +-3 +-1 +· · · +10 +0.4 – 14 +100 +0.20+6.98e−06 +−4.06e−06 +0.29+0.00 +−0.00 +-1.10+0.00 +−0.00 +· · · +· · · +· · · +· · · +2 +0.48 +-2.7 +-1 +· · · +10 +0.4 – 14 +100 +0.20+1.72e−03 +−1.64e−03 +0.48+0.03 +−0.02 +-0.99+0.17 +−0.18 +· · · +· · · +· · · +· · · +3 +0.48 +-2.3 +-1 +· · · +10 +0.4 - 14 +100 +0.20+3.54e−03 +−3.31e−03 +-0.85+0.93 +−1.56 +-1.34+0.47 +−1.14 +· · · +· · · +· · · +· · · +4 +0.48 +-2 +-1 +· · · +10 +0.4 – 14 +100 +0.20+5.30e−03 +−1.75e−03 +-1.11+0.97 +−0.75 +-1.57+0.79 +−0.85 +· · · +· · · +· · · +· · · +5 +0.48 +-1 +-1 +· · · +10 +0.4 – 14 +100 +0.20+1.63e−03 +−2.39e−03 +-0.76+0.97 +−0.81 +-1.22+0.31 +−0.96 +· · · +· · · +· · · +· · · +6 +-2 +-2.7 +-1 +· · · +10 +0.4 – 14 +100 +0.20+2.80e−03 +−2.10e−03 +-2.02+0.09 +−0.11 +-0.91+0.28 +−0.22 +· · · +· · · +· · · +· · · +7 +1 +-2.7 +-1 +· · · +10 +0.4 – 14 +100 +0.20+2.12e−04 +−7.09e−04 +1.29+0.43 +−0.28 +-0.92+0.02 +−0.07 +· · · +· · · +· · · +· · · +8 +0.48 +-2.7 +-2 +· · · +10 +0.4 – 14 +100 +0.20+2.50e−05 +−2.63e−05 +0.44+0.00 +−0.01 +-2.02+0.01 +−0.01 +· · · +· · · +· · · +· · · +9 +0.48 +-2.7 +-0.3 +· · · +10 +0.4 – 14 +100 +0.20+4.43e−03 +−3.44e−03 +-1.22+0.95 +−1.06 +-0.75+0.54 +−0.37 +· · · +· · · +· · · +· · · +10 +0.48 +-2.7 +-1 +· · · +1 +0.4 – 14 +100 +0.20+1.62e−04 +−1.63e−04 +0.48+0.00 +−0.00 +-1.00+0.02 +−0.02 +· · · +· · · +· · · +· · · +11 +0.48 +-2.7 +-1 +· · · +30 +0.4 – 14 +100 +0.21+8.61e−04 +−3.76e−03 +0.77+0.85 +−0.28 +-0.50+0.11 +−0.37 +· · · +· · · +· · · +· · · +12 +0.48 +-2.7 +-1 +· · · +10 +0.4 – 14 +10 +0.21+7.33e−04 +−2.85e−03 +0.96+0.70 +−0.44 +-0.47+0.10 +−0.29 +· · · +· · · +· · · +· · · +13 +0.48 +-2.7 +-1 +· · · +10 +1 – 14 +100 +0.20+2.47e−03 +−3.95e−04 +0.51+0.10 +−0.01 +-0.76+0.25 +−0.04 +· · · +· · · +· · · +· · · +14 +0.48 +-2.7 +-1 +· · · +10 +0.4 - 14 +100 +0.20+7.53e−04 +−9.24e−04 +0.42+1.04 +−1.15 +-0.62+0.07 +−0.08 +· · · +-2.39+0.35 +−0.33 +124.55+56.56 +−82.38 +1264.49+151.39 +−121.10 +15 +0.48 +-2.7 +-1 +· · · +1 +0.4 – 14 +100 +0.20+1.39e−05 +−1.44e−05 +0.20+0.01 +−0.02 +-0.71+0.00 +−0.00 +· · · +-2.54+0.02 +−0.01 +194.23+1.25 +−0.88 +1043.45+4.14 +−1.91 +16 +0.48 +-2.3 +-1 +· · · +10 +0.4 – 14 +100 +0.21+7.71e−04 +−1.02e−03 +0.25+1.16 +−1.24 +-0.09+0.06 +−0.10 +· · · +-1.21+0.52 +−0.50 +175.25+31.87 +−86.34 +1237.49+508.48 +−528.66 +17 +0.48 +-2 +-1 +· · · +10 +0.4 – 14 +100 +0.20+6.21e−03 +−4.08e−03 +-1.84+2.85 +−0.97 +-2.88+2.82 +−3.98 +· · · +-0.95+0.55 +−1.03 +150.58+44.47 +−122.39 +1000.70+706.26 +−352.40 +18 +0.48 +-2.7 +-1 +10−12 +10 +0.4 – 14 +100 +0.21+2.88e−04 +−1.72e−04 +0.62+0.03 +−0.02 +-0.86+0.11 +−0.12 +0.11+0.03 +−0.03 +· · · +· · · +· · · +19 +0.48 +-2.7 +-1 +0.1 +10 +0.4 – 14 +100 +0.20+4.02e−04 +−6.63e−04 +0.67+0.59 +−0.12 +-0.96+0.23 +−0.24 +0.37+0.09 +−0.07 +· · · +· · · +· · · +20 +0.48 +-2.7 +-1 +0.5 +10 +0.4 – 14 +100 +0.20+1.82e−04 +−2.84e−04 +0.39+0.12 +−0.10 +-1.03+0.33 +−0.46 +0.74+0.13 +−0.13 +· · · +· · · +· · · +21 +0.48 +-2.7 +-1 +0.5 +10 +0.4 – 14 +100 +0.20+3.96e−04 +−4.70e−04 +0.56+0.96 +−1.19 +-0.69+0.23 +−0.33 +fixed to 0.5 +-1.69+0.32 +−0.36 +103.11+70.32 +−69.47 +1443.65+373.77 +−310.00 +22 +0.48 +-2.7 +-1 +0.5 +10 +0.4 – 14 +100 +0.20+3.87e−04 +−3.48e−04 +0.60+0.91 +−0.98 +-0.82+0.34 +−0.44 +0.68+0.15 +−0.21 +-1.77+0.32 +−0.36 +111.53+67.82 +−73.41 +1553.21+276.23 +−399.58 +23 +0.48 +-2.7 +-1 +0.5 +10 +0.4 – 14 +100 +0.20+7.03e−04 +−6.06e−04 +0.42+1.00 +−1.02 +-0.11+0.07 +−0.09 +fixed to 0 +-1.62+0.42 +−0.42 +130.80+52.94 +−86.47 +1071.37+399.26 +−381.53 +24 +0.48 +-2.7 +-1 +0.5 +10 +0.4 – 14 +100 +0.20+6.13e−04 +−9.64e−04 +· · · +-1.90+0.81 +−1.36 +0.78+0.12 +−0.18 +-2.27+1.22 +−1.13 +327.32+120.86 +−245.79 +1370.26+430.51 +−535.90 +25 +0.48 +-2.7 +-1 +0.5 +10 +0.4 – 14 +100 +0.20+3.80e−04 +−3.47e−04 +· · · +-0.95+0.32 +−0.47 +0.74+0.14 +−0.16 +-1.72+0.38 +−0.35 +48.60+51.83 +−32.93 +1577.37+285.82 +−401.02 +26 +0.48 +-2.3 +-1 +0.1 +10 +0.4 – 14 +100 +0.21+4.38e−04 +−8.41e−04 +-2.27+0.67 +−0.46 +-3.80+2.23 +−3.28 +0.78+0.13 +−0.18 +-1.31+0.82 +−1.06 +86.15+65.71 +−61.63 +1136.49+542.96 +−444.41 +27 +0.48 +-2.3 +-1 +0.5 +10 +0.4 – 14 +100 +0.20+4.50e−04 +−6.77e−04 +-2.20+0.60 +−0.48 +-4.33+2.36 +−3.92 +0.72+0.17 +−0.21 +-1.29+0.80 +−1.11 +62.56+65.77 +−45.02 +1052.87+540.75 +−342.32 +28 +0.48 +-2.3 +-1 +0.5 +1 +0.4 – 14 +100 +0.20+4.66e−05 +−4.72e−05 +0.46+0.08 +−0.17 +-1.20+0.08 +−0.10 +0.90+0.02 +−0.01 +-2.31+0.08 +−0.11 +187.18+15.16 +−13.86 +976.55+33.41 +−41.97 + +YunMa: Enabling Spectral Retrievals of Exoplanetary Clouds +15 +5. DISCUSSION +5.1. Transit spectroscopy using YunMa +To understand the performances of YunMa in detail, +we performed retrieval experiments for over a hundred +cases, with different chemistry models, atmospheric and +cloud scenarios for super-Earths/sub-Neptunes, hot- +Jupiters and brown dwarfs. A variety of cloud species +were modelled and analysed. This paper presents a se- +lection of representative examples. We have validated +the A-M cloud size distribution in YunMa against pre- +vious literature simulating NH3 clouds in Jupiter’s at- +mosphere, KCl clouds on artificial large exoplanets and +brown dwarfs, and H2O clouds on K2-18 b. We have also +tested a number of numerical settings, including fitting +methods, tolerances and retrieval samplings. +In YunMa, the mixing ratios of the condensable gas +and the condensate are strongly correlated, therefore +when cloud forms, the condensable species in the gas +phase decreases. Also, a balance is imposed between the +upward turbulent mixing and the downward sedimenta- +tion velocity from the A-M approach (equation 2). In +the current YunMa, at each pressure level, the particle +number density represents the total number density of +particles with different radii. The Earth measurements +shown in Fig. 4 of Ackerman & Marley (2001) suggest +a bimodal distribution of the particle sizes at the same +pressure level. YunMa is able to use radius bins with +their respective number densities to represent more pre- +cisely the cloud distribution in the spectral simulation. +For the retrieval calculations, however, we had to sim- +plify this information to reduce the computing time. +The cloud opacity is determined by the cloud particle +size and number density; different particle sizes have ab- +sorption and scattering peaks at different wavelengths. +Optically thick clouds cause transit depths with neg- +ligible or no modulations as a function of wavelength: +the atmosphere below the cloud deck is, in fact, unde- +tectable while the atmosphere sounded above the cloud +deck is more rarefied. Retrieving information about the +atmospheric composition and structure is very difficult +in the most extreme cases. +For an atmosphere with +optically thin clouds, the absorption features due to +radiative-active gases (water vapour here) are detectable +but less prominent compared with a clear atmosphere. +The abundances of these gases can be retrieved largely +from their absorption features. If these are condensable +species, their abundances further constrain the cloud +microphysics. Both the wavelength-dependent features +and the overall transit depth help the retrieval perfor- +mance. +fsed is the ratio between the sedimentation velocity +and the turbulent convective velocity. +From the def- +inition, a higher fsed means a shorter sedimentation +timescale, as we fixed K to a constant value. A higher +sedimentation efficiency leads to larger offsets to the +downward draft, constraining the upward supplement of +water vapour and cloud formation. On the contrary, for +small fsed, the sedimentation timescale is much longer +compared to the diffusion timescale, so the condensation +continues at lower pressure to balance the downward +sedimentation and upward turbulent mixing; therefore, +the cloud region expands. fsed is sensitive to the cloud +particles’ nucleation rate (Gao et al. 2018). +It has a +close relationship with the condensate particle size and +can be expressed by the particle radius in the lognormal +distribution power-law approximation: +fsed = +� ∞ +0 r3+α dn +dr dr +rαw +� ∞ +0 r3 dn +dr dr , +(11) +which indicates that small fsed encourages small cloud +particle formation. +The upward transport is stronger +than the sedimentation when the cloud particles are +small and vice-versa, which is in line with the experi- +mental results in Figure B4 (a, b). The spectrum is sen- +sitive to fsed when this parameter has values between +10−1 and 10−3, as shown in Fig. B4 (c), indicating the +detectability of fsed in this interval. +In our experiments, the particle radii are typically 1– +10 µm, so they do not block the Rayleigh scattering +slope caused by H2 and H2O at the optical wavelengths. +For smaller radii (e.g., cyan line in Fig. B3), they are +expected to contribute more to the optical spectrum, +although theoretically it would be hard to form particles +at very small sizes according to the nucleation theory in +cloud formation (Gao et al. 2018). For any particle radii +even smaller presented in this paper, it is just for model +test purposes in extreme cases, and we make no efforts +to show their detailed analysis here. +Radiative-inactive gases such as N2, if present, can +change the atmospheric scale height. +The increase of +scale height decreases the transit depth and dampens its +spectroscopic features, as illustrated in Fig. 3. As men- +tioned in previous sections, the presence of radiative- +inactive gases cannot be detected directly through spec- +troscopic signatures. Opaque clouds may behave sim- +ilarly to inactive gases in mitigating spectroscopic fea- +tures, leading to potential degeneracy in retrieval exper- +iments. Illustrated in Fig. 6, the comparison between +Case 22 and 23 in Table 3 suggests the potential degen- +eracy between Rp, the baseline condensable gas abun- +dance, T-p profile, and the N2 abundance. In Case 23, +we force XN2 = 0 in the retrieval to monitor how YunMa + +16 +Ma et al. +compensates for the missing gases in the atmosphere. +Adjustment of Rp translates the transit depth without +impact on spectroscopic features. The baseline condens- +able gas abundance and T-p profile are correlated to the +formation of clouds. The results indicate that when N2 +is absent in prior, the mixing ratio of water vapour – the +radiative-active gas – is significantly increased to com- +pensate for the missing molecular weight, while more +clouds are formed to further reduce the spectroscopic +features. The decrease of Tsurf helps in the same way. +While the potential degeneracy exists from analysis and +the results suggest the model behaviour in the case of +missing radiactive-inactive gases, the model chose from +statistics the scenario closer to the ground truth in Case +22, showing the model’s potential in retrieving clouds in +heavy atmospheres when it is not opaque. +As mentioned before, the cloud formation and T-p +profile are correlated in YunMa. +Therefore, the pres- +ence of optically thin clouds can help to constrain the +T-p profile in retrievals: for instance, pc, Tc and Tsurf +are well retrieved in Case 22. On the contrary, the T-p +profile is not well constrained if clouds are completely +absent (Case 25) or the microphysics part is removed +from the retrieval (Case 24). +5.2. Cloud formation with the next-generation +facilities’ data +With the high-quality transit spectra offered by the +next-generation facilities, the uncertainties, wavelength +coverage and spectral resolution will be significantly im- +proved compared to most current data, and a simple +opaque cloud model is insufficient for the transit study of +next-generation data according to our results; the mod- +elling of cloud radiative transfer and microphysics such +as YunMa is needed. In our experiments, we chose a +nominal 10 ppm as the observational uncertainty and +found that clouds can be well-characterised in most ex- +periments. With the next-generation data and YunMa, +there are still limits in retrieving the featureless spec- +tra. A smaller uncertainly may help in retrieving cloud +parameters in the most difficult cases: +we therefore +adopted an unrealistic 1 ppm to test their detectability +in an ideal case (Case 10, 15 and 7). With 1 ppm, the +atmospheres were constrained well, although the spec- +tra were featureless. Our experiments also suggest that +broad wavelength coverage is paramount to characteris- +ing clouds well: here, optical wavelengths play a critical +role when combined with infrared spectral coverage. +5.3. Numerical instabilities in cloud microphysics +simulations +We have selected the explicit Runge-Kutta method +of order 8 (DOP853, Hairer et al. 1988) to solve equa- +tion (2) as it delivered the most stable numerical per- +formance for our experiments. However, numerical in- +stabilities may occur when a large relative (rtol) and +absolute (atol) tolerances are chosen. +Sometimes the +ODE solver cannot converge for qt and indicates “no +clouds” as a solution due to the numerical instability. +Caution is also needed in estimating the cloud mixing +ratio (qc), as qt might be two orders of magnitude larger +than qc and numerical errors could be injected when qs +is subtracted from qt. When solving the ODE with too +large tolerance, qt might converge to a certain value, but +qc would be estimated as negligible according to equa- +tion (2 and 1). In other words, even though when the +ODE solution for the gas phase seems numerically sta- +ble, it might not be precise and accurate enough. +In +those cases, the retrieval performances are affected, as +shown in Fig. B5, where multiple islands of solutions in +the posteriors are visible; these are caused by numerical +instabilities, and the issue is more obvious for larger tol- +erance values. After many tests, we have decided to use +“DOP853” in solving the ODE with rtol = 1 × 10−13 +and atol = 10−16, which guarantee numerically stable +results. +6. CONCLUSIONS +YunMa is a state-of-art cloud model optimised for +the interpretation of the next generation of exoplane- +tary atmospheric data, as provided by e.g., JWST, Ro- +man, Twinkle and Ariel. These facilities will provide an +unprecedented amount of high-quality data, where the +cloud formation process and cloud scattering properties +can no longer be ignored. +YunMa cloud microphysics is based on the model pub- +lished by Ackerman & Marley (2001), while the scat- +tering properties of clouds are calculated through the +open-source B-H Mie code. When coupled to the Tau- +REx framework (Al-Refaie et al. 2021), YunMa be- +comes a very versatile model which can simulate tran- +sit and eclipse spectra for a variety of cloudy exoplan- +ets with different masses, atmospheric compositions and +temperatures. Most importantly, YunMa+TauREx can +be used as a spectral retrieval framework optimised for +cloudy atmospheres. Here, the sedimentation efficiency, +fsed, is a critical parameter to control the cloud micro- +physics in the retrieval. +We have validated YunMa against previous work +which also adopted the A-M approach. We have vali- +dated the radiative transfer calculations in YunMa, in- +cluding cloud scattering, against PyMieScatt. Finally, +we have validated YunMa against other cloud micro- +physics models published in the literature. + +YunMa: Enabling Spectral Retrievals of Exoplanetary Clouds +17 +We have run over one hundred retrieval experiments +with YunMa, with different cloud compositions (e.g., +KCl, MgSiO3, Fe clouds). In this paper, we have pre- +sented and discussed 28 cases of water clouds in the +atmosphere of a temperate super-Earth, K2-18b-like. +Through these experiments, we have learnt that YunMa +is capable of retrieving accurately cloud formation pa- +rameters, as well as atmospheric trace gases’ mixing ra- +tios and thermal profiles when clouds are not so opaque +to mask all the atmospheric features at most wave- +lengths. +The optical wavelengths contain important +scattering information of the cloud particles and are +therefore critical for the correct extraction of the cloud +formation parameters. +While the current version of YunMa can retrieve only +one size of cloud particles at a time, the future version +of YunMa will include the possibility of retrieving a dis- +tribution of the cloud particle sizes. 2-D cloud models, +which include horizontal convection, will soon be within +reach of computing speed and might be considered in +future versions of YunMa. +The +work +presented +in +this +paper +was +partially +supported +by +UKSA, +grant +ST/X002616/1 +and +ST/W00254X/1, and ExoMolHD ERC grant 883830. +QC is supported by the ESA Research Fellowship Pro- +gram. +The authors acknowledge the use of the UCL +Kathleen High-Performance Computing Facility (Kath- +leen@UCL) and associated support services in complet- +ing this work. This work utilised the Cambridge Ser- +vice for Data-Driven Discovery (CSD3), part of which +is operated by the University of Cambridge Research +Computing on behalf of the STFC DiRAC HPC Facil- +ity (www.dirac.ac.uk). The DiRAC component of CSD3 +was funded by BEIS capital funding via STFC capi- +tal grants ST/P002307/1 and ST/R002452/1 and STFC +operations grant ST/R00689X/1. DiRAC is part of the +National e-Infrastructure. +We thank Dr Andrew S. Ackerman for the construc- +tive communication on the A-M model and for shar- +ing the original code, Dr Kai-Hou Yip and Sam Wright +for suggestions on the numerical methods and Dr Yui +Kawashima for the cloud and haze model discussion. +APPENDIX +A. SUPPLEMENTARY EQUATIONS +In Section 2.1.1, the SVP can be estimated with the +Clausius-Clapeyron equation: +es = e0 exp +� +ℓ +RSV +� 1 +T0 +− 1 +T +�� +, +(A1) +where ℓ is the latent heat of evaporation, RSV is the spe- +cific gas constant for the vapour, T is the atmospheric +temperature, and T0 is the temperature at vapour pres- +sure e0. We use the results from laboratory measure- + +18 +Ma et al. +ments of these parameters for the different chemical +species when available. +β is the Cunningham slip factor: +β = 1 + 1.26NKn, +(A2) +The Knudsen number (NKn) is the ratio between the +molecular mean free path and the droplet radius. +YunMa adopts two ways to estimate η. One is that +Lavvas et al. (2008) suggested to use: +η = 1 +3ρaV λa, +(A3) +where V is the thermal velocity of gaseous components, +and λa is the mean free path. Another uses the definition +by Rosner (2012), which is also adopted by Ackerman +& Marley (2001): +η = 5 +16 +√πmkBT +πd2 +(kBT/ϵ)0.16 +1.22 +, +(A4) +where d is the diameter of a gas particle and ϵ is the +atmospheric Lennard-Jones potential well depth. When +using the A-M approach, vf and the particle size are +positively correlated using η either from Rosner (2012) +or Lavvas et al. (2008). +In Section 2.1.2, assuming a lognormal cloud particle +size distribution, the geometric mean (rg) is defined as: +rg = e +� ∞ +0 +lnr dn +dr dr +� ∞ +0 +dn +dr dr . +(A5) +The power law approximation allows representation of +fsed using the particle size distribution. +fsed ≈ +� ∞ +0 r3+α dn +dr dr +rαw +� ∞ +0 r3 dn +dr dr , +(A6) +where n is the accumulated number density as defined +in Section 2 and σg is the geometric standard deviation +of the lognormal particle radius distribution. Through +an integration of the lognormal distribution, Ackerman +& Marley (2001) derived that: +rg = rw f +1 +α +sed exp +� +−α + 6 +2 +ln2σg +� +, +(A7) +where the σg is the geometric standard deviation of the +particle radii. +The effective mean radius (reff) is the area-weighted +average radius defined by Hansen & Travis (1974) to +approximately represent the scattering properties of the +whole size distribution by a single parameter when the +particle radius is larger than the radiation wavelength. +To derive reff, we first recall that in lognormal distribu- +tion, the t-th raw moment is given by: +mt = N0rg exp +�(tσ)2 +2 +� +. +(A8) +Therefore reff, which is the area-weighted average ra- +dius, can be estimated through: +reff = +� ∞ +0 rπr2 dn +dr dr +� ∞ +0 πr2 dn +dr dr = rw f +1 +α +sed exp +� +−α + 1 +2 +ln2σg +� +. +(A9) +Similarly, the total number density for the particles is +estimated by using the volume-weighted mean: +N = 3ερaqc +4πρpr3g +exp +� +−9 +2 ln2σg +� +. +(A10) +B. SUPPLEMENTARY FIGURES +(a) +(b) +Figure B1. Validation of YunMa (solid lines) against the A- +M model in Gao et al. (2018) (dashed lines). Top: condensate +mixing ratios. Bottom: cloud particle effective mean radii. + +10-2. +Pressure (bar) +10-1 +101. +102 +10-1 +100 +Cloud particle effective radius (μm) +K= 102 m?s-1, fsed = 0.125 (YunMa) +K= 102 m2s-1, fsed =0.125 (Ga0 et al. 2018) +K= 103 m?s-1, fsed = 0.093 (YunMa) +K= 103 m?s-1, fsed = 0.093 (Ga0 et al. 2018) +K= 102 m?s-1, fsed =0.125 (YunMa) +K= 102 m²s-1, fsed = 0.125 (Ga0 et al. 2018)10-2 +Pressure (bar) +10 +101. +102 +10-9 +10-8 +10-7 +Cloud molecular mixing ratioYunMa: Enabling Spectral Retrievals of Exoplanetary Clouds +19 +Figure B2. Comparison of YunMa’s results against simu- +lations by Charnay et al. (2021 b) for the 100 × solar metal- +licity scenario (see Fig. 6 (a) of the substellar region). 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Bottom +panels: residuals obtained by subtracting the extinction coefficients as estimated by the two codes, YunMa and PyMieScatt. +The maximum discrepancy, corresponding to the largest particle radius simulated here (r = 1 × 102), is negligible, i.e. ±0.002. +Cho, J. Y.-K., Skinner, J. W., & Thrastarson, H. T. 2021, +The Astrophysical Journal Letters, 913, L32, +doi: 10.3847/2041-8213/abfd37 +Chubb, K. L., Rocchetto, M., Yurchenko, S. N., et al. 2021, +A&A, 646, A21, doi: 10.1051/0004-6361/202038350 + +101. +10-2, +coefficient ( +10-8. +xtinction +10-11 +r = 1.000e+02 μm (PyMieScatt) +r = 1.425e-03 μm (PyMieScatt) +r = 1.000e+02 μm (YunMa) +r = 1.425e-03 μm (YunMa) +r = 2.424e+00 μm (PyMieScatt) +r = 3.455e-05 μm (PyMieScatt) +r = 2.424e+00 μm (YunMa) +r = 3.455e-05 μm (YunMa) +10-14 +r = 5.878e-02 μm (PyMieScatt) +r = 8.377e-07 μm (PyMieScatt) +r = 5.878e-02 μm (YunMa) +r = 8.377e-07 μm (YunMa) +100 +1010.002 +0.000 +-0.002 +r=1.000e+02μm +1e-7 +0.0 +-0.5 +1.0 +r = 2.424e+00 μm +1e-5 +0.0 +Residual +-0.5 +-1.0- +r = 5.878e-02 μm +1e-8 +0 +-1 +r = 1.425e-03 μm +1e-13 +-1 +-2 +r = 3.455e-05 μm +1e-18 +0 +-2 +r = 8.377e-07 μm +100 +101 +Wavelength (μm)YunMa: Enabling Spectral Retrievals of Exoplanetary Clouds +21 +(a) +(b) +(c) +Figure B4. +Test results for reff (top left), N (top right) and the transit spectrum (bottom) with different sedimentation +efficiency fsed. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 1 Ahmed Faris Al-Refaie ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='1 Quentin Changeat ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 1 Billy Edwards ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 5,' 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+page_content=' Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 181-8588,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Japan 3European Space Agency (ESA) ESA Office,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Space Telescope Science Institute (STScI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 3700 San Martin Drive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Baltimore MD 21218,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' United States of America 4SRON,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Netherlands Institute for Space Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Niels Bohrweg 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' NL-2333 CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Leiden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The Netherlands 5AIM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' CEA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Universit´e Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Universit´e de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' F-91191 Gif-sur-Yvette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' France ABSTRACT In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' we present YunMa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' a model which enables the study of cloud microphysics and radiative properties in exoplanetary atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' YunMa simulates the vertical distribution and sizes of cloud particles and their corresponding scattering signature in transit spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We validated YunMa against results from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' When coupled to the TauREx 3 platform, an open Bayesian framework for spectral retrievals, YunMa enables the retrieval of the cloud properties and parameters from transit spectra of exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The sedimentation efficiency (fsed), which controls the cloud microphysics, is set as a free parameter in retrievals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We assess the retrieval performances of YunMa through 28 instances of a cloudy super- Earth’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' This work also highlights the need for cloud radiative transfer and microphysics modelling to retrieve next-generation data of exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Keywords: Exoplanets(498) — Exoplanet atmospheres(487) — Transmission spectroscopy(2133) — Atmospheric clouds(2180) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' INTRODUCTION Thousands of exoplanets have been detected since the late 20th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' During the past decade, transit spec- troscopy has become one of the most powerful tech- niques for studying exoplanets’ atmospheres in-depth (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', reviews by Tinetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Burrows 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Mad- husudhan 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Data recorded from space-borne in- struments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Hubble, Spitzer and James Webb Space Telescopes) or from the ground have revealed important information about exoplanet atmospheric chemistry and dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Tsiaras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Wel- banks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Changeat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Edwards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' JWST Transiting Exoplanet Community Early Release Science Team 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Venot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Roudier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021) and may provide insight into planetary in- Corresponding author: Sushuang Ma sushuang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='20@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='uk terior composition and formation (Madhusudhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Charnay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' A number of spectral retrieval models have been de- veloped by different teams to interpret the atmospheric data and quantify their information content;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' these in- clude e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Madhusudhan & Seager (2009), Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2012), TauREx 3 (Al-Refaie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021), NEMESIS (Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2008), CHIMERA (Line et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2013), AR- CiS (Min et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Ormel & Min 2019), PICASO (Robbins-Blanch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Batalha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2019), BART (Harrington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2022), petitRADTRANS (Molli`ere, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2020), HELIOS (Kitzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2020), PO- SEIDON (MacDonald & Madhusudhan 2017), HyDRA (Gandhi & Madhusudhan 2018), SCARLET (Benneke 2015), PLATON II (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2020), and Pyrat-Bay (Cubillos & Blecic 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Up to date, most of the re- trieval studies of exoplanetary atmospheres are highly parameterised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' This approach has been very sensible arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='13708v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='EP] 31 Jan 2023 ID2 Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' given the relatively poor information content of cur- rent atmospheric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' However, a number of papers in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Caldas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Changeat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Changeat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2022) have cautioned against this approach when applied to data recorded with next- generation facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Clouds are omnipresent in planetary, exoplanetary and brown dwarf atmospheres (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', review by Helling 2022) and have often been detected in exoplanet atmospheric data (Kreidberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Stevenson 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Tsiaras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Their presence imposes additional complexity and uncertainties in the interpretation of exoplanet atmospheric spectra (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Changeat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Tsiaras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Mai & Line 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Models simulating the formation and radiative prop- erties of clouds and hazes have been published in the lit- erature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Exo-REM (Baudino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Charnay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2018), Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2020), Windsor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2022) and Kawashima & Ikoma (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Due to the – currently limited – observational con- straints and computational resources available to simu- late the complexity of clouds, retrieval studies of cloudy atmospheres are still in their infancy (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Fortney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' For instance, many studies have adopted wavelength-independent opaque clouds, where all the ra- diation beneath the cloud top is blocked from reaching the telescope, and retrieve the vertical location of clouds (Boucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Brogi & Line 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Wakeford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2018) used a grey, uniform cloud in the ATMO Retrieval Code (ARC) (Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Drummond, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Tremblin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Other models constrain from radiative transfer the uniform cloud par- ticle sizes without being estimated through cloud mi- crophysics models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' For instance, Benneke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2019 a) have initially estimated the particle sizes in the at- mosphere of GJ 3470 b using Mie-scattering theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Extended from this highly parametric approach, cloud scattering parameters and inhomogeneous coverage were also retrieved: NEMESIS was used by Barstow (2020) and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2022) to retrieve the cloud’s opacity, scattering index, top and base pressures, particle sizes and shape factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Pinhas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2019) run POSEIDON to constrain the cloud’s top pressure and coverage frac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2022) adopted PICASO to extract the cloud’s base pressure, optical thickness, single scatter- ing albedo, scattering asymmetry and coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Lueber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2022) extended the use of Helios-r2 to retrieve non-grey clouds, with extinction efficiencies estimated from Mie theory calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The model Aurora (Wel- banks & Madhusudhan 2021) presents inhomogeneities in cloud and haze distributions by separating the atmo- sphere horizontally into four distinct areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The data provided by the next-generation telescopes will be greatly superior in quality and quantity, allow- ing us to obtain more stringent constraints to our under- standing of clouds in exoplanetary atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Transit spectra of exoplanets recorded from space by the James Webb Space Telescope (JWST, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='6–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='3 µm, Bean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Greene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Gardner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2006), Ariel (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='8 µm, Tinetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Tinetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021) and Twinkle (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 µm, Edwards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2019) at rela- tively high spectral resolution and/or broad wavelength coverage will open the possibility of integrating self- consistent, cloud microphysics approaches into atmo- spheric retrieval codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' A good example of such models is ARCiS (ARtful modelling Code for exoplanet Science, Min et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Ormel & Min 2019), which simulates cloud formation from diffusion processes and parametric coagulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' ARCiS also generates cloudy transit spec- tra from Mie theory (Fleck & Canfield 1984) and Dis- tribution of Hollow Spheres (DHS, Min, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Molli`ere, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2019), and can be used to retrieve the cloud diffusivity and nuclei injection from transit spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In this work, we present a new optimised model to study cloud microphysical processes directly integrated into a spectral retrieval framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We consider clouds as a thermochemical product, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' the aggregation of condensates in the atmosphere, while hazes form pho- tochemically (Kawashima & Ikoma 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The cloud distribution depends on the atmospheric conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Be- ing generated thermochemically, clouds form and diffuse depending on the atmospheric thermal structure and, in return, contribute to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' They also depend on the mix- ing profiles of the condensable gases in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Clouds act as absorbers and/or scatterers and therefore may dampen the atomic and molecular spectroscopic features and change the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Based on studies of the Earth and Solar System’s plan- etary atmospheres, Lewis (1969) published a 1-D cloud model optimised to describe tropospheric clouds in gi- ant planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' This model assumes that the fall speeds of all condensates are equivalent to the updraft veloc- ities, and only vapour is transported upward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Lunine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (1989) included a correlation between cloud par- ticle sizes, downward sedimentation and upward turbu- lent mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Based on previous models by Lewis (1969), Carlson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (1988), Lunine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (1989) and Marley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (1999), Ackerman & Marley (2001) proposed a new method to estimate the mixing ratio and vertical size distribution of cloud particles (A-M model hereafter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In the A-M model, the sedimentation timescale is esti- YunMa: Enabling Spectral Retrievals of Exoplanetary Clouds 3 mated through cloud microphysics, taking into account the atmospheric gas kinetics and dynamical viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The model assumes an equilibrium between upward tur- bulent mixing and sedimentation, where the turbulent mixing is derived from the eddy diffusion in the atmo- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The key assumptions of the A-M model are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Clouds are distributed uniformly in the horizontal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Condensable particles rain out at (su- per)saturation while maintaining a balance of the upward and downward drafts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' It does not consider the cloud cover variations caused by precipitation or the microphysics be- tween different types of clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The A-M model was originally proposed for giant exo- planets and brown dwarfs and was tested on Jupiter’s ammonia clouds, demonstrating that this approach is applicable to a broad range of temperatures and plane- tary types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Another popular 1-D cloud microphysics model is the Community Aerosol and Radiation Model for Atmo- spheres (CARMA), initially developed for the Earth’s stratospheric sulfate aerosols (Turco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Toon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' CARMA is a time-dependent cloud mi- crophysics model which solves the discretised continuity equations for aerosol particles starting from nucleation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2018) extended the use of CARMA to sim- ulate clouds on giant exoplanets and brown dwarfs by including additional condensates predicted to form in hot atmospheres and compared the results with the A- M model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The A-M model, while able to provide the cloud particle sizes and number density distributions, is of intermediate numerical complexity and, therefore, potentially adaptable to be included in retrieval codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In addition to the original implementation by Ackerman & Marley (2001), Virga (Rooney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2022) simulates the cloud’s particle size distribution from the A-M ap- proach and estimates separately the sedimentation effi- ciency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' PICASO (Robbins-Blanch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Batalha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2019) adopts Virga to simulate cloudy exoplan- etary atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2022) couples MIT GCM and Virga to include clouds in 3-D models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Xuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2022) present high-resolution spectroscopic sim- ulations with petitTRANS (Molli`ere, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' There, clouds are modelled from the A-M theory and spectra are simulated with Mie theory and DHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' None of these models, however, are currently available for re- trieval simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' To simulate inhomogeneities for cloud formation in the horizontal direction, we would need to consider global circulation atmospheric effects, such as those modelled in Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' An example of 3-D at- mospheric model with clouds is Aura-3D (Welbanks & Madhusudhan 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Nixon & Madhusudhan 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The retrieval part for Aura-3D is highly parametrised, both for the atmospheric and cloud parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Helling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2022, 2019) have simulated global cloud distributions by generating inputs to their kinetic cloud model from pre-calculated 3D Global Circulation Models (GCMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Unfortunately, these complex models require excessive computing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In addition, the data expected to be observed in the near future are unlikely to constrain the large number of parameters needed in a 3-D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Therefore, while theoretical studies with 3-D models are very important to progress in our understanding of clouds in exoplanetary atmospheres and as benchmarks, they are currently less useful for interpreting available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In this paper, we present a new cloud retrieval model, YunMa, optimised for transit spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In YunMa, we built the cloud model based on Ackerman & Marley (2001) and simulated the cloud contribution to transit spectra using extinction coefficients as calculated by the open-source B-H Mie code (Bohren & Huffman 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' YunMa is fully integrated into the TauREx 3 retrieval platform (Al-Refaie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Al-Refaie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Al-Refaie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2019) and, for the first time, provides cloud microphysics capabilities into a retrieval model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We describe the model in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Section 3, we detail the experimental setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Section 4, we validate particle size distributions and spectroscopic simulations against previous studies published in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Af- ter validation, we show new spectral and retrieval sim- ulations obtained with YunMa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Section 5, we dis- cuss our results and assumptions and identify possible improvements to the model to be considered in future developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' MODEL DESCRIPTION YunMa estimates the vertical distribution of the cloud particle sizes (VDCP hereafter, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' B1 b and B4 a, b in Appendix B) based on A-M model and their contribution to the radiative transfer calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The YunMa module has been integrated into the TauREx 3 retrieval platform: the combined YunMa-TauREx model is able to constrain the VDCP from observed at- mospheric spectra, as described in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Modelling the cloud particle size distribution YunMa model contains a numerical realisation of the A-M microphysical approach to simulate the VDCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We 4 Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 1 a pictorial representation of the A-M approach: it assumes that clouds form with different VDCP to maintain the balance between the upward tur- bulent mixing and downward sedimentation of the con- densable species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Depending on the atmospheric T-p profile, multiple cloud layers may form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Sketch of the A-M micophysical approach adopted in YunMa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Cloud particles may form when the mix- ing ratio of the condensable gas exceeds the saturation mix- ing ratio, which is derived from its saturation vapour pressure (SVP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The vertical distribution of the cloud particle sizes (VDCP) is derived from the balance between the sedimen- tation of the cloud particles and the atmospheric turbulent mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Cloud mixing profile Cloud particles start forming when the partial pres- sure of a certain gas exceeds the saturation vapour pres- sure (SVP): the formation strongly depends on the at- mospheric thermal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The condensation process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' occurring when the partial pressure exceeds the SVP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' is estimated by comparing the molecular mixing ratio of the gas phase with its saturation vapour mixing ratio: qc(z) = max[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' qt(z) − (Sc + 1) qs(z)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (1) where qc is the mixing ratio of the condensed species,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' z is the altitude,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' qs is the mixing ratio where the con- densable gas saturates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' qt is the total mixing ratio of a condensable chemical species,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' including both the con- densate and gas phases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' and Sc is the supersaturation factor which persists after condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' qs can be es- timated from the ratio between the SVP of a certain chemical species and the atmospheric pressure at the same altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Note that in this paper, the mixing ratio refers to the volume fraction of a chemical species in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In the A-M approach, the turbulent mixing of the con- densate and vapour is assumed to be in equilibrium with the sedimentation of the condensate: −K(z) ∂qt(z) ∂z − fsed w∗(z) qc(z) = 0, (2) where K (m2 s−1) represents the vertical eddy diffusion coefficient, and w∗ (m s−1) is the convective velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' fsed is the ratio between the mass-weighted droplet sed- imentation velocity and w∗, defined as: fsed = � ∞ 0 vf dm dr dr ερaw∗qc ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (3) here ρa is the atmospheric mass density, which can be estimated through the Ideal Gas Law, ρp is the mass density of a condensed particle, ε is the ratio between the molecular weights of the condensates and the at- mosphere, and vf is the sedimentation velocity which will be explained later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The first term in equation (2) describes the upward vertical draft derived from the macroscopic eddy diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The second term describes the downward sedimentation, which is in equi- librium with the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The eddy diffusion coefficient (K) is one of the key parameters affecting cloud formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In free convection (Gierasch & Conrath 1985), it can be estimated as: K = H 3 � L H � 4 3 � RF µρacp � 1 3 , (4) where H, µ and cp are, respectively, the atmospheric scale height, mean molecular weight and specific heat capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' F = σT 4 eff is the approximated radiative flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The turbulent mixing length (L) is the scale height of the local stability in eddy diffusion, as opposed to the atmospheric scale height (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In YunMa, we use the value of K as estimated by the atmospheric chemistry models embedded in TauREx 3 (Al-Refaie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Al-Refaie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021) and derive L, accordingly, from equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The convective velocity scale (w⋆) men- tioned above can also be estimated as a ratio between K and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The sedimentation velocity, denoted by vf, is the speed at which a cloud particle settles within a hetero- geneous mixture due to the force of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' vf can be Space/Anothercloud layer Horizontally homogeneous Clouddeck Turbulent mixing Cloud· Partialpres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Vapour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Particles VSSVP Sedimentation Cloudbase Clearatmosphere Surface/AnothercloudlayerYunMa: Enabling Spectral Retrievals of Exoplanetary Clouds 5 estimated through viscous fluid physics: vf = 2 9 βgr2∆ρ η , (5) where ∆ρ is the difference between ρp and ρa, β is the Cunningham slip factor, and η is the atmospheric dy- namical viscosity (See Appendix A for more details of SVP, β and η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Particle size and number density Following the A-M approach, we assume spherical cloud particles with radii r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The particle radius at w⋆, denoted as rw, can be obtained using these relationships between vf and w⋆: vf(rw) = w⋆, (6) and vf = w⋆( r rw )α, (7) where α corresponds to the sedimentation velocity de- crease in viscous flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In A-M, the particle size distri- bution was constrained by in-situ measurements of Cal- ifornian stratocumulus clouds, which followed a broad lognormal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The assumption of lognormal distribution allows estimating the geometric mean ra- dius (rg), the effective radius (reff) and the total cloud particle number density (N), using the detailed defini- tions and derivations listed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Cloud contribution in transit spectra To estimate the wavelength-dependent cloud contri- bution to transit spectra, YunMa adopts the scatter- ing theory and absorption cross sections as described in Bohren & Huffman (2008, BH-Mie hereafter), assuming spherical cloud particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The cross-section of the cloud particles (kλ) at each wavelength (λ) and particle size are estimated through the extinction coefficient (Qext) of the corresponding wavelength and particle size, de- rived by BH-Mie from the refractive indices of the cloud particles: kλ = Qext πr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (8) We used the water ice refractive indices reported in War- ren & Brandt (2008) for our simulations of the tem- perate super-Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We show some examples of at- mospheres with water ice clouds in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Post- experimental tests were conducted to avoid the contam- ination of liquid water particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We simulate the cloud optical depths from the parti- cle sizes and number densities along each optical path, which passes the terminator at altitude zter, with a path length szter of each atmospheric layer: τλ = � ztop zter � ∞ 0 kλ dn dr dr dszter dz dz, (9) where n is the accumulated number density of particles with a radius smaller than r and ztop is the altitude at the top of the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The contribution of the clouds to the transit spectra, ∆Fc, can be estimated as: ∆Fc = 2 � ztop zbottom(Rp + z)(1 − e−τλ) dz R2s , (10) where zbottom is the altitude at Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' While YunMa has the capability to include any customised cloud particle size distribution in the spectral simulations, in this pa- per, we aim at model testing and, for simplicity, we use a single radius bin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' uniform cloud particle size rc = rg (see equation A5 in Appendix A) for each atmospheric layer in the radiative-transfer simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' YunMa-TauREx: retrieval of cloudy atmospheres We integrate the YunMa VDCP and τλ simulations in the Tau Retrieval of Exoplanets framework (Tau- REx 3, Al-Refaie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Al-Refaie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Al- Refaie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2019), which allows atmospheric retrieval simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' TauREx 3 combined to YunMa allow us to perform retrievals which include cloud microphysical processes and cloud scattering properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Parameters estimated by TauREx 3 include atmospheric T-p and chemical profiles, planetary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', mass and radius) and stellar (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', temperature and metallicity) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The radiative transfer calculations executed by TauREx 3 consider molecular and atomic absorptions, Rayleigh scattering and collisionally induced absorptions (CIA) of H2-H2 and H2-He pairs from Cox (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' YunMa uses as initial condition the gas mixing ratio profiles provided by TauREx 3 chemistry models (qt, equation 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In this paper, for simplicity, we assume the baseline chemical abundances are constant with al- titude instead of a more complex chemical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' YunMa then adjusts the gas phase mixing ratios, atmo- spheric mean molecular weight and atmospheric density in the TauREx 3 chemistry models as a result of the for- mation of clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' To simulate transit spectra and per- form retrievals, we use the atmospheric grids and optical paths defined in TauREx 3 and add the cloud opaci- ties as estimated by YunMa B-H Mie to the absorptions caused by the chemical species, using the methods ex- plained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The retrievals were tested on 80 Intel(R) Xeon(R) Gold 6248 CPU @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='50GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' METHODOLOGY In this paper, we use YunMa to perform retrieval sim- ulations of small temperate planets, where we expect a considerable amount of H2O to be present in the at- mosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' For simplicity, we consider only water clouds 6 Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Priors for spectral retrieval exper- iments using YunMa of all the cases listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Parameter Unit Ground Mode Priors Rp RN Je 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='20 factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='75 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='25 fsed · · Table 3 log 10−3 – 102 XH2O · · Table 3 log 10−12 – 1 pc bar Table 3 log 10−4 – 1 Tc K 200 linear 0 – 500 Tsurf K 1000 linear 500 – 2000 XN2 · · Table 3 linear 10−12 – 1 Note—XH2O represents the water vapour mixing ratio forming in the atmosphere and we do not consider super- saturation cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The planetary parameters are inspired by K2-18 b (Tsiaras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Charnay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021 b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021), which is a suitable candidate for cloud model testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We list all the priors of our experiment in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In this work, we set K as a constant and estimate η using the approximation proposed by Ros- ner (2012, equation A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We include both scattering and absorption due to water clouds based on Bohren & Huffman (2008), Rayleigh scattering of all the gas species and CIA of H2-H2 and H2-He pairs, which are enabled by TauREx 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We fine-tune the abundance of N2 to adjust the scale heights of the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' H2 and He act as the filling gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We use the POKAZATEL dataset for 1H16 2 O (Polyansky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2018) from the Exo- Mol database1 (Tennyson & Yurchenko 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Tennyson & Yurchenko 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Chubb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021) to estimate the water vapour absorption and Rayleigh scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The CIA data is from HITRAN2 (Karman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We use the PHOENIX library (Husser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2013) to simu- late the stellar atmospheres spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' For the numerical parameter settings, after a num- ber of tests, we decided to use the explicit Runge-Kutta method of order 8 (DOP853, Hairer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 1988) with relative tolerance (rtol) of 1 × 10−13 and absolute tol- erance (atol) of 1 × 10−16 to solve the partial differ- ential equation (2) for all the experiments presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We have opted for a logarithm sampling to retrieve most of the atmospheric parameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', fsed, XH2O and pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We have used a linear sampling, instead, for N2 to obtain a better numerical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The priors are sufficiently unconstrained to avoid biases gen- erated by excessive pre-knowledge, as discussed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' in 1 https://exomol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='com 2 https://hitran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='org Changeat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' After a number of tests, we have chosen to use 400 live points for 3-dimensional re- trievals and 1000 for more dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' To begin with, we run a sensitivity study with YunMa about the planetary and instrumental parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We set the planetary radius (Rp), fsed and XH2O as free pa- rameters in our 3-dimensional retrieval tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We list in Table 1 the planetary parameters adopted in the simu- lations, the prior ranges and the sampling modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The simulations are conducted with 80 atmospheric layers, from 10 bar to 10−6 bar, which encompass the typical observable atmospheric range for super-Earths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We se- lect Case 2 in Table 3 as the nominal case, and test the model sensitivity to the key parameters in the retrievals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Case 2, clouds dampen the gas spectroscopic features but do not obscure them entirely (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The nom- inal fsed refers to the value adopted in A-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The water SVP (both liquid and ice) used are taken from Appendix A in A-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' SENSITIVITY STUDIES TO KEY ATMOSPHERIC PARAMETERS We have performed sensitivity studies to test how the model behaves when changing some of the key atmo- spheric parameters, including pc, fsed, XH2O, XN2 (Case 1–9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Isothermal T-p profiles, as commonly used in tran- sit retrieval studies, are too simplistic for cloud studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We first assume T-p profiles with a dry adiabatic lapse rate in the troposphere, a moist adiabatic lapse rate in the cloud-forming region and a colder isothermal pro- file above the tropopause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' To be compatible with the computing requirements in retrieval, we simplify it to a ”two-point” profile (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 5 in Section 4) in Tau- REx while keeping the lapse rate at the cloud deck, since the cloud formation is more sensitive to it than the T-p profile in the deeper atmosphere from our preliminary sensitivity tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We define as Tc and pc the temperature and pressure at the tropopause and Tsurf the tempera- ture at 10 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' By tuning pc, the altitude at which the cloud forms and the cloud’s optical thickness will be al- tered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Here we test pc from 10−3 to 10−1 bar (Case 1–5), which is a much broader range than the one considered in previous literature about K2-18 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Case 6, we set the sedimentation efficiency fsed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='01, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' the down- ward sedimentation of the cloud particles is relatively slow compared to the net upward molecular mixing of the condensable species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' By contrast, in Case 7 where fsed = 10, we have a larger downward draft velocity scale compared to the upward one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Case 8 and 9, we mod- ulate the amount of condensable gas, here represented by the water vapour mixing ratio (XH2O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' YunMa: Enabling Spectral Retrievals of Exoplanetary Clouds 7 SENSITIVITY STUDIES TO DATA QUALITY The new generation of space-based facilities, such as JWST and Ariel, will deliver unprecedentedly high- quality data in terms of wavelength coverage, signal-to- noise ratio and spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We select as nominal case transit spectra covering 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='4–14 µm, at a spectral resolution of 100, with 10 ppm uncertainty across wave- lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We chose 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='4 µm as the blue cut-off to maximise the information content about Rayleigh scattering and 14 µm as the red cut-off to maximise the information content about water vapour and atmospheric tempera- ture for the type of planets considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We then test how YunMa’s performances degrade when we com- promise with any of those parameters, for instance: un- certainties of 30 ppm in Case 11 and spectral resolution of 10 in Case 12, which should have similar effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We move the blue cut-off at longer wavelengths in Case 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' RETRIEVALS OF ATMOSPHERIC THERMAL PROFILES We have run retrieval experiments to test how YunMa performs with increasingly complex model assumptions and which parameters may be problematic in these re- trievals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' YunMa’s ability to retrieve pc, Tc and Tsurf were studied in Cases 14–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Case 15, we use a higher quality spectrum as input and compare the results with Case 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' ADDITION OF N2 IN RETRIEVALS Inactive, featureless gases, such as N2, inject much uncertainty in the retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Cases 18–20, we include different amounts of the inert and featureless gas N2 in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' N2 may exist in super-Earths’ at- mospheres, as it happens for Solar System planets at similar temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Being N2 heavier than H2O, we adjust the mean molecular weight and the scale heights accordingly by modulating XN2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Heavier atmospheres have smaller scale heights: the spectral features are less prominent and harder to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We first try to retrieve only Rp, fsed, XH2O and XN2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' DEGENERACY BETWEEN CLOUDS AND HEAVY ATMOSPHERE In Cases 14–20, N2 and the T-p profile were retrieved separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Here we attempt to retrieve the cloud mi- crophysics parameters (fsed) together with XN2, the T-p profile and other atmospheric parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Case 21 re- trieves these parameters except XN2 fixed to 50% for comparison with Case 22 to investigate the degeneracy imposed by the uncertainty of XN2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Case 22, we per- form the full retrieval including both XN2 and the T-p profile as free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Similar to N2, decreases the scale height and compresses the transit spectrum, the existence of clouds mitigates the spectral features, and the difference between these two scenarios may be dif- ficult to distinguish from the current data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' One hypothesis is that if we do not include N2 among the priors, the model will add clouds to compensate for the missing N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Case 23, we test this hypothesis by forc- ing XN2 to zero and then monitor the cloud parameters in the posteriors obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' COMPARISON OF CLOUD RETRIEVAL MODELS Here forward spectra are simulated with YunMa cloud microphysics and then retrieved with another simplified cloud retrieval framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Case 24, clouds are de- scribed as an opaque cloud deck across wavelengths in TauREx 3, which is commonly used to retrieve data from the last decades with narrow wavelength coverage, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', HST/WFC3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Case 25, we deliberately omit cloud parameters in the retrieval priors and learn if and how other parameters can compensate for those missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' RETRIEVALS OF FEATURELESS SPECTRA In Cases 26–28, we retrieve the atmospheric parame- ters from spectra with minimal spectroscopic features, where spectral standard deviation (σspec) is less than 1 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The spectra are featureless due to heavy atmo- spheres and cloud contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Case 28, we unreal- istically decrease the uncertainty to 1 ppm at resolving power 100 to evaluate YunMa’s performance with ide- alised data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Model validation We validated our model against Ackerman & Mar- ley (2001) by comparing the condensate mixing ratios of Jovian ammonia clouds (qc) with different values of fsed, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The two sets of results are consistent and the small differences in qc translate into ∼ 10−2 ppm in transit depth, which is completely neg- ligible compared to typical observational noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We further validated our model against the results from Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2018) by comparing the KCl cloud molecular mixing ratio and particle-size profile (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' B1 in Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2018) used CARMA to simulate cloud microphysics in exoplanets and brown dwarfs with Teff = 400 K and log g = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='25, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='25 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='25 (in cgs units), corresponding to planetary masses of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='72 MJ, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='47 MJ, and 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='54 MJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In the best fit between CARMA to A-M model, the fsed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='093 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='025 for the cases with K = 104, 103 and 102 m2 s−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The mixing length is derived from con- stant eddy diffusion, as described in equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The 8 Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Validation of the YunMa cloud microphysics model against the Jovian ammonia clouds in Ackerman & Marley (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' qc corresponding to different sedimentation efficiencies (fsed) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Solid lines: results from YunMa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Dashed lines: results from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 1 in Ackerman & Marley (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Dotted line: T-p profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' results agree with each other within 8 %, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' ∼ 10−3 ppm difference in the transit depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We compared simulations from YunMa with the re- sults from Charnay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2021 b, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' B2 in Ap- pendix B), which include the horizontal effects generated by global circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Being YunMa a 1D model, we can- not reproduce the exact same results, but we can have an approximated comparison with their simulations at the substellar point, where the horizontal winds are weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We assume 100 × solar composition, with fsed = 3 and K = 106 cm2s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Both models predict clouds forming in the region between 3 × 10−2 and 1 × 10−2 bar and a cloud molecular mixing ratio ∼ 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We have validated the BH-Mie module in YunMa against PyMieScatt, an open-source model simulating atmospheric particle scattering properties (Sumlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2018), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Cloud particle radii were selected in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='1-100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The largest discrep- ancy in Qext is within ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='002, which corresponds to an average of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='01 ppm in the planetary transit depth of the nominal scenario in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Simulated transit spectra with YunMa We present here the transmission spectra generated with YunMa of a cloudy super-Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 3 shows five examples at resolving power of 100 which corresponds to the ground truths of some of the retrieval cases in Table 3: pc = 2 × 10−3 bar with XN2 = 0 (blue, Case 2, 10, 11, 14 and 15) and = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 (purple, Case 20–25);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' pc = 1 × 10−2 bar with XN2 = 0 (green, Case 3 and 16) and = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 (yellow, Case 27 and 28) and one case of opaque cloud with pc = 1 × 10−2 bar in the H2/He dominated atmosphere (red, Case 4 and 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The plan- etary and atmospheric parameters are listed in Table 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' All the simulations contain baseline 10% H2O abundance across the atmosphere, which is then altered by the cloud formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The rest of the atmosphere is N2 and H2/He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We select fsed = 3 for all the scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The simulation results are summarised in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In the experiment without N2 and pc set to 2 × 10−3 bar, clouds form at high altitude, where the atmospheric density (ρa) is low compared to the cases where pc = 5 × 10−3 bar and = 1 × 10−2 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Here the sedimen- tation velocity (vf) is small with small cloud particle radii and number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The cloud contribution (blue dash-dotted line) has a mean transit depth of 2660 ppm and σspec of 17 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' It is an optically thin cloud which does not completely block the spectral features shaped by water vapour absorption (blue dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' When pc = 1 × 10−2 bar, clouds form at relatively low altitudes, where the atmospheric density (ρa) is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Here vf is large and the cloud particles have relatively large radii and number density, which increase the opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The cloud contribution (red dash-dotted line) has a mean flux depth of 2727 ppm and σspec of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='62 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Since the clouds are optically thick, they contribute significantly to the mean transit depth and obscure the spectral fea- tures of water vapour (red dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' However, the water vapour features are still able to show due to the low altitude of the clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Still, the spectral deviation is only 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='16 ppm, where the spectroscopic features have a high chance of being hidden by the observational un- certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' pc = 5 × 10−3 bar is an intermediate case regarding the simulated cloud altitude and opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The simulation suggests that the intermediate combination of these two cloud properties does not result in more significant atmospheric features than in other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In atmospheres with relatively high mean molecular weight – and therefore small scale height – for the same value of pc, the transit depth is smaller, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In the case with XN2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 and pc = 2 × 10−2 and = 5 × 10−2 bar, the mean value of the transit depths are ∼ 200 ppm smaller than them in a H2/He dominated atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Here the cloud particles form at higher ρa and therefore have larger particle size and larger number density compared to those formed in the H2/He domi- nated atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The spectrum has σspec of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='58 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='62 ppm, which are negligible compared to the obser- vational uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Besides pc and XN2, we also have tested different fsed to understand how this parameter controls the cloud mi- crophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The particle radii, rc, number density and transit spectra across all the cloud pressure levels and Temperature (K) 110 120 130 140 150 160 10-1 Pressure (bar) 3 × 10-1 4 × 10-1 6 × 10-1 100 10-8 10-7 10-6 10-5 10-4 Cloud molecular mixing ratio (qc) fsed = O (YunMa) fsed = 1 (A-M 01) fsed = 10 (YunMa) fsed = 0 (A-M 01) fsed = 3 (YunMa) fsed = 10 (A-M 01) fsed = 1 (YunMa) fsed = 3 (A-M 01) TYunMa: Enabling Spectral Retrievals of Exoplanetary Clouds 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 1 2 5 10 Wavelength ( m) 2500 2550 2600 2650 2700 2750 2800 Transit depth (ppm) pc = 2 × 10 3 bar all contribution pc = 2 × 10 3 bar gas phase H2O absorption pc = 2 × 10 3 bar absorption and scattering of H2O cloud pc = 2 × 10 3 bar with XN2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 all contribution pc = 2 × 10 3 bar with XN2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 gas phase H2O absorption pc = 2 × 10 3 bar with XN2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 absorption and scattering of H2O cloud pc = 5 × 10 3 bar all contribution pc = 5 × 10 3 bar gas phase H2O absorption pc = 5 × 10 3 bar absorption and scattering of H2O cloud pc = 5 × 10 3 bar with XN2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 all contribution pc = 5 × 10 3 bar with XN2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 gas phase H2O absorption pc = 5 × 10 3 bar with XN2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 absorption and scattering of H2O cloud pc = 1 × 10 2 bar all contribution pc = 1 × 10 2 bar gas phase H2O absorption pc = 1 × 10 2 bar absorption and scattering of H2O cloud Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Simulated transit spectra of cloudy super-Earths using YunMa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Solid line: total transit depth with all contributions included;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' dot-dashed line: water ice clouds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' faint-dotted line: water vapour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Blue, green and red lines: H2/He dominated, cloudy atmospheres with different pc (see legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Purple and yellow lines: heavier cloudy atmospheres with 50% N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' obtained with different fsed are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Here we note that, from the results, the cloud particle sizes in- crease with fsed while the number densities at each layer behave reversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Also, which is easy to understand, the more atmospheric layers with clouds, the larger the op- tical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Retrieval results We show in this section how YunMa performs with dif- ferent model assumptions and ground truth parameters (GTPs), following the approach described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' GTPs and priors are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The posterior median values and their standard deviations obtained for all the simulated cases are summarised in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Cases 1–5 test the effects of different pc to the tran- sit spectra and retrievals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The transit spectra of Cases 2 and 4 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 3 (blue and red solid lines respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The larger is pc, the more opaque mean- while, the lower altitude becomes the clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The sig- nificance of spectroscopic features owns to both factors, and generally speaking, the more significant features are, the easier it is to retrieve the atmospheric parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We choose pc = 2 × 10−3 bar as nominal case and show the corresponding posterior distributions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Cases 2, 6 and 7 test the impact on the transit spec- Rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='4 log(H2O) log(H2O) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='99+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='60 log(fsed) log(fsed) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='48+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='196 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='208 Rp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 (derived) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='4 log(H2O) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='60 log(fsed) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 (derived) (derived) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='90+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='54 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Example of cloud retrieval using YunMa inte- grated in TauREx 3 (Case 2 in Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The sedimenta- tion efficiency (fsed), which is the main parameter control- ling the cloud microphysics, is well recovered together with the vapour water mixing ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' tra and retrievals of the sedimentation efficiency (fsed), which controls the cloud microphysics in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In all three cases, GTPs for fsed, XH2O and Rp are well 10 Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Atmospheric and cloud parameters included in YunMa simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The corresponding transit depths are also reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' pc 2 × 10−3 (bar) 5 × 10−3 (bar) 1 × 10−2 (bar) 2 × 10−3 (bar) 5 × 10−3 (bar) XN2 · · · · · · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 Parameter Unit All contribution, mean ppm 2770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='17 2750 2729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='82 2558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='65 2553.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='71 All contribution, std (σspec) ppm 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='72 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='59 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='62 Cloud contribution, mean ppm 2659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='65 2729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='60 2726.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='53 2534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='78 2542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='78 Cloud contribution, std ppm 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='85 MMW (bottom of the atmosphere) g mol−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='88 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='88 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='88 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='73 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='73 Atmospheric pressure (cloud base) bar 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='34 × 10 −3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='40 × 10 −3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='43 × 10 −2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='34 × 10 −3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='40 × 10 −3 Atmospheric pressure (cloud deck) bar 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='54 × 10 −4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='34 × 10 −3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='28 × 10 −3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='54 × 10 −4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='33 × 10 −3 Cloud MMR (cloud base) · · 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='35 × 10 −4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='51 × 10 −5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='00 × 10 −4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='92 × 10 −4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='35 × 10 −4 Cloud MMR (cloud deck) · · 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='56 × 10 −8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='92 × 10 −7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='28 × 10 −8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='49 × 10 −7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='33 × 10 −6 vf m s−1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='54 – 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='53 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='07 - 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='67 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='43 – 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='41 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='14 – 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='97 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='37 – 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='28 rc µm 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='03 – 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='39 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='79 – 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='88 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='13 – 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='35 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='99 – 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='09 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='53 – 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='66 N (cloud base) m−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='03 × 104 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='19 × 104 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='47 × 104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='28 × 104 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='22 × 10 3 N (cloud deck) m−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='45 × 101 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='72 × 101 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='24 × 101 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='28 × 101 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='16 × 10 1 Note—MMR and MMW: molecular mixing ratio and mean molecular weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The pressure at the bottom of the atmosphere is assumed to be 10 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' within the retrieved parameter space, with more or less peaked and symmetric posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' When fsed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='01 and 3, the posterior distributions of fsed are closer to normal distribution centred at the retrieved solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' When fsed = 10, the upper limit is not well constrained in the posterior distribution of fsed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' This effect is not unexpected, as high fsed scenarios tend to have negligible impact on the planetary transit depth due to thinner cloud layers and smaller number density (N) compares to a low fsed scenario, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', the fsed = 10 and 100 cases in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' From the experiments on the baseline water abundance (XH2O), in atmospheres where water can condense, when the XH2O is less than 1 × 10−3, clouds cannot form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' When XH2O = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='01 (Case 8) a thin cloud with low opacity may form, the water vapour spectral features are well visible and the retrieval performs well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' By contrast, a higher mixing ratio of the condensable gas increases the partial pressure and con- tributes to the condensation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' This is, for in- stance, the case of XH2O = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 (Case 9) where the cloud is thick and largely blocks the spectral features, making the retrieval of the atmospheric parameters difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' From the experiments on observational data qual- ity, Case 10’s Bayesian evidence (4568.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='52) compared to the ones calculated for Case 2 (3755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='80) and Case 11 (3368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='29) showcases how the retrieval performance im- proves when the observational uncertainties are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Case 12 performs better than Case 2, as the spectral resolution of the transit spectrum used as input to the retrieval is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Case 13, we omitted the informa- tion contained in the optical wavelengths, which means that we have less information about the cloud scatter- ing properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The retrieval performances are degraded compared to Case 2, which includes the optical wave- lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Case 14–17, we retrieved the T-p profiles as free pa- rameters for different pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Our results show that both the T-p profiles and cloud parameters can be constrained, although the retrieved gas phase mixing ratio and fsed have large standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Case 18 compared to Case 2 showcases how the addition of the spectrally in- active N2 injects uncertainty in the retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Differ- ent amounts of N2 (XN2), are considered in Cases 18– 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Despite the minimal spectral features of the cloudy heavy atmospheres due to N2 injection, it is still able to retrieve the atmospheric parameters in simple cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Cases 21–25 we retrieve both the T-p profile and XN2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The corresponding transit spectra are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 3 (purple lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Case 21 shows how the atmo- spheric parameters, with the exception of XN2, can be retrieved in a heavy atmosphere with XN2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' When we include the uncertainty of XN2 (Case 22), the GTPs for Rp, XH2O, fsed, Tsurf and Ttop (see the retrieved T-p profile in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 5) still fall into the 2σ confidence range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' fsed is significantly less constrained in Case 22 than Case 21 due to the uncertainty of XN2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Cases 22–24 are compared with each other in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Po- tential degeneracy could happen according to Case 23: when we omit N2 among the priors, the retrieval tries to compensate for the missing radiative-inactive gas by YunMa: Enabling Spectral Retrievals of Exoplanetary Clouds 11 decreasing Rp and Tsurf, while increasing XH2O, and the Bayesian evidence of Case 23 (3757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='08) is close to Case 22 (3757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In the experiment of model comparison, if we use the simple opaque cloud retrieval model in Tau- REx 3 (Case 24) instead of YunMa for the not opaque case, the cloud and other atmospheric parameters are significantly more degenerated and less constrained than using YunMa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Without the cloud in the prior (Case 25), the results show a lack of constraints on the T-p profile while comparable performance on other parameters with Case 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 0 250 500 750 1000 1250 1500 1750 Temperature (K) 10 6 10 5 10 4 10 3 10 2 10 1 100 101 Pressure (bar) Fitted profile Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Retrieved T-p profile of Case 22 in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The isothermal temperature (Tc), surface temperature (Tsurf) and the pressure where the isothermal profile starts (pc) are re- trieved using YunMa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The shaded area indicates the stan- dard deviation of the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Cases 26–28 focus on retrieval of plain spectra (pc = 5 × 10−2 bar in a heavy atmosphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The transit spec- trum for Case 27 and 28 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 3, yellow line: the spectral signal is very small compared to the observational uncertainty (10 ppm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' As expected, the retrieval performance is limited unless the observational uncertainty is unrealistically decreased to 1 ppm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 12 Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Retrieval posteriors for Case 22 (purple), 23 (orange) and 24 (green) in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Blue crosses indicate the ground truth parameters and the vertical dashed lines in histograms indicate the 1σ and 2σ confidence ranges of the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' These three retrievals use the same transit spectrum as input (thin cloud with XN2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 3) but different retrieval assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Case 22 (purple): H2O and N2 are included as priors and the cloud formation is simulated by YunMa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Case 23 (orange): same as Case 22 except that N2 is not included among the priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Case 24 (green): same as Case 22 except that the cloud is simulated by a simpler model from TauREx, in which the atmosphere becomes opaque below the cloud deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The only retrieved cloud parameter for the simpler model is the cloud deck pressure, which is not shown here for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' YunMa cloud with XN, as free parameter in retrieval YunMa cloud with fixed X, = O in retrieval Flat-line cloud with Xn, as free parameter in retrieval Juns T = 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='55+%99 L 20 log(Pc) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='23±8:3 log(Pc) log(H20) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='82±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='44 log(H20) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='4 N2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='68±8:31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='2 fsed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='60±0:91 seo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='6 Tsurf Tc log(Pc) log(H20) fsed N2YunMa: Enabling Spectral Retrievals of Exoplanetary Clouds 13 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' YunMa retrieval experiments with thick clouds and XN2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 (orange line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Observational uncertainty = 10 ppm (blue line, Case 27 in Table 3) and = 1 ppm (brown line, Case 28 in Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Red lines indicate GTPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='20±0:00 Tsurf = 976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='57+41:46 pc = 5 × 10-2 bar, Error = 10 ppm Te= 187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='18±15:86 pc = 5 × 10-2 bar, Error = 1 ppm T 180 120 6 log(Pc) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='69+8:18 4 log(Pc) log(H20) = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='20±8:18 0% ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (02H)60) 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' log(fsed)= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='46±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='19 log(fsed) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='4 N2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='90±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='0 6 10 % .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 1 Tsurf Te log(Pc) log(H20) log(fsed) N214 Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' YunMa retrieval experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The ground truth parameters (GTPs) assumed in the simulations are listed in the left columns of the table and the retrieved posteriors on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The retrieval priors are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In each case, we test the model sensitivity of the atmospheric parameters of fsed, pc, XH2O, XN2, and the observational parameters of the error, wavelength coverage (λ) and the spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We choose the nominal value of pc as 2 × 10−3 bar, where the cloud is not either too thin to be detected or too thick to block the spectroscopic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Case GTPs Posteriors log(fsed) log(pc) log(XH2O) XN2 Error λ Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Rp log(fsed) log(XH2O) XN2 log(pc) Tc Tsurf log(bar) (ppm) (µm) RN Je log(bar) (K) (K) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='48 3 1 · · 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='4 – 14 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='20+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='98e−06 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='06e−06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='29+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='10+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='00 · · · · · · · · 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='18+15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='16 −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='86 976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='55+33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='41 −41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='97 YunMa: Enabling Spectral Retrievals of Exoplanetary Clouds 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' DISCUSSION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Transit spectroscopy using YunMa To understand the performances of YunMa in detail, we performed retrieval experiments for over a hundred cases, with different chemistry models, atmospheric and cloud scenarios for super-Earths/sub-Neptunes, hot- Jupiters and brown dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' A variety of cloud species were modelled and analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' This paper presents a se- lection of representative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We have validated the A-M cloud size distribution in YunMa against pre- vious literature simulating NH3 clouds in Jupiter’s at- mosphere, KCl clouds on artificial large exoplanets and brown dwarfs, and H2O clouds on K2-18 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We have also tested a number of numerical settings, including fitting methods, tolerances and retrieval samplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In YunMa, the mixing ratios of the condensable gas and the condensate are strongly correlated, therefore when cloud forms, the condensable species in the gas phase decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Also, a balance is imposed between the upward turbulent mixing and the downward sedimenta- tion velocity from the A-M approach (equation 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In the current YunMa, at each pressure level, the particle number density represents the total number density of particles with different radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The Earth measurements shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 4 of Ackerman & Marley (2001) suggest a bimodal distribution of the particle sizes at the same pressure level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' YunMa is able to use radius bins with their respective number densities to represent more pre- cisely the cloud distribution in the spectral simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' For the retrieval calculations, however, we had to sim- plify this information to reduce the computing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The cloud opacity is determined by the cloud particle size and number density;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' different particle sizes have ab- sorption and scattering peaks at different wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Optically thick clouds cause transit depths with neg- ligible or no modulations as a function of wavelength: the atmosphere below the cloud deck is, in fact, unde- tectable while the atmosphere sounded above the cloud deck is more rarefied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Retrieving information about the atmospheric composition and structure is very difficult in the most extreme cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' For an atmosphere with optically thin clouds, the absorption features due to radiative-active gases (water vapour here) are detectable but less prominent compared with a clear atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The abundances of these gases can be retrieved largely from their absorption features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' If these are condensable species, their abundances further constrain the cloud microphysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Both the wavelength-dependent features and the overall transit depth help the retrieval perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' fsed is the ratio between the sedimentation velocity and the turbulent convective velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' From the def- inition, a higher fsed means a shorter sedimentation timescale, as we fixed K to a constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' A higher sedimentation efficiency leads to larger offsets to the downward draft, constraining the upward supplement of water vapour and cloud formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' On the contrary, for small fsed, the sedimentation timescale is much longer compared to the diffusion timescale, so the condensation continues at lower pressure to balance the downward sedimentation and upward turbulent mixing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' therefore, the cloud region expands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' fsed is sensitive to the cloud particles’ nucleation rate (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' It has a close relationship with the condensate particle size and can be expressed by the particle radius in the lognormal distribution power-law approximation: fsed = � ∞ 0 r3+α dn dr dr rαw � ∞ 0 r3 dn dr dr , (11) which indicates that small fsed encourages small cloud particle formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The upward transport is stronger than the sedimentation when the cloud particles are small and vice-versa, which is in line with the experi- mental results in Figure B4 (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The spectrum is sen- sitive to fsed when this parameter has values between 10−1 and 10−3, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' B4 (c), indicating the detectability of fsed in this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In our experiments, the particle radii are typically 1– 10 µm, so they do not block the Rayleigh scattering slope caused by H2 and H2O at the optical wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' For smaller radii (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', cyan line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' B3), they are expected to contribute more to the optical spectrum, although theoretically it would be hard to form particles at very small sizes according to the nucleation theory in cloud formation (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' For any particle radii even smaller presented in this paper, it is just for model test purposes in extreme cases, and we make no efforts to show their detailed analysis here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Radiative-inactive gases such as N2, if present, can change the atmospheric scale height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The increase of scale height decreases the transit depth and dampens its spectroscopic features, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' As men- tioned in previous sections, the presence of radiative- inactive gases cannot be detected directly through spec- troscopic signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Opaque clouds may behave sim- ilarly to inactive gases in mitigating spectroscopic fea- tures, leading to potential degeneracy in retrieval exper- iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 6, the comparison between Case 22 and 23 in Table 3 suggests the potential degen- eracy between Rp, the baseline condensable gas abun- dance, T-p profile, and the N2 abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Case 23, we force XN2 = 0 in the retrieval to monitor how YunMa 16 Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' compensates for the missing gases in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Adjustment of Rp translates the transit depth without impact on spectroscopic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The baseline condens- able gas abundance and T-p profile are correlated to the formation of clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The results indicate that when N2 is absent in prior, the mixing ratio of water vapour – the radiative-active gas – is significantly increased to com- pensate for the missing molecular weight, while more clouds are formed to further reduce the spectroscopic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The decrease of Tsurf helps in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' While the potential degeneracy exists from analysis and the results suggest the model behaviour in the case of missing radiactive-inactive gases, the model chose from statistics the scenario closer to the ground truth in Case 22, showing the model’s potential in retrieving clouds in heavy atmospheres when it is not opaque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' As mentioned before, the cloud formation and T-p profile are correlated in YunMa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Therefore, the pres- ence of optically thin clouds can help to constrain the T-p profile in retrievals: for instance, pc, Tc and Tsurf are well retrieved in Case 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' On the contrary, the T-p profile is not well constrained if clouds are completely absent (Case 25) or the microphysics part is removed from the retrieval (Case 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Cloud formation with the next-generation facilities’ data With the high-quality transit spectra offered by the next-generation facilities, the uncertainties, wavelength coverage and spectral resolution will be significantly im- proved compared to most current data, and a simple opaque cloud model is insufficient for the transit study of next-generation data according to our results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' the mod- elling of cloud radiative transfer and microphysics such as YunMa is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In our experiments, we chose a nominal 10 ppm as the observational uncertainty and found that clouds can be well-characterised in most ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' With the next-generation data and YunMa, there are still limits in retrieving the featureless spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' A smaller uncertainly may help in retrieving cloud parameters in the most difficult cases: we therefore adopted an unrealistic 1 ppm to test their detectability in an ideal case (Case 10, 15 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' With 1 ppm, the atmospheres were constrained well, although the spec- tra were featureless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Our experiments also suggest that broad wavelength coverage is paramount to characteris- ing clouds well: here, optical wavelengths play a critical role when combined with infrared spectral coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Numerical instabilities in cloud microphysics simulations We have selected the explicit Runge-Kutta method of order 8 (DOP853, Hairer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 1988) to solve equa- tion (2) as it delivered the most stable numerical per- formance for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' However, numerical in- stabilities may occur when a large relative (rtol) and absolute (atol) tolerances are chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Sometimes the ODE solver cannot converge for qt and indicates “no clouds” as a solution due to the numerical instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Caution is also needed in estimating the cloud mixing ratio (qc), as qt might be two orders of magnitude larger than qc and numerical errors could be injected when qs is subtracted from qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' When solving the ODE with too large tolerance, qt might converge to a certain value, but qc would be estimated as negligible according to equa- tion (2 and 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In other words, even though when the ODE solution for the gas phase seems numerically sta- ble, it might not be precise and accurate enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In those cases, the retrieval performances are affected, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' B5, where multiple islands of solutions in the posteriors are visible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' these are caused by numerical instabilities, and the issue is more obvious for larger tol- erance values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' After many tests, we have decided to use “DOP853” in solving the ODE with rtol = 1 × 10−13 and atol = 10−16, which guarantee numerically stable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' CONCLUSIONS YunMa is a state-of-art cloud model optimised for the interpretation of the next generation of exoplane- tary atmospheric data, as provided by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', JWST, Ro- man, Twinkle and Ariel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' These facilities will provide an unprecedented amount of high-quality data, where the cloud formation process and cloud scattering properties can no longer be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' YunMa cloud microphysics is based on the model pub- lished by Ackerman & Marley (2001), while the scat- tering properties of clouds are calculated through the open-source B-H Mie code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' When coupled to the Tau- REx framework (Al-Refaie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021), YunMa be- comes a very versatile model which can simulate tran- sit and eclipse spectra for a variety of cloudy exoplan- ets with different masses, atmospheric compositions and temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Most importantly, YunMa+TauREx can be used as a spectral retrieval framework optimised for cloudy atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Here, the sedimentation efficiency, fsed, is a critical parameter to control the cloud micro- physics in the retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We have validated YunMa against previous work which also adopted the A-M approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We have vali- dated the radiative transfer calculations in YunMa, in- cluding cloud scattering, against PyMieScatt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Finally, we have validated YunMa against other cloud micro- physics models published in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' YunMa: Enabling Spectral Retrievals of Exoplanetary Clouds 17 We have run over one hundred retrieval experiments with YunMa, with different cloud compositions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', KCl, MgSiO3, Fe clouds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In this paper, we have pre- sented and discussed 28 cases of water clouds in the atmosphere of a temperate super-Earth, K2-18b-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Through these experiments, we have learnt that YunMa is capable of retrieving accurately cloud formation pa- rameters, as well as atmospheric trace gases’ mixing ra- tios and thermal profiles when clouds are not so opaque to mask all the atmospheric features at most wave- lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The optical wavelengths contain important scattering information of the cloud particles and are therefore critical for the correct extraction of the cloud formation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' While the current version of YunMa can retrieve only one size of cloud particles at a time, the future version of YunMa will include the possibility of retrieving a dis- tribution of the cloud particle sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2-D cloud models, which include horizontal convection, will soon be within reach of computing speed and might be considered in future versions of YunMa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The work presented in this paper was partially supported by UKSA, grant ST/X002616/1 and ST/W00254X/1, and ExoMolHD ERC grant 883830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' QC is supported by the ESA Research Fellowship Pro- gram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The authors acknowledge the use of the UCL Kathleen High-Performance Computing Facility (Kath- leen@UCL) and associated support services in complet- ing this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' This work utilised the Cambridge Ser- vice for Data-Driven Discovery (CSD3), part of which is operated by the University of Cambridge Research Computing on behalf of the STFC DiRAC HPC Facil- ity (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='dirac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The DiRAC component of CSD3 was funded by BEIS capital funding via STFC capi- tal grants ST/P002307/1 and ST/R002452/1 and STFC operations grant ST/R00689X/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' DiRAC is part of the National e-Infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We thank Dr Andrew S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Ackerman for the construc- tive communication on the A-M model and for shar- ing the original code, Dr Kai-Hou Yip and Sam Wright for suggestions on the numerical methods and Dr Yui Kawashima for the cloud and haze model discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' SUPPLEMENTARY EQUATIONS In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='1, the SVP can be estimated with the Clausius-Clapeyron equation: es = e0 exp � ℓ RSV � 1 T0 − 1 T �� , (A1) where ℓ is the latent heat of evaporation, RSV is the spe- cific gas constant for the vapour, T is the atmospheric temperature, and T0 is the temperature at vapour pres- sure e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' We use the results from laboratory measure- 18 Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' ments of these parameters for the different chemical species when available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' β is the Cunningham slip factor: β = 1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='26NKn, (A2) The Knudsen number (NKn) is the ratio between the molecular mean free path and the droplet radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' YunMa adopts two ways to estimate η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' One is that Lavvas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2008) suggested to use: η = 1 3ρaV λa, (A3) where V is the thermal velocity of gaseous components, and λa is the mean free path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Another uses the definition by Rosner (2012), which is also adopted by Ackerman & Marley (2001): η = 5 16 √πmkBT πd2 (kBT/ϵ)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='22 , (A4) where d is the diameter of a gas particle and ϵ is the atmospheric Lennard-Jones potential well depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' When using the A-M approach, vf and the particle size are positively correlated using η either from Rosner (2012) or Lavvas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='2, assuming a lognormal cloud particle size distribution, the geometric mean (rg) is defined as: rg = e � ∞ 0 lnr dn dr dr � ∞ 0 dn dr dr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (A5) The power law approximation allows representation of fsed using the particle size distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' fsed ≈ � ∞ 0 r3+α dn dr dr rαw � ∞ 0 r3 dn dr dr , (A6) where n is the accumulated number density as defined in Section 2 and σg is the geometric standard deviation of the lognormal particle radius distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Through an integration of the lognormal distribution, Ackerman & Marley (2001) derived that: rg = rw f 1 α sed exp � −α + 6 2 ln2σg � , (A7) where the σg is the geometric standard deviation of the particle radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The effective mean radius (reff) is the area-weighted average radius defined by Hansen & Travis (1974) to approximately represent the scattering properties of the whole size distribution by a single parameter when the particle radius is larger than the radiation wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' To derive reff, we first recall that in lognormal distribu- tion, the t-th raw moment is given by: mt = N0rg exp �(tσ)2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (A8) Therefore reff, which is the area-weighted average ra- dius, can be estimated through: reff = � ∞ 0 rπr2 dn dr dr � ∞ 0 πr2 dn dr dr = rw f 1 α sed exp � −α + 1 2 ln2σg � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (A9) Similarly, the total number density for the particles is estimated by using the volume-weighted mean: N = 3ερaqc 4πρpr3g exp � −9 2 ln2σg � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (A10) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' SUPPLEMENTARY FIGURES (a) (b) Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Validation of YunMa (solid lines) against the A- M model in Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2018) (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Top: condensate mixing ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Bottom: cloud particle effective mean radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 10-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Pressure (bar) 10-1 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 102 10-1 100 Cloud particle effective radius (μm) K= 102 m?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='s-1, fsed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='125 (YunMa) K= 102 m2s-1, fsed =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='125 (Ga0 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2018) K= 103 m?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='s-1, fsed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='093 (YunMa) K= 103 m?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='s-1, fsed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='093 (Ga0 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2018) K= 102 m?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='s-1, fsed =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='125 (YunMa) K= 102 m²s-1, fsed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='125 (Ga0 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2018)10-2 Pressure (bar) 10 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 102 10-9 10-8 10-7 Cloud molecular mixing ratioYunMa: Enabling Spectral Retrievals of Exoplanetary Clouds 19 Figure B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Comparison of YunMa’s results against simu- lations by Charnay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2021 b) for the 100 × solar metal- licity scenario (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 6 (a) of the substellar region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The results are consistent, despite being generated by a 1-D cloud microphysical model in this work and the MIT General Cir- culation Model (Campin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2022) in the case of Charnay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' (2021 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' REFERENCES Ackerman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', & Marley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' S.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2022, ApJS, 260, 3, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='3847/1538-4365/ac5cc2 Charnay, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', B´ezard, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Baudino, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2018, ApJ, 854, 172, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='3847/1538-4357/aaac7d Charnay, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Blain, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', B´ezard, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021 b, Astronomy & Astrophysics, 646, A171 Charnay, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Tobie, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Lebonnois, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', & Lorenz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2022, A&A, 658, A108, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='1051/0004-6361/202141898 Temperature (K) 300 400 500 600 700 Cloud MMR (Charnay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021) Cloud location (Charnay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021) Condensate volume mixing ratio qc 10-3 Vapour volume mixing ratio qt Pressure (bar) 10 10-1 Temperature 100 10-7 10-6 10-5 10-4 10-3 10-2 Molecular mixing ratio20 Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Figure B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Validation of radiative transfer simulations obtained with YunMa against the open source code PyMieScatt (Sumlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The extinction coefficients for cloud particles with different sizes are estimated from the theory of Bohren & Huffman (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' To address the computational limitations of retrievals, we pre-calculated the extinction coefficients used in equation (8) to estimate the cross-sections of the cloud particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The pre-calculated list includes values for particle radii from 1 × 10−7 to 1 × 10−2 µm, equally spaced in the logarithm space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Here we show only six examples in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Bottom panels: residuals obtained by subtracting the extinction coefficients as estimated by the two codes, YunMa and PyMieScatt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The maximum discrepancy, corresponding to the largest particle radius simulated here (r = 1 × 102), is negligible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Cho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Skinner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', & Thrastarson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021, The Astrophysical Journal Letters, 913, L32, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='3847/2041-8213/abfd37 Chubb, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Rocchetto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Yurchenko, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021, A&A, 646, A21, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='1051/0004-6361/202038350 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 10-2, coefficient ( 10-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' xtinction 10-11 r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='000e+02 μm (PyMieScatt) r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='425e-03 μm (PyMieScatt) r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='000e+02 μm (YunMa) r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='425e-03 μm (YunMa) r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='424e+00 μm (PyMieScatt) r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='455e-05 μm (PyMieScatt) r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='424e+00 μm (YunMa) r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='455e-05 μm (YunMa) 10-14 r = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='878e-02 μm (PyMieScatt) r = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='377e-07 μm (PyMieScatt) r = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='878e-02 μm (YunMa) r = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='377e-07 μm (YunMa) 100 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='002 r=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='000e+02μm 1e-7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='0 r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='424e+00 μm 1e-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='0 Residual 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='0- r = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='878e-02 μm 1e-8 0 1 r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='425e-03 μm 1e-13 1 2 r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='455e-05 μm 1e-18 0 2 r = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='377e-07 μm 100 101 Wavelength (μm)YunMa: Enabling Spectral Retrievals of Exoplanetary Clouds 21 (a) (b) (c) Figure B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Test results for reff (top left), N (top right) and the transit spectrum (bottom) with different sedimentation efficiency fsed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' The cloud layer shrinks with the increasing of fsed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Cox, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2015, Allen’s Astrophysical Quantities (Springer) Cubillos, P.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2016, A&A, 594, A69, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='1051/0004-6361/201628799 Edwards, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Rice, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Zingales, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2019, Experimental Astronomy, 47, 29 Edwards, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Changeat, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Tsiaras, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='00649.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 1984, Journal of Computational Physics, 54, 508, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='1016/0021-9991(84)90130-X Fortney, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Barstow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', & Madhusudhan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2021, in ExoFrontiers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Big Questions in Exoplanetary Science, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Madhusudhan, 17–1, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='1088/2514-3433/abfa8fch17 Gandhi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', & Madhusudhan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2018, MNRAS, 474, 271, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='1093/mnras/stx2748 Gao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', Marley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=', & Ackerman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 2018, The Astrophysical Journal, 855, 86 10-6 10-5 10-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Pressure (bar) 10 0-3 10-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' 10-1 100 1011 10-6 10-4 10-2 100 102 Cloud particle radius (rc, μm) fsed = 10-6 fsed = 10-2 fsed = 3 fsed = 10 fsed = 10010-6 10-5 10-4 Pressure (bar) 10 0-3 10-2 10-1 100 1011 10-2 103 108 1013 1018 1023 1028 Cloud particle number density (N) fsed = 10-2 fsed = 10-6 fsed = 3 fsed = 10 fsed = 1002800 2750 Transit depth (ppm) 2700 2650 2600 2550 2500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='5 10 Wavelength (μm) fsed = 10-6 All contribution fsed = 3 All contribution fsed = 10 All contribution fsed = 100 All contribution fsed = 10-6 H20 cloud fsed = 10-2 H20 cloud fsed = 3 H20 cloud fsed = 10 H20 cloud fsed = 100 H20 cloud22 Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' log(P1) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='27+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='06 195 198 201 204 T1 T1 = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='14+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='625 log(H2O) log(H2O) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='69+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='75 log(fsed) log(fsed) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='49+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='55 log(P1) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='0 (derived) 195 198 201 204 T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='625 log(H2O) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='75 log(fsed) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='0 (derived) (derived) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='48+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content='05 Figure B5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Example of numerical instability in cloud formation simulations when large relative (rtol) and absolute (atol) tolerances are chosen in solving equation (2) with Runge-Kutta method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Blue plots: rtol = 1 × 10−8 and atol = 1 × 10−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' Red plots: rtol = 1 × 10−12 and atol = 1 × 10−15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btFST4oBgHgl3EQfCziU/content/2301.13708v1.pdf'} +page_content=' In these tests, the retrieval performances are affected by numerical instability, as illustrated in this figure where multiple islands of solutions in the posteriors are clearly visible.' 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b/c9FQT4oBgHgl3EQfizaw/content/tmp_files/2301.13351v1.pdf.txt @@ -0,0 +1,1794 @@ +A FAST ALGEBRAIC MULTIGRID SOLVER AND ACCURATE +DISCRETIZATION FOR HIGHLY ANISOTROPIC HEAT FLUX I: +OPEN FIELD LINES ∗ +GOLO A. WIMMER†, BEN S. SOUTHWORTH†, THOMAS J. GREGORY‡, AND +XIANZHU TANG† +Abstract. +We present a novel solver technique for the anisotropic heat flux equation, aimed +at the high level of anisotropy seen in magnetic confinement fusion plasmas. Such problems pose +two major challenges: (i) discretization accuracy and (ii) efficient implicit linear solvers. +We si- +multaneously address each of these challenges by constructing a new finite element discretization +with excellent accuracy properties, tailored to a novel solver approach based on algebraic multigrid +(AMG) methods designed for advective operators. We pose the problem in a mixed formulation, +introducing the heat flux as an auxiliary variable and discretizing the temperature and auxiliary +fields in a discontinuous Galerkin space. The resulting block matrix system is then reordered and +solved using an approach in which two advection operators are inverted using AMG solvers based on +approximate ideal restriction (AIR), which is particularly efficient for upwind discontinuous Galerkin +discretizations of advection. To ensure that the advection operators are non-singular, in this paper +we restrict ourselves to considering open (acyclic) magnetic field lines. We demonstrate the proposed +discretization’s superior accuracy over other discretizations of anisotropic heat flux, achieving error +1000× smaller for anisotropy ratio of 109, while also demonstrating fast convergence of the proposed +iterative solver in highly anisotropic regimes where other diffusion-based AMG methods fail. +Key words. Anisotropic heat flux, anisotropic diffusion, auxiliary operator, algebraic multigrid, +AIR +1. Introduction. The heat flux of plasmas in magnetic confinement fusion ex- +hibits strong anisotropy [15]. Heat can be conducted at a rate many orders of magni- +tude greater parallel to magnetic field lines than it is perpendicular to the magnetic +field; this ratio +κ∥ +κ⊥ can exceed 1010 in realistic experiments [13]. If a numerical sim- +ulation does not adequately resolve the anisotropy of the heat flux, the resulting +simulated plasma confinement times may be spuriously short. Unfortunately, many +numerical approaches suffer from the large parallel component of heat flux polluting +the perpendicular direction, leading to non-physically large perpendicular heat flux. +In addition, anisotropic heat flux in a magnetized plasma has a very restrictive CFL +condition because of the large κ∥, so for realistic applications an implicit solver is +typically necessary. In the isotropic limit with κ∥ = κ⊥, the problem reduces to the +heat equation, and classical geometric and algebraic multigrid (AMG) solvers [28] +are known to be effective. Conversely, the highly anisotropic regime, κ∥ ≫ κ⊥, is +notoriously difficult for multilevel and iterative solvers, and efficient parallel solvers +for realistic applications remain a largely open question. +One common approach to address pollution of flux in the perpendicular direction +∗Corresponding author email address: gwimmer@lanl.gov +Funding: G. A. W., T. G., and X. T. have been supported by the U.S. Department of Energy +Office of Fusion Energy Sciences and Office of Advanced Scientific Computing Research under the +Tokamak Disruption Simulation (TDS) Scientific Discovery through Advanced Computing (SciDAC) +project, as well as the Base Theory Program, both at Los Alamos National Laboratory (LANL) under +contract No. 89233218CNA000001. B. S. S. was supported by the Laboratory Directed Research +and Development program of Los Alamos National Laboratory as a Nicholas C. Metropolis Fellow +and under project number 20220174ER. The computations have been performed using resources of +the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy +Office of Science User Facility operated under Contract No. DE-AC02-05CH11231. +†Theoretical Division, Los Alamos National Laboratory, USA. +‡Imperial College London +1 +arXiv:2301.13351v1 [math.NA] 31 Jan 2023 + +is to make use of meshes aligned with the magnetic field lines [8, 14, 15]. However, +this can only apply for configurations in which the magnetic field lines form nested +flux surfaces, which is not the case, for instance, for tokamak disruptions that can +have volume-filling stochastic magnetic field lines [10]. Mesh-alignment also places +limitations on software and algorithm flexibility. +Several other avenues have been +explored focusing on increased accuracy of heat flux discretizations, thereby allowing +for general, non-field-aligned meshes. Recently, authors have considered higher order +discretizations [12] and adaptive mesh refinement [29], where the mesh is refined +specifically in areas where large error due to anisotropy is expected. One can also make +use of an auxiliary variable, explicitly resolving the heat flux along magnetic field lines +as a variable in its own right. This formulation has been shown to better resolve the +heat flux after discretization, reducing the aforementioned parallel-to-perpendicular +cross-contamination. One such auxiliary variable implementation, provided by Gunter +et al. [13], uses a mixed finite element method, with the temperature field defined on +a continuous Lagrange element of order n, and the auxiliary heat flux variable on a +discontinuous Lagrange element of order n − 1. +For large sparse matrices that arise in finite element discretizations, almost all fast +and parallelizable solvers are iterative and multilevel in nature. The basic idea is to +combine a relatively cheap procedure to attenuate error on one “level” in the hierarchy +(“relaxation”), with a coarse-representation that represents the complementary error +in a smaller solution space (“coarse-grid correction”). Relaxation is typically a local +procedure, such as Jacobi, Gauss-Seidel, or (overlapping) Schwarz, which attenuates +high-frequency error in a geometric sense, and error associated with large eigenvalues +in a spectral sense. The fundamental difficulty with developing fast solvers for strongly +anisotropic diffusion is that there exists a large set of eigenmodes that are smooth +in the parallel direction and high frequency in the perpendicular direction, which +have small eigenvalues. +Such modes cannot be effectively attenuated by standard +local relaxation procedures due to the small associated eigenvalues, and are equally +difficult to represent in a coarse solution space due to the high frequency character in +the perpendicular direction. +One solution to the aforementioned problems is line-relaxation, where one does +a block relaxation aligned with one dimension. E.g., in 2D, for each discrete point +in the x-dimension, one solves the fully coupled problem in the y-dimension. If the +mesh is aligned with the anisotropy, one can perform line relaxation in the direction +of anisotropy; for non-grid aligned anisotropy, one can use alternating line relax in +grid-aligned directions, e.g., in x and y in 2D, or attempt to find an ordering of mesh +points that is approximately aligned with the anisotropy. In 3D, things become more +complicated, typically requiring plane relaxation. Line relaxation was a key compo- +nent of the multilevel solvers recently used in [12] for anisotropic heat flux in plasma. +The downsides of line relaxation are its high cost, need for structured grids to define +lines/planes on, preferably aligned with anisotropy, and in particular its poor perfor- +mance in parallel. A number of papers have also tried to address anisotropic diffusion +without line relaxation through algebraic solvers, primarily variations in AMG (e.g., +see [4, 6, 11, 20, 25]). Although there has been some success in this regard, there +is a disconnect between “hard” anisotropic problems considered in solvers papers, +and accurate discretizations of the anisotropic heat flux that actually arises in, e.g., +magnetic confinement fusion problems. +This paper conjointly addresses the challenges in discretization accuracy and +solver efficiency for strongly anisotropic heat flux with open field lines. Following [13], +we pose the anisotropic heat flux in mixed form using an auxiliary variable, which +2 + +offers superior accuracy in the strongly anisotropic regime compared with direct dis- +cretizations of the second-order diffusion operator. This formulation also uncovers the +purely anisotropic diffusion operator as a combination of two linear advection opera- +tors. The 2 × 2 block system resulting from a mixed discretization typically has mass +matrices and the isotropic diffusion term on the diagonal, with the advection oper- +ators in the off-diagonal blocks. Although robust multilevel methods for (physically +realistic) anisotropic diffusion remains a largely open question, significant progress +has been recently made developing fast and robust AMG methods for linear advec- +tion based on an approximate ideal restriction (AIR) [19, 21]. +Thus, rather than +follow the traditional approach approximating a Schur complement that represents +the anisotropic diffusion operator, we instead swap the rows of the block system so +that the advection operators are on the diagonal, and apply a block triangular pre- +conditioning, with the diagonal advection blocks inverted using AIR-AMG. Given the +transport point of view of this solver scheme, it is well-constructed to deal with the +anisotropy of the problem: the higher the anisotropy, the more efficient this approach +is compared to classical ones. +The remainder of this work is structured as follows: in Section 2 we introduce the +problem and motivate our solver strategy. The proposed solver strategy has several +implications on the discretization that must be taken into consideration. In particular, +the advection blocks need to be square and invertible, which requires care in choice +of finite element space, enforcement of boundary conditions, and constraints on the +anisotropic field lines. In addition, we must choose a discretization that AIR is an +effective solver for – as for many problems, some FEM formulations are much easier to +solve than others. Section 3 describes a new DG-upwind-based spatial discretization +of the auxiliary formulation that conforms with each of these constraints. The full +solver strategy is then introduced in Section 4, including a spectral analysis of the +(advection-)preconditioned Schur complement. +Numerical results are presented in +Section 5. From an accuracy perspective, the new mixed DG discretization is shown +to significantly outperform a non-mixed CG and DG formulation, as well as the mixed +CG system developed in [13]. +In addition, in the regime of large anisotropy, the +proposed solver strategy is shown to be fast in terms of iteration counts and wall- +clock times, whereas other classical diffusion-based AMG schemes demonstrate very +poor scaling or simply fail to converge. Finally, in Section 6, we review our results +and discuss future work. +2. Motivation. +2.1. Anisotropic Heat Transport. Following Gunter et al. [13], we formulate +the evolution of temperature T within a plasma through use of a heat flux q, with +forcing S: +∂T +∂t − ∇ · q = S, +(2.1a) +q = κ∥∇∥T + κ⊥∇⊥T, +(2.1b) +over some bounded domain Ω, where κ∥ denotes the conductivity parallel to magnetic +field lines and κ∥ denotes the conductivity perpendicular to field lines. In this work, +we will consider pure Dirichlet boundary conditions for T: +(2.2) +T|∂Ω = TBC. +In magnetic confinement fusion applications, the conductivities may depend on the +magnetic field and the temperature. For implicit discretizations, the resulting non- +3 + +linear problem can be solved, for example, with a quasi-Newton iteration scheme, +using the given iteration’s known value for T in the conductivities. For simplicity, +in what follows, we assume constant conductivities and note that the framework fol- +lows equally for non-constant ones (up to the required nonlinear iterative procedure). +Similarly, ∇∥ and ∇⊥ are the components of the gradient along and orthogonal to +magnetic field lines, respectively: +(2.3) +∇∥(·) := b · ∇(·))b, +∇⊥ := ∇ − ∇∥, +for b = B/|B|, where B is the magnetic field along which the anisotropy is defined. +Heat flux in the context of magnetic confinement fusion is highly anisotropic, diffusing +significantly faster along magnetic field lines as opposed to perpendicular to them. As +mentioned in the introduction, we can expect to see +κ∥ +κ⊥ ∼ 1010 and larger. Next to a +split of the heat flux into a parallel and perpendicular component, it is also possible +to do a split into an isotropic part, and a remaining purely anisotropic (parallel) +part. For this purpose, we define a purely anisotropic conductivity as κ∆ := κ∥ − κ⊥. +Diffusion in the plasma is isotropic when κ∆ = 0, and is predominately anisotropic if +κ∆ ≫ κ⊥. Defining κ∆ in this way allows us to represent the heat flux as +(2.4) +q = κ∆∇∥T + κ⊥∇T. +Considering the directional derivative b · ∇T as an auxiliary variable, we may rewrite +(2.1) as +∂T +∂t − √κ∆∇ · (bζ) − ∇ · (κ⊥∇T) = S, +(2.5a) +ζ = √κ∆b · ∇T, +(2.5b) +where we split up the factor of κ∆ evenly between equations (2.5a) and (2.5b) in +view of the solver strategy to be presented below. In this form, the PDE system has +a more conventional advection-diffusion character. Gunter et al. [13] demonstrated +that this mixed system – solved for T and ζ – may lead to higher accuracy than a +single-variable model which solves for T only, by resolving the parallel heat flux term +ζ more accurately. +Finally, we note the equation’s two limiting cases in order to motivate the solver +strategy presented in this work: for κ∥ = κ⊥, we find that κ∆ = 0 and the system +reduces to a (forced) heat equation given by +∂T +∂t = ∇ · (κ⊥∇T) + S, +(2.6) +and we can expect classical multigrid methods to perform well when solving implicit +discretizations of (2.6). However, for κ⊥ = 0, we instead have a purely anisotropic +system of equations +∂T +∂t − √κ∆∇ · (bζ) = S, +(2.7a) +ζ = √κ∆b · ∇T, +(2.7b) +in which the dynamics are governed by two transport operators. +4 + +2.2. Solver considerations. Written in block operator form, (2.5) can be for- +mulated as +(2.8) +� +∂ +∂t + κ⊥∆ +−√κ∆∇ · +� +b(·) +� +−√κ∆b · ∇ +I +� �T +ζ +� += +�S +0 +� +, +where ∆ denotes the Laplacian, and I the identity operator. Upon discretizing in +space with a suitable finite element method and applying implicit integration in time +(including diagonally implicit Runge Kutta or linear multistep methods), the resulting +discrete linear systems may take the form +(2.9) +� 1 +∆tMT + κ⊥L +√κ∆GT +b +−√κ∆Gb +Mζ +� �T n+1 +ζn+1 +� += +�FT +Fζ +� +, +where MT and Mζ are mass matrices, L denotes the discrete Laplacian, Gb the discrete +scalar-form transport operator, and FT , Fζ correspond to forcing terms as well as +terms on the known time level n. +Consider the limiting case of a steady-state (i.e. no time derivative) and purely +anisotropic problem (κ⊥ = 0). Plugging this value of κ⊥ into (2.9) and swapping +the block rows in the corresponding equations yields the equivalent block-triangular +system +(2.10) +�−√κ∆Gb +Mζ +0 +√κ∆GT +b +� �T n+1 +ζn+1 +� += +�Fζ +FT +� +. +Here, we have decoupled the implicit purely anisotropic diffusion equation into the +solution of two successive linear advection equations. Although it may seem unusual +to take a parabolic problem and reformulate in terms of hyperbolic operators, in +this case there is a fundamental lack of efficient, parallelizable iterative solvers for the +parabolic purely anisotropic setting, while there do exist fast AMG methods for linear +advection [19, 21]. Note that we opted for a multiplicatively even split of κ∆ between +Gb and GT +b in view of future work considering temperature dependent conductivities. +While the split is not necessary for the constant values of κ∆ considered here, it may +avoid that one solve is more challenging than the other when such a temperature +dependency is included. +The above limiting case motivates the more general solver strategy proposed in +this paper (and detailed in Section 4), focused on inverting transport operators rather +than directly trying to solve the highly anisotropic diffusion equations. For this frame- +work to be viable, we must construct the transport operator to be invertible, which +can be achieved by (i) choosing the same finite element space for T and ζ, and (ii) +assuming open magnetic field lines throughout the domain, that is, the magnetic field +is acyclic. +The former ensures that Gb is square, and the latter ensures that the +operator is nonsingular – in contrast, if the magnetic field has closed field lines, then +constant values of T along any closed field line constitute a non-trivial kernel. In addi- +tion, we must choose a transport discretization that is amenable to the fast reduction +based AMG solver, AIR, developed in [19, 21]. Because AIR is reduction based, it is +most effective on discretizations with low matrix connectivity, a property particuarly +enjoyed by upwind discontinuous Galerkin discretizations, where matrix connectivity +only occurs to immediately adjacent elements that are upwind with respect to the +magnetic field. The following section constructs a suitable discretization, and the +solver strategy is expanded in Section 4. +5 + +3. Finite element discretization. To solve the anisotropic diffusion system +(2.5), we discretize the transport term +(3.1) +∇ · +�√κ∆b(·) +� +, +using the classical DG-upwind method (see e.g. [16]). For this purpose, we start with +an operator of the form +Lb(θ; φ) := − ⟨√κ∆θb, ∇φ⟩ + +� +Γ +�√κ∆φb · n�˜θ dS + +� +∂Ωout +√κ∆φb · nθ dS, +(3.2) +for L2-inner product ⟨·, ·⟩, and any functions θ, φ in the given kth polynomial order +DG space VDG +k +(Ω), which we will specify further below in this section. The set ∂Ωout +denotes the outflow subset of the boundary ∂Ω relative to B. Further, Γ denotes the +set of all interior facets of the mesh, and we applied jump and upwind facet operations +defined by +(3.3) +�ψ� := ψ+ − ψ−, +˜ψ := +� +ψ+ if b+ · n+ < 0, +ψ− otherwise, +for any scalar field ψ. n denotes the facet normal vector, and the two sides of each +mesh facet are arbitrarily denoted by + and − (and hence n+ = −n−). Given (3.2), +we then define two transport operators according to +Lb,T (ζh; φ) := Lb(ζh; φ) + +� +∂Ωin +√κ∆φ(b · n)ζin dS +∀φ ∈ VDG +k +, +(3.4a) +Lb,ζ(ψ; Th) := Lb(ψ; Th) − +� +∂Ωout +√κ∆TBC(b · n)ψ dS +∀ψ ∈ VDG +k +, +(3.4b) +where ∂Ωin denotes the inflow part of the boundary, defined analogously to ∂Ωout, +and ζin denotes a known value to be specified at the end of this section. Note that this +setup of known inflow boundary integrals ensures that Lb,T and Lb,ζ are non-singular +for problems where all magnetic field lines are open. An intuition for this choice of +operators and boundary integrals is given in Appendix A. +Given these operators, we can formulate the system to be solved as finding +(Th, ζh) ∈ (VDG +k +× VDG +k +) such that +� +φ, ∂Th +∂t +� +− Lb,T (ζh; φ) − IP(Th; φ) += − +� +∂Ω +κBCφ(Th − TBC) dS + ⟨φ, S⟩ +∀φ ∈ VDG +k +, +(3.5a) +⟨ψ, ζh⟩ + Lb,ζ(ψ; Th) = 0 +∀ψ ∈ VDG +k +, +(3.5b) +for penalty parameter κBC, and where IP(Th; φ) is the discretized Laplacian using an +interior penalty method (see e.g. [5]): +IP(Th; φ) = − ⟨∇φ, κ⊥∇Th⟩ + +� +Γ +�Th� {κ⊥∇φ} dS + +� +Γ +�φ� {κ⊥∇Th} dS ++ +� +∂Ω +κ⊥n · ∇φ(Th − TBC) dS+ +� +∂Ω +κ⊥n · ∇Thφ dS− +� +Γ +κ⊥κp +he +�φ� �Th� dS, +(3.6) +6 + +for any test function φ ∈ VDG +k +, penalty parameter κp, and where he denotes the +given facet’s length scale. For the numerical results below, we used as a length scale +|K+|+|K−| +2|∂K| +, where |K±| denote the facet’s adjacent cell’s volumes, and |∂K| denotes +the facet area. Further, the average operation is given by +(3.7) +{u} := 1 +2 +� +u+ + u−� +· n+, +for any vector field u. +Finally, the non-dimensional interior penalty parameter in +IP(Th; φ) is set to κp = 2 in the numerical results section below. The discussion for +the penalty parameter κBC for exterior facets (first term on right-hand side of (3.5a)) +is more intricate and we postpone it to the end of this section. +In the numerical results section below, we consider 3D extruded meshes based +on prism cells. Such cells are of interest in magnetic confinement fusion applications, +where tokamak meshes may be created starting from a triangular mesh for the poloidal +plane (i.e. a vertical cross section), which is then extruded to create a 3D mesh. While +the extrusion would be curved in the context of tokamak meshes, here we restrict +ourselves to straight extrusions for simplicity1. +We then set VDG +k +(Ω) = VDG +2 +(Ω) +to the corresponding second polynomial order discontinuous Galerkin space. +For +prisms, the latter space can be constructed via a finite element which is given by +the tensor product of the standard DG finite element dP2 for triangles, together with +the analogous one for intervals. Further, we discretize B in the second order div- +conforming Nedelec space Nc2f. There is no specific reason to this choice of finite +element space other than related work on magnetic confinement fusion discretizations +by the authors. With regards to the choice of magnetic field space, we generally found +the scheme’s accuracy to be only significantly affected by the latter space’s polynomial +degree, which should be at least equal to the temperature space’s polynomial degree. +Remark 1. Structural properties. The non-discretized anisotropic diffusion +operator in (2.5) acts to diffuse temperature and may create a heat flux across the +domain’s boundary in the direction of the magnetic field B. This can easily be seen +by taking the L2-inner product of (2.5) with T, and setting κ⊥ = 0. After integrating +by parts, this yields +(3.8) +1 +2 +d +dt∥T∥2 +2 = −∥ζ∥2 +2 + +� +∂Ω +√κ∆ζb · nTBC dS, +up to contributions due to the forcing S. Further, the anisotropic transport operator +does not impact the system’s total energy other than at the boundary. This can by +seen by integrating equation (2.5) (with κ⊥ = 0) and applying integration by parts +again. We then have +(3.9) +d +dt +� +Ω +T dx = +� +∂Ω +√κ∆ζb · n dS, +where in this context, the integral over T plays the role of total energy. +Similarly, for the newly introduced DG discretization (3.5) with the anisotropic +part only (with κ⊥ = 0), we can set φ = Th in (3.5a) and further ψ = ζh in (3.5b), +1In exploratory numerical tests, we found our framework to also work in curved meshes, albeit +with an additional error that depends on the finite element space and mesh degrees in the direction +of the curved extrusion. +7 + +which leads to +1 +2 +d +dt∥Th∥2 +2 = − ∥ζh∥2 +2 − +� +∂Ω +κBCTh(Th − TBC) dS ++ +� +∂Ωin +√κ∆ζin(b · n)Th dS + +� +∂Ωout +√κ∆ζh(b · n)TBC dS, +(3.10) +where we substituted Lb(ζh, Th) from Lb,T (ζh, Th) in (3.5a) by the corresponding term +from (3.5b). This behavior is consistent in the sense that when Th is equal to the +strong solution T, and ζin is set to the strong directional gradient √κ∆b · ∇T, these +terms are equal to the right-hand side of (3.8). Note that the correct discrete diffusive +behavior (3.10) may initially seem counterintuitive, since we obtained it by combining +two transport operators that apply upwinding in the same direction (along b). This can +be resolved by observing that while we upwind the trial function for Lb,T in (3.4a), we +instead upwind the test function for Lb,ζ in (3.4b). The latter in turn can be considered +as downwinding the trial function. Altogether, the upwinded and downwinded trial +functions then combine to the above diffusive behavior. +Finally, to obtain a discrete version of (3.9), we set φ = 1 in (3.5a), which leads +to +d +dt +� +Ω +Th dx = +� +∂Ωout +√κ∆ζh(b · n) dS + +� +∂Ωin +√κ∆ζin(b · n) dS +(3.11) +− +� +∂Ω +κBC(Th − TBC) dS + +� +Γ +�√κ∆b · n�˜ζh dS. +For the boundary integrals, as before this is consistent with the non-discretized version +(3.9) when Th is equal to the strong solution T. +However, this time, there is an +additional interior facet integral term (last term in (3.11)), which need not vanish +since b = B/|B| need not have continuous normal components, even if B is a div- +conforming finite element field (i.e. has continuous normal components across facets). +However, as for the boundary integrals, the term is weakly consistent in the sense that +for a strong solution with a continuous field B, the term vanishes. +We end this section with a brief description of the time discretization to be used in +the numerical results section, as well as terminology for the next section, the choice of +ζin, and a discussion on the penalty terms. The space discretization (3.5) is coupled to +an implicit midpoint rule. Given the fully discretized scheme, we denote the operator +Lb in matrix form as −√κ∆Gb, where the change in sign and factorization of √κ∆ +are to aid the transport solver discussion below. Similarly, we denote the discrete +Laplacian operator IP in matrix form as −κ⊥L. Further, we denote the VDG +2 +mass +matrix by M. Finally, for the implicit system, the boundary penalty term (first term +on righ-hand side of (3.5a)) leads to a mass matrix defined along the boundary only, +and we denote the latter by MBC. +The need for specifying ζin as a value independent of the unknown ζh to be +solved for is a shortcoming of the proposed space discretized formulation (3.5), which +arises from employing DG-upwinded transport operators for both Th and ζh. +We +therefore require known inflow boundary values for both fields for the operators to +be invertible, while only values for either Th (Dirichlet) or ζh (Neumann) are given. +One possible way to circumvent this is to set ζin = √κ∆b · ∇ ¯Th, where ¯Th is the +midpoint-in-time discretization of Th. In particular, this is an expression independent +8 + +of ζh, and therefore does not change the resulting transport block −√κ∆Gb which +we are required to invert. While we found this approach to work well in terms of +solution accuracy, it adds a contribution with overall factor κ∆ to the top-left block +of the matrix system (2.9), thereby rendering our transport operator based solver +motivation (2.10) invalid. Instead, we initially set the inflow flux ζin equal to the +projection of √κ∆b · ∇T|t=0 into VDG +2 +, and subsequently set it equal to the most +recent solution for ζh of the resulting mixed system. In other words, ζin can be seen +as an explicit time discretization of ζh. Generally, we found this setup to be stable, +likely due to the interior penalty term that weakly enforces the Dirichlet boundary +conditions. We further found the resulting heat flux error at the inflow boundary +generally to be small, including test cases where the heat flux varies in time at the +latter boundary. In practice, this shortcoming can likely further be remedied at no +additional cost by incorporating updates to the value of ζin in an outer nonlinear +iterative procedure. As mentioned in Subsection 2.1, the latter procedure is needed +in magnetic confinement fusion simulations where the conductivities κ⊥, κ∥ depend +on the temperature Th. +Given this choice of ζin, both additional boundary integrals in (3.4) are absorbed +into the right-hand side vectors FT and Fζ, respectively, leading to an overall discrete +2 × 2 block system of the form +(3.12) +� 1 +∆tM + κ⊥L + MBC +√κ∆GT +b +−√κ∆Gb +M +� �T n+1 +h +ζn+1 +h +� += +�FT +Fζ +� +, +whose structure is equal to (2.9) described in our motivation section Subsection 2.2, up +to the additional boundary penalty term. The latter term can be considered as part +of the discrete Laplacian L, in which case the correct penalty parameter formulation +is +(3.13) +κBC = ˜κBC +κ⊥ +he +, +for some non-dimensional parameter ˜κBC. The top-left block of (3.12) can then be +grouped according to +1 +∆tM + κ⊥(L + ˜κBCMBC,h−1 +e ), where we factored ˜κBCκ⊥ out of +MBC to highlight the matrix grouping. Further, we denote the remaining operator as +MBC,h−1 +e , noting that it contains a factor of h−1 +e . However, this may be impractical +for the anisotropic limit in which κ⊥ may be negligible, thereby not ensuring a strong +enough weak enforcement of the boundary conditions. In particular, the transport +operators Lb,T and Lb,ζ alone are not sufficient to enforce the boundary conditions. +We therefore instead consider MBC independently of L, and use a penalty parameter +of the form +(3.14) +κBC = ˜κBC +he +∆t, +where again ˜κBC is non-dimensional. In the numerical results section below, we set +˜κBC = 20. Given the factor of +1 +∆t, we group MBC together with the mass matrix, +leading to an overall mixed system +(3.15) +� 1 +∆t(M + ˜κBCMBC,he) + κ⊥L +√κ∆GT +b +−√κ∆Gb +M +� �T n+1 +h +ζn+1 +h +� += +�FT +Fζ +� +, +where again we factored ˜κBC/∆t out of MBC to highlight the matrix grouping. This +time, we denote the remaining matrix as MBC,he, since it contains a factor of he. +9 + +4. Transport solvers for anisotropic diffusion. For standard block diagonal, +triangular, and LDU preconditioners applied to block 2 × 2 linear systems, conver- +gence of Krylov and fixed-point iterations are fully defined by the preconditioning +of the Schur complement [26]. In the standard form (2.9) of an auxiliary-variable +discretization of (2.1) or (2.5), the natural (1,1) Schur complement is effectively a +discrete representation of the continuous anisotropic diffusion equation; as discussed +in Section 1, it is very difficult to construct effective preconditioners for such equa- +tions. Building on the discussion in Subsection 2.2, we reorder the discrete 2×2 block +system in (3.15) to take the equivalent form +(4.1) +� +−√κ∆Gb +M +1 +∆t(M + ˜κBCMBC,he) + κ⊥L +√κ∆GT +b +� �T n+1 +ζn+1 +� += +�Fζ +FT +� +. +In doing this reordering, the (2,2) Schur complement defining convergence takes the +form (note, the (1,1) Schur complement is the negative transpose of the (2,2)) +S22 :=√κ∆GT +b + +� 1 +∆t(M + ˜κBCMBC,he) +� +(√κ∆Gb)−1M + κ⊥L(√κ∆Gb)−1M. +(4.2) +Note that MBC,he can be seen as a mass matrix over boundary facets, which is scaled +by a factor of he. Altogether, we expect its eigenspectrum to be akin to that of a mass +matrix over cells. For simplicity, we further assume ˜κBC to be O(1), so that from the +point of view of the spectral analysis to follow, M + ˜κBCMBC,he can be replaced by +M without a qualitative change in the analysis. We then have +(4.3) +S22 ≈ √κ∆GT +b + +� 1 +∆tM + κ⊥L +� +(√κ∆Gb)−1M. +This section provides a spectral analysis on preconditioning (4.2) by inverting the +diagonal transport operator to support the proposed strategy. The analysis is based +on the observation that the preconditioned Schur complement is similar via a simple +mass-matrix similarity transform to the identity plus two symmetric positive definite +(SPD) operators. By nature of this similarity, we expect spectral analysis of the result- +ing SPD operators to provide excellent measures of preconditioning (a property which +is not always the case with advective operators and inverses). The purely anisotropic +case is discussed in Subsection 4.1 and the general case in Subsection 4.2. We then +use a Lanczos algorithm to compute eigenvalue bounds for the preconditioned Schur +complement for realistic problems in Subsection 4.3 and demonstrate consistency with +the theory. Finally, we detail the solver in practice in Subsection 4.4. +4.1. The purely anisotropic case. Recall in this paper we are particularly +interested in the strongly anisotropic regime, where κ∆ ≫ κ⊥. If we consider the +limit of κ⊥ = 0, the approximate Schur complement (4.3) simplifies to +(4.4) +S22 := √κ∆GT +b + +1 +∆tM(√κ∆Gb)−1M. +Now consider preconditioning S22 with the diagonal transport block: +(4.5) +(√κ∆GT +b )−1S22 = I + +1 +∆tκ∆ G−T +b +MG−1 +b M. +Observe that our preconditioned Schur complement is a perturbation of the identity, +and (formally) similar to the SPD operator +(4.6) +M 1/2(√κ∆GT +b )−1S22M −1/2 = I + +1 +∆tκ∆ M 1/2G−T +b +MG−1 +b M 1/2. +10 + +Next, we uncover the eigenvalue structure of the second term on the right-hand side +of (4.5). Observe that M −1GT +b M −1Gb corresponds to a discrete anisotropic diffusion +operator, which is similar to M −1GbM −1GT +b . Moreover, recall that b is a unit vector, +and the eigenvalues of a discrete diffusion operator span a range of [1/c2, 1/c1h2], +for mesh spacing h and O(1) constants c1, c2, independent of h. Thus, the inverse +G−T +b +MG−1 +b M that appears in (4.5), and by similarity the second term of (4.6), is +an SPD operator with eigenvalues in the range [c1h2, c2]. Altogether, this implies +that the preconditioned Schur complement (4.5) has real-valued eigenvalues that are +bounded below by one and lie in the range +(4.7) +σ +� +(√κ∆GT +b )−1S22 +� +⊂ 1 + +1 +∆tκ∆ [c1h2, c2]. +In particular, note that for reasonable values of ∆t and particularly for large κ∆, we +achieve excellent preconditioning of the Schur complement by simply inverting the +diagonal advection blocks, largely independent of any additional constants scalings +that depend on discretization. +Of course, in the limit of a steady-state problem, +∆t → ∞, we have exactness, as used to motivate the solver strategy in Subsection 2.2. +4.2. The general case. For κ⊥ > 0, the analysis above largely holds, but we +have additional perturbations in our preconditioned Schur complement, where +(√κ∆GT +b )−1S22 = I + +1 +∆tκ∆ G−T +b +MG−1 +b M + κ⊥ +κ∆ G−T +b +LG−1 +b M. +(4.8) +As before, the above operator is similar to an SPD operator, this time given by +M 1/2(√κ∆GT +b )−1S22M −1/2 = I + +1 +∆tκ∆ M 1/2G−T +b +MG−1 +b M 1/2 ++ κ⊥ +κ∆ M 1/2G−T +b +LG−1 +b M 1/2. +(4.9) +Note that both of the perturbation terms in (4.9) are SPD, and thus the precondi- +tioned Schur complement again has positive real-valued eigenvalues, bounded below +by one. As done already for the second-to-last term in (4.8), we aim to establish a +connection of the last term with a diffusion operator. Observe that by similarity, we +have +(4.10) +G−T +b +LG−1 +b M ∼ G−1 +b MG−T +b +L = +� +G−1 +b MG−T +b +M +�� +M −1L +� +, +where the latter term, M −1L, corresponds to a discrete representation of the isotropic +diffusion operator, and the first term, G−1 +b MG−T +b +M, corresponds to the inverse of the +anisotropic diffusion operator as discussed in the previous section. To understand the +product of these operators, suppose we are in 2D and b is grid-aligned, say in the +x-dimension. Then (4.10) is a discrete representation of ∂−1 +xx (∂xx +∂yy) = 1+∂−1 +xx ∂yy. +On a unit domain with zero Dirichlet boundary conditions, the eigenfunctions and +eigenvalues of the Laplacian are {ujk, π2(j2 + k2)}, where ujk = 2 sin(jπx) sin(kπy), +and similarly for ∂xx only in the (j, x) dimensions. Thus, if we consider the high- +est frequency mode represented by our grid in the y-dimension, k = 1/h, and the +smoothest mode in the x-dimension, j = 1, we have the eigenpair +(4.11) +(1 + ∂−1 +xx ∂yy)2 sin(πx) sin( 1 +hπy) = +� +1 + +1 +h2 +� +2 sin(πx) sin( 1 +hπy). +A complete analysis generalized to specific FEM discretizations, non-unit-square do- +mains, more complex boundary conditions, or non-grid aligned magnetic field b is +11 + +a more complex endeavor and outside the scope of this work. However, a similar +result will still hold, namely that modes that are highly oscillatory orthogonal to b +and smooth in the direction of b will typically scale like 1 + c3/h2, with the c3/h2 +arising due to oscillatory functions that are not damped by the anisotropic inverse. +Altogether, we then have an estimate for the spectrum contained in the real-valued +subset +(4.12) +σ +� +κ⊥ +κ∆ M 1/2G−T +b +LG−1 +b M 1/2� +⊂ +� +0, κ⊥c3 +κ∆h2 +� +, +for some constant c3. +To conclude, the analysis in the previous two subsections indicates that for suffi- +ciently small ratios of +(4.13) +c1 +∆tκ∆ +⪅ 1, +κ⊥c3 +κ∆h2 ⪅ 1, +for constants c1, c3 independent of ∆t, κ∆, h, we expect excellent preconditioning of +the Schur complement. Numerical results in Section 5 confirm these observations in +practice, with rapid convergence obtained in solving (4.1). On the other hand, for +more isotropic regimes O(κ∥) ≈ O(κ⊥), and for setups with smaller time steps or κ∆ +such that O(∆t) ≈ O(κ−1 +∆ ), we expect our solver performance to deteriorate since +the transport operator alone will no longer be a good preconditioner for the Schur +complement. Future work will consider more general methods that are also robust in +the isotropic or mixed regimes. However, we note that in strongly magnetized plasmas, +we are primarily concerned with extreme anisotropies for which the methods proposed +here are robust. +Finally, in terms of the time step and therefore the first ratio in (4.13), we set +∆t = 10−3 for the numerical tests in Section 5. In comparison, the local diffusive time +scale h2/κ∥ – which is an important quantity in tokamak applications – lies in an ap- +proximate range h2/κ∥ ∈ [10−4, 10−13] for the resolutions and parallel conductivities +considered below. However, the latter scale is not the only factor determining the +choice of time step and in extension the scheme’s temporal accuracy. Other factors +include the problem’s nonlinearity in the case of T-dependent conductivities and the +temperature profile. Overall, we found our fixed choice of time step to work well in +terms of accuracy for the test cases considered below. +4.3. Eigenvalue bounds in practice. To further investigate the parameter +regimes (4.13), we include a brief numerical study to analyze the largest eigenvalue of +the components of the preconditioned Schur complement (√κ∆GT +b )−1S22, with S22 +given by (4.2). In the following, we denote the second-to-last term in (4.2) by S22,M, +and the last one by S22,L. We have tested each operator using the mesh described +in Appendix B, as well as a periodically extruded 3D version thereof. We consider +3 levels of mesh refinement, with 2 extruded cells in 3D for the lowest refinement. +In each refinement, the resolution is doubled in each dimension. For each resolution, +we tested the behavior for differing choices of time step ∆t ∈ [10−3, 10−2] and +κ∥ +κ⊥ ∈ +[103, 106, 109], with κ⊥ = 1. Finally, the magnetic field B is defined as the curved +and slanted one used in Subsection 5.4 below. +The eigenvalues are computed by +implementing the preconditioned Schur complement’s components as actions applied +to an input vector xin. The actions are then passed to an implicitly restarted Lanczos +method [17] as implemented in SciPy2. The resulting largest eigenvalues for the 2D +2The number of Lanczos vectors is capped at 10 for the highest resolution 3D, together with +a relative tolerance of 10% for computational speedup, noting that the latter resolution leads to +12 + +∆t +κ∥/κ⊥ +(√κ∆GT +b )−1S22,M +(√κ∆GT +b )−1S22,L +(√κ∆GT +b )−1S22 − I +103 +7.8e0 +4.6e0 +2.9e0 +5.6e-1 +1.4e0 +3.7e0 +7.8e0 +4.7e0 +4.3e0 +10−3 +106 +7.8e-3 4.6e-3 2.9e-3 +5.6e-4 1.4e-3 3.7e-3 +7.8e-3 4.7e-3 4.3e-3 +109 +7.8e-6 4.6e-6 2.9e-6 +5.6e-7 1.4e-6 3.7e-6 +7.8e-6 4.7e-6 4.3e-6 +103 +7.8e-1 4.6e-1 2.9e-1 +5.6e-1 +1.4e0 +3.7e0 +8.0e-1 +1.5e0 +3.7e0 +10−2 +106 +7.8e-4 4.6e-4 2.9e-4 +5.6e-4 1.4e-3 3.7e-3 +8.0e-4 1.5e-3 3.7e-3 +109 +7.8e-7 4.6e-7 2.9e-7 +5.6e-7 1.4e-6 3.7e-6 +8.0e-7 1.5e-6 3.7e-6 +Table 4.1 +Largest eigenvalues of preconditioned Schur complement components for 2D mesh. S22,M and +S22,L correspond to second and third terms on right-hand side of (4.2), respectively. +The three +values for each given anisotropy ratio correspond to the three spatial refinement levels. +∆t +κ∥/κ⊥ +(√κ∆GT +b )−1S22,M +(√κ∆GT +b )−1S22,L +(√κ∆GT +b )−1S22 − I +103 +6.6e2 +3.9e2 +2.4e2 +3.6e1 +1.0e2 +2.7e2 +6.6e2 +4.0e2 +3.4e2 +10−3 +106 +6.6e-1 3.9e-1 2.4e-1 +3.6e-2 1.0e-1 2.7e-1 +6.6e-1 4.0e-1 3.4e-1 +109 +6.6e-4 3.9e-4 2.4e-4 +3.6e-5 1.0e-4 2.7e-4 +6.6e4 +4.0e-2 3.4e-4 +103 +6.6e1 +3.9e1 +2.4e1 +3.6e1 +1.0e2 +2.7e2 +6.6e1 +1.1e2 +2.8e2 +10−2 +106 +6.6e-2 3.9e-2 2.4e-2 +3.6e-2 1.0e-1 2.7e-1 +6.6e-2 1.1e-1 2.8e-1 +109 +6.6e-5 3.9e-5 2.4e-5 +3.6e-5 1.0e-4 2.7e-4 +6.6e-5 1.1e-4 2.8e-4 +Table 4.2 +Largest eigenvalues of preconditioned Schur complement components for 3D mesh. S22,M and +S22,L correspond to second and third terms on right-hand side of (4.2), respectively. +The three +values for each given anisotropy ratio correspond to the three spatial refinement levels. +and 3D setups are given in Tables 4.1 and 4.2, respectively; a larger leading eigenvalue +will lead to a less efficient solver. +The leading eigenvalues λmax +SM of (√κ∆GT +b )−1S22,M and λmax +SL +of (√κ∆GT +b )−1S22,L +evolve according to their leading factors in (4.13), except increasing the resolution +additionally decreases λmax +SM . Upon further investigation, we found this to be related +to the inclusion of the boundary penalty term MBC in the Schur complement S22. +Without it, the value λmax +SM is independent of the spatial resolution as expected – for +instance, in 2D and for ∆t = 10−3, κ∥/κ⊥ = 103, we get λmax +SM = 0.84 for all three +resolutions. Finally, we found that as we increase the spatial resolution, the largest +eigenvalues that are obtained when including MBC converge to the one when the latter +operator is excluded. +We also observe that either λmax +SM or λmax +SL +may be larger, depending on the choice +of h and ∆t (for fixed conductivities). Additionally, the observed preconditioned Schur +complement’s largest eigenvalues (ignoring the additive factor of 1 from the identity) +are slightly less than the sum λmax +M ++ λmax +L +. Finally, we find both (√κ∆GT +b )−1S22,L +and (√κ∆GT +b )−1S22,M to have considerably larger largest eigenvalues in 3D than in +2D. In further studies including different discretization and initial condition setups, +we found this likely to be due to the degree of the magnetic field’s alignment with +respect to the 3D mesh, rather than discretization factors such as the choice of finite +approximately 4.5 × 105 degrees of freedom. +13 + +element, type of cell and cell aspect ratio. +4.4. The solver in practice. In practice, we apply a block lower triangular +preconditioner of the form +(4.14) +� +−√κ∆Gb +0 +1 +∆t(M + ˜κBCMBC) + κ⊥L +√κ∆GT +b +�−1 +, +as a preconditioner to an outer flexible GMRES Krylov iteration solving the system in +(4.1), with an outer tolerance set to 10−8. We use lower triangular as opposed to upper +triangular so as to directly include the isotropic operator in our preconditioning, even +if is it not being inverted (and experimental results have supported this, with lower +triangular preconditioning converging 10-20% faster than upper triangular). Following +[26], we do not use block LDU preconditioning, noting that it typically provides +minimal improvement in convergence compared with block triangular, at the expense +of an additional (approximate) solve of a diagonal block. Inner solves of (√κ∆Gb)−1 +and (√κ∆Gb)−T are approximated using AIR-AMG [19, 21] as a preconditioner for +right-preconditioned GMRES. We choose an inner relative residual stopping tolerance +of min{10−3, 10−3/∥b∥}, where b is the current right-hand side. This is effectively +enforcing both an absolute and relative residual tolerance. We do this because when +using block preconditioning with inexact Schur complements, it is almost always more +computationally efficient to use approximate rather than exact inner solves (e.g., +Navier Stokes results in [26] suggest a tolerance of about 10−3). +However, when +(∆tκ∆)−1 or κ⊥/κ∆ is not trivially small (in our experience ⪆ 10−4), the right-hand +side provided to the lower triangular solve can be very large, e.g., O(104). In such +cases, we find that a relative stopping tolerance of 10−3 doesn’t even lead to a residual +< 1 in norm, and the outer iteration fails to converge. Thus, for each inner solver we +enforce achieving at least three digits in both absolute and relative residual tolerance. +Block AIR is applied using the PyAMG library [3], with the discretization ma- +trices grouped by DG element blocks. Ruge-Stuben coarsening [24] with second pass +is applied on a block level using a classical strength of connection with tolerance +θC = 0.01 based on a hard minimum, {−aij ≥ θC maxk |aik|}, and calculated on +a block basis as implemented in PyAMG. No pre-relaxation is used and FFC-block- +Jacobi post relaxation is applied with no damping. We use one-point interpolation and +distance-one block AIR [19] with strong connections determined through a classical +strength of connection as above with tolerance θR = 0.25. +5. Numerical results. Having described the scheme and solver strategy, we +next test their properties in terms of convergence and efficiency. To set the scene, we +first describe other discretizations and a standard solver strategy to compare against, +as well as implementation details. +5.1. Comparison to other discretizations and solvers. For the test cases +below, we will consider three additional discretizations. This includes the CG-primal +system (i.e. +without the auxiliary heat flux term) for T in the standard second +polynomial order continuous Galerkin space VCG +2 +. For prisms, the latter is constructed +analogously to the DG case, using the tensor product of the standard CG element P2 +for triangles and the analogous 1D element. In this case, all gradient operations are +well-defined, and the discretization is given by +� +η, ∂Th +∂t +� ++⟨b · ∇η, κ∆b · ∇Th⟩+⟨∇η, κ⊥, ∇Th⟩ = ⟨η, S⟩ +∀η ∈ VCG +2 +. +(5.1) +14 + +Additionally, we consider the VCG +2 +-VDG +1 +mixed finite element formulation based +on Gunter et al. [13], which reads +� +η, ∂Th +∂t +� ++ ⟨b · ∇η, √κ∆ζh⟩ + ⟨∇η, κ⊥∇Th⟩ = ⟨η, S⟩ +∀η ∈ VCG +2 +, +(5.2a) +⟨φ, ζh − √κ∆b · ∇Th⟩ = 0 +∀φ ∈ VDG +1 +. +(5.2b) +Finally, we consider a DG-primal system for T in VDG +2 +. For this purpose, we use a +discretization similar to the conductivity tensor approach in [12, 29], except that we +use a split isotropic-anisotropic formulation of the continuous equations in order to +have a scheme that is equal to the mixed DG formulation up to the treatment of the +purely anisotropic term. The temperature equation is then given by +� +φ, ∂Th +∂t +� +− IPb(Th; φ) − IP(Th; φ) = ⟨φ, S⟩ +∀φ ∈ VDG +2 +, +(5.3) +where the isotropic term IP is discretized according to (3.6), and the anisotropic +term IPb is given by (A.2). The interior penalty parameter for the resulting isotropic +DG-Laplacian is set as for the mixed DG setup, while the one for the anisotropic +DG-Laplacian is set to 10. As before, the above space discretizations are discretized +in time using the implicit midpoint rule. The exception for the inflow boundary term +involving ζin – as discussed at the end of Section 3 – does not apply here, since the +above space discretizations do not contain such a term. +Next to alternative discretizations, we also consider an alternative solver strategy +for the DG and CG mixed systems, which is given by the standard block matrix +form (2.9) with the natural (1,1) Schur complement setup as a precondioner to an +outer flexible GMRES Krylov iteration. The lower right block – which corresponds to +the DG auxiliary variable mass matrix – is solved using a conjugate gradient Krylov +iteration with block Jacobi as a precondioner. The Schur complement is solved for +using a preconditioner ˜S−1 = +� 1 +∆tMT + κ⊥L) + κ∆GT +b [diag(Mζ)]−1Gb +�−1 to an inner +GMRES Krylov iteration, with the preconditioner solved for using BoomerAMG from +hypre [9]. The multigrid options are left to the default ones. The outer and inner +tolerances are set to 10−8 and 10−3, respectively. +5.2. Implementation. The discretization and solver were implemented using +Firedrake [23], which heavily relies on PETSc [2]. A parallel version of AIR is im- +plemented in hypre, which can be called in Firedrake through the PETSc interface. +However, the interface currently does not contain the full set of options available to +tune AIR. As mentioned in Subsection 4.4, we therefore instead call AIR through +PyAMG. The corresponding Python functions are then used in the Firedrake frame- +work via a custom preconditioner interface. The resulting setup can only be used in +serial, and the solver efficiency results below therefore show wall-clock times for serial +runs. +The diffusion-based multigrid based solver strategy described in the previous sub- +section uses a classical AMG solver as implemented in hypre’s BoomerAMG [9]. The +hypre implementation allows for more options and is more geared towards efficiency +than the PyAMG version of classical multigrid; indeed we found the BoomerAMG +implementation to generally require slightly fewer iterations, as well as less wall-clock +time. Note, we also tested several AMG methods implemented in PyAMG specifically +designed for anisotropic diffusion, including [20, 22, 25], but again found the hypre +15 + +classical AMG implementation to consistently win in terms of iteration count and +wall-clock time. +For the convergence results, all systems were solved with a direct solver. DG +based discretizations are solved using mumps [1], as this is less taxing on the memory +(DG spaces have more degrees of freedom than the corresponding CG spaces). For +CG, mumps did not consistently converge at higher anisotropy ratios and higher +resolutions, and so superlu-dist [18] was used instead. For problems where both of +these choices worked, the results were the same up to round-off; the choice of solver +is for computational speedup and does not affect the solution accuracy. +Next to the DG scheme described in Section 3, as well as the above schemes used +for comparison purposes, we also attempted an AIR based strategy for a mixed CG +discretization. However, we found such a scheme difficult to implement in view of the +strong Dirichlet boundary conditions. In typical finite element codes, it is non-trivial +to extract the resulting symmetric transport operators, noting that we need the latter +operators for our block-row swapped solver strategy. Other than the implementation +difficulty, we further thought it justified to skip the mixed CG discretization together +with the AIR framework, since the mixed DG formulation leads to a lower connectivity +than the CG one, which is advantageous for the latter framework. +5.3. Convergence Studies. Here, we test the convergence order of our novel +space discretization, using a variation of a test case considered in [13] and originally +presented in the paper introducing the magnetohydrodynamic model NIMROD [27]. +The original test case from the literature is posed in 2 dimensions; for the purposes +of our problem, we extrude this into 3D in order to avoid having to deal with the +magnetic field singularity (with |B| = 0) present in the 2D problem, which we found +to be problematic for the upwinding scheme. Since such singularities do not occur +in physically realistic scenarios, we do not consider this as a weakness of our scheme. +The domain is described by Ω = [0, Lx] × [0, Lx] × [0, Lz], which is periodic in the +z-direction. We take Lx = 1, Lz = 5. The initial temperature field is given by +(5.4) +T0 = sin +�πx +Lx +� +sin +�πy +Lx +� +, +with TBC = 0 on ∂Ω. The magnetic field B is chosen to align with the contours of T0 +in the two base dimensions and be constant in the third; namely that +(5.5) +B = (Bx, By, Bz) = (−∂yT, ∂xT, 5), +which ensures that (Bx, By) = ∇⊥T0 for 2D curl ∇⊥ = (−∂y, ∂x). Finally, as in the +original test case, we add a counter forcing S given by −κ⊥∆T0, which is included in +order to ensure a steady state test case. The error can then be measured relative to +the initial condition expression (5.4) as +(5.6) +eT (t) = ∥Th(t) − T0∥2 +∥T0∥2 +. +As described in Section 3, in meshing the domain, we use a triangular mesh for +[0, Lx] × [0, Lx], which is then extruded in the third dimension. Since the initial con- +ditions are highly symmetric, we use a perturbed version of a regular triangular mesh. +This breaks any symmetry in the problem due to possible alignments of the initial +conditions with the mesh, to ensure that our convergence study really is analyzing the +effect of the discretization on the error, as opposed to symmetries in the particular +problem chosen. The base mesh is depicted in Figure B.1 in Appendix B. +16 + +Three resolutions are tested, starting from resolution ∆x = Lx/7 for the base +mesh, which is then twice refined by splitting cells. For the resolution in the extruded +direction, we note that due to the setup of the initial conditions, we find that 2 cells +are sufficient to resolve the latter direction, and increasing the number of cells will not +decrease the overall error. To keep the overall computational cost low, we therefore +use 2 cells in the extruded direction for all three base resolutions. Finally, we use a +time step of ∆t = 10−3 for 100 time steps, giving a final time of tmax = 0.1. We +test anisotropy ratios of 103, 106 and 109 – implementationally, we set κ⊥ = 1, such +that the anisotropy ratio is equivalent to κ∥. The resulting convergence plots are +presented Figure 5.1, where we consider eT (t) averaged over the last two time steps. +This is done since for the lower anisotropy ratio 103, we found all four discretizations +to exhibit a small degree of oscillatory behavior in the error across time steps, likely +due to an interplay between the isotropic diffusion and counter-forcing terms. +Fig. 5.1. Relative L2 error (5.6) convergence plots for the two non-mixed and mixed discretiza- +tions, respectively. The anisotropy ratios are given from left to right by 103, 106, and 109. Orange +circles denote the novel mixed DG scheme (3.5), green diamonds the mixed CG scheme (5.2), red +triangles the primal DG scheme (5.3), and purple triangles the primal CG scheme (5.1). The black +lines indicate first to third order convergence. +For an anisotropy ratio of 103, all four schemes show similar errors and conver- +gence rates. For 106, all schemes still display second to third order convergence rates; +however, the errors for the newly introduced scheme (3.5) are approximately two or- +ders of magnitude smaller than for the other three schemes. Finally, for the high +degree of anisotropy 109, scheme (3.5) still displays favorable error values and conver- +gence rates, while the remaining schemes exhibit relative errors of order 1. Inspecting +the runs’ field development, we found these large errors likely to be due to a large +spurious contribution of the heat flux’ parallel component in the perpendicular direc- +tion, thereby rendering the latter direction overly diffusive. The mixed DG scheme, +on the other hand, is able to avoid such an excessive spurious diffusion likely due to a +better representation of cross-cell fluxes through the upwind formulation. Note that +the results for the mixed CG scheme are considerably worse than in [13]. We found +this to be unrelated to using a 3D version of the original test case; instead, the differ- +ence seems to be related to the use of triangles in the base mesh in this work, versus +quadrilaterals in [13]. Switching to quadrilaterals, in additional tests not shown here, +we found the performance of the mixed CG scheme to be drastically improved; albeit +with an error that was still larger than the one for the mixed DG scheme. +17 + +100 +10-3. +10-1 +10-1 +10-2 +10-4 +Th E P2 +Th E dP2 +10-2, +10-3 +(Th, h)E P2 × dP1 +(Th, Sh)E dP2 X dP2 +α △x1 +10-4 +10-5. +α △x2 +10-3, +α △x3 +10-1 +6×10-2 4×10-2 +10-1 +6 × 10-2 +4 × 10-2 +6×10-24×10-2 +10-1 +Horizontal cell side lengthAxFinally, we again stress a caveat of the newly introduced scheme (3.5), given by +the heat flux guess ζin at the inflow boundary. While non-dynamical test cases – +such as the one considered here – allow for an analytic solution given by the initial +condition, they may conceal problems related to dynamics that vary in time. That +said, in further tests with quadrilateral meshes, a non-steady dynamical evolution, +and an anisotropy ratio of 103, we found little difference between the mixed CG and +the mixed DG discretizations. +5.4. Solver Studies. While we used a modification of the test case from [27] +for the convergence results, here we consider a new test case, which in particular has +open field lines only. We consider the same base mesh and domain as in the previous +test case. Again, the base mesh starts with resolution ∆x = Lx/7, and we use 2 cells +in the extruded direction. As before, we refine twice by splitting cells. However, this +time, we also refine the number of cells in the extruded direction in order to ensure +a truly three-dimensional solver study. For the highest refinement level, this leads to +approximately 105 and 4.5 × 105 degrees of freedom for VCG +2 +and VDG +2 +, respectively. +The initial temperature field is given by +(5.7) +T0 = 1 + 1 +20 +� +1 − cos +�2πy +Lx +�� +sin +�πx +Lx +� ++ x + y +10, +with TBC defined according to the values of T0 on ∂Ω. As in the convergence study, the +magnetic field B is chosen to align with the contours of T0 in the two base dimensions. +This leads to field lines which are slightly slanted with respect to the domain, as well +as curved towards the domain’s center. Further, B is again chosen to be constant in +the third dimension; this time, we set Bz = 15 +2 . Finally, since here we only consider +the solver performance and not convergence rates, there is no need for an analytic +solution and the associated counter-forcing; hence we set S = 0. In the following, +we consider results for a) the mixed DG-scheme (3.5) together with the AIR solver +strategy described in Subsection 4.4, b) the mixed DG-scheme (3.5) together with +the classical AMG based solver strategy described in Subsection 5.1, as well as c) the +mixed CG scheme (5.2) together with the aforementioned classical AMG based solver +strategy. As before, the time step is set to ∆t = 10−3, and we consider iteration counts +as well as wall-clock time from the 2nd to the 5th time step, i.e. a total of four time +steps. Note that we disregard the first time step as a simple way of avoiding to count +wall-clock time associated with form assembly. Further, runs which either exceed a +specified average wall-clock time or outer iteration count per time step (approx. 1500 +seconds and 104 iterations, respectively) are discarded. The resulting averages per +time step for the inner iteration counts, as well as wall-clock times, for anisotropy +ratio κ∥/κ⊥ = 10k, k = 2, ..., 10, are depicted in Figures 5.2 and 5.3, respectively. +As expected for the mixed CG scheme solved using classical AMG, we find a +steady increase in solver iteration counts and according wall-clock times as the ratio +of anisotropy increases. For the highest refinement level, runs with anisotropy ratio +107 or higher did not meet the imposed cutoff criteria. +Passing from the first to +the second refinement level, runs at the highest three anisotropy ratios experience a +wall-clock time increase by a factor of approximately 100. Surprisingly, for the mixed +DG scheme solved using classical AMG, we find the increase in solver iteration count +and wall-clock times to eventually stall as we increase the ratio of anisotropy. For the +lowest refinement level, the times stall at approximately 10 seconds, while for the next +higher one, they do at about 240 seconds – which indicates a roughly 24-fold increase +across the two refinement levels. For this setup, most of the highest refinement level +18 + +Fig. 5.2. +Average total inner iteration counts per time step using setup described in Sub- +section 5.4. Top row: counts with respect to anisotropy ratio. Bottom row: counts with respect to +refinement level. Left column: mixed CG scheme solved using classical AMG. Center column: mixed +DG scheme solved using classical AMG. Right column: mixed DG scheme solved using AIR. +runs did not meet the imposed time limit. In comparison to the mixed CG scheme +solved using classical AMG, this is likely due to the much higher number of degrees +of freedom associated with the DG scheme. +For the mixed DG scheme solved using our newly proposed solver strategy in- +cluding AIR, we instead find a decrease in iteration counts and wall-clock times as +the anisotropy ratio increases. As described in Section 4, this is as expected, as such +higher ratios fit better into our solver strategy’s favorable parameter regime (4.13). +For the highest refinement level, runs with anisotropy ratio 103 or lower did not meet +the imposed time limit. In particular, the advection-based block preconditioner strug- +gles with lower anisotropy ratios, while it performs very well for ratios 108 and higher. +For instance, at the highest anisotropy ratio 1010 and the medium refinement level, +the mixed CG formulation solved with classical AMG requires an average of approx- +imately 104 iterations, while the mixed DG formulation solved with our new AIR +based strategy only requires approximately 50, i.e. around 200 times fewer. Given +the aforementioned rates of increase for the number of iterations as we increase the +19 + +(Th, h)E P2 X dP1 +(Th,Sh)EdP2XdP2 +(Th,h)EdP2XdP2 +CLASSICAL +CLASSICAL +AIR +104 +104 +103 +103 +102 +102 +101 +101 +103 +106 +109 +103 +106 +109 +103 +106 +109 +Kμ/K +104 +104 +103 +103 +102 +102 +101 +101 +0 +1 +2 +0 +1 +2 +0 +1 +2 +refinement level +ref. level 0 +￥-K|/K11E+2 +-K/K11E+5 +...K/KI1E+8 +ref. level 1 +--+-K|/K1 1E+3 +-→-K|/K11E+6 +.·K/KI1E+9 +ref. level 2 +K/KI1E+4 +-+·K/KI1E+7 +-·K//K11E+10Fig. 5.3. +Average wall-clock times in seconds per time step using setup described in Sub- +section 5.4. Top row: counts with respect to anisotropy ratio. Bottom row: counts with respect to +refinement level. Left column: mixed CG scheme solved using classical AMG. Center column: mixed +DG scheme solved using classical AMG. Right column: mixed DG scheme solved using AIR. +refinement level, we expect this gap to rapidly increase for higher resolutions. Simi- +larly, for the same anisotropy ratio and refinement level, we find the mixed DG/AIR +based run to be around 45 times faster than the mixed CG/classical AMG based run; +this holds true even though the DG discretization has far more degrees of freedom +than the CG one, and we use a classical AMG based code implementation that is +more streamlined for efficiency than our AIR based one. Again, given the rates of +increase in wall-clock time as we increase the refinement level, we expect this gap +to be larger still for higher resolutions, provided that the AMG based approach still +converges. In addition, recent work has proposed a modified AIR [7] that reduces +total computational time compared with ℓAIR as implemented in hypre [19, 21] by +several times, suggesting further multiplicative speedups possible. +Finally, we find the difference in iteration count for the mixed DG scheme at +the three highest anisotropy ratios to not be as pronounced between the AIR- and +classical AMG based solver strategies. Picking again the example of an anisotropy +ratio of 1010 at the medium refinement level, we find that the AIR based strategy +20 + +(Th, Sh)E P2 X dPi +(Th,Sh)EdP2XdP2 +(Th,h)EdP2XdP2 +CLASSICAL +CLASSICAL +AIR +103 +103 +102 +102 +101 +101 +100 +100 +103 +106 +109 +103 +106 +109 +103 +106 +109 +Kμ/K1 +103 +103 +102 +102 +101 +101 +100 +100 +0 +1 +2 +0 +2 +0 +1 +2 +refinement level +ref.level 0 +￥K//K11E+2 +-K//K11E+5 +.K/KI1E+8 +ref. level 1 +--+- K|/KI 1E+3 +3-→-K|/K11E+6 +...K/KI1E+9 +ref.level 2 +·K/KI1E+4 +-+·K//K11E+7 +-+K//K11E+10requires about 10 times fewer iterations than the classical AMG based one. +It is +unclear why the classical AMG based strategy performs this well in terms of iteration +count for the mixed DG scheme. However, considering the wall-clock time, we find +that the mixed DG/classical AMG based run is again around 30 times slower than +the AIR based one, with significant degradation as the mesh is further refined. +We end the discussion with observations on the AIR strategy’s solver behavior. +First, we found the number of inner iterations to remain very confined throughout +the range of anisotropy ratios that we tested in this section. For the highest refine- +ment level and an anisotropy ratio of 102, the inner solves for the transport operator +and Schur complement take approximately 10 and 22 iterations per outer iteration, re- +spectively, with the larger number of Schur complement iterations due to the modified +inner tolerance proposed in Subsection 4.4. For 106, we obtain 10 and 19, respectively. +Finally, for 1010, the counts are 10 and 14, respectively. In other words, the inner +solver count for the transport operator remains the same, while for the Schur com- +plement, it slowly decreases. +This is because as κ⊥L becomes a more significant +off-diagonal contribution to the Schur complement’s right-hand side as κ∆ decreases, +we require a larger residual reduction to meat an absolute residual tolerance. +Second, considering the relatively low inner iteration counts throughout the range +of anisotropy ratios, we note that it is the outer solver that begins to struggle at +anisotropy ratios of 105 or less. Recall that the Schur complement is preconditioned +using the transport operator (see (4.5)), which we expect to work well when we are +within the valid parameter regime (4.13) for our solver. The latter regime increas- +ingly ceases to be satisfied for ratios 105 or less, as can be inferred from the largest +eigenvalue Table 4.2. For a ratio 105, the largest eigenvalues of the preconditioned +Schur complement are 7.6, 5 and 4.4 for the three spatial refinements, respectively. +They then further increase inversely proportionally to the anisotropy ratio. Consis- +tent with the spectral analysis (see Table 4.1), the proposed method is efficient in +2D for smaller anisotropies, on the order of 103, but results are not presented due to +space, and because we are interested in 3D-simulations in practice. +These considerations aside, we find that altogether, for the very high anisotropy +ratios > 106 typically considered in magnetic confinement fusion, our novel mixed +DG/AIR based scheme performs very well, and clearly outperforms the classical AMG +based approaches. +6. Conclusion. Confined plasma simulations may exhibit high anisotropy, which +needs to be resolved accurately in order to avoid cross-contamination of heat flux from +components parallel to the magnetic field into components perpendicular to the mag- +netic field. This cross-contamination in turn can lead to spuriously short simulated +magnetic confinement times. In addition, such heat flux requires implicit integration, +and for large-scale simulation requires efficient iterative solvers. In this work, we pre- +sented a novel DG space discretization based on the heat flux as an auxiliary variable, +and two upwind transport operators. It is coupled to a novel solver strategy based +on the AIR AMG method, where the block matrix system resulting from an implicit +time discretization is block-row swapped, and is solved for using a Schur complement +approach. For high ratios of anisotropy, the corresponding A11 block and Schur com- +plement are then both akin to DG upwind transport operators, which can be solved +for efficiently using AIR. +We compared the proposed spatial discretization to a CG and a DG direct dis- +cretization, as well as another CG mixed discretization which also uses the heat flux +as an auxiliary variable. Further, we compared the AIR based solver strategy with +21 + +a standard one based on classical AMG. In the first test case based on (extruded) +prism elements, we showed that for moderate rates of anisotropy, our novel space dis- +cretization has comparable errors and an equal convergence rate when compared to +the three other discretizations. Further, for higher rates of anisotropy, our discretiza- +tion retains the low errors and favorable convergence rate (third-order convergence +for 2nd-order basis functions), while the remaining discretizations’ errors and rates +deteriorate. For anisotropy ratios of 109, our proposed discretization achieves error +≈ 1000 times smaller than the other discretizations tested. In the second set of tests, +we demonstrated the newly proposed AIR based solver strategy to be highly efficient +at anisotropy ratios 106 and higher. We found the latter strategy to take up to 200 +times fewer solver iterations and 45 times less wall-clock time than the classical AMG +approach at a moderate space resolution, in spite of using a more streamlined and +efficient classical AMG implementation than the one for AIR. Additionally, we found +the AIR based solver strategy to run within our imposed time and iteration count +cut-off criteria at higher resolutions, where this was no longer the case for the AMG +approach. +A major shortcoming of the proposed solver methodology is that it requires all +magnetic field lines to be open. Ongoing work is focused on developing a generalized +solver strategy for the proposed discretization that is robust for closed field lines. +Appendix A. Intuition for DG-upwind operators. +The DG-upwind trans- +port operator based discretization of the anisotropic heat flux can be motivated start- +ing from an interior penalty formulation. For the anisotropic heat flux term +(A.1) +∇ · +� +κ∆b(b · ∇T) +� +, +one such interior penalty formulation is given analogously to (3.6) by +IPb(Th; φ) = − ⟨b · ∇φ, κ∆b · ∇Th⟩ +(A.2a) ++ +� +Γ +�(b · n)Th� {κ∆b · ∇φ} dS + +� +Γ +�(b · n)φ� {κ∆b · ∇Th} dS +(A.2b) +− +� +Γ +κ∆κp +he +�(b · n)φ� �(b · n)Th� dS − +� +∂Ω +2κ∆κp +he +φ(Th − TBC) dS +(A.2c) ++ +� +∂Ω +κ∆(b · n)(b · ∇φ)(Th − TBC) dS + +� +∂Ω +κ∆(b · n)(b · ∇Th)φ dS. +(A.2d) +Next, we reformulate the central difference facet integrals (A.2b) to upwinded ones, +further appropriately replace occurrences of √κ∆b · ∇Th by ζh, and drop the interior +facet penalty term. We then obtain, after some rewriting, +� +IPb(Th; φ) = − ⟨b · ∇φ, κ∆b · ∇Th⟩ − +� +∂Ω +2κ∆κp +he +φ(Th − TBC) dS +(A.3a) ++ +� +Γ +�√κ∆Thb · n� √κ∆ � +b · ∇φ dS + +� +Γ +�√κ∆φb · n� ˜ζh dS +(A.3b) ++ +� +∂Ω +√κ∆(Th − TBC)(b · n)√κ∆(b · ∇φ) dS + +� +∂Ω +√κ∆φ(b · n)ζh dS. +(A.3c) +22 + +Finally, we modify the boundary terms (A.3c) in accordance with the upwind modifi- +cation; to so so, we set Th = TBC and ζh = ζin along ∂Ωin. In particular, the portion +along ∂Ωin of the first integral in (A.3c) then vanishes, and only the one along ∂Ωout +remains. Additionally, we reformulate the penalty parameter in the boundary term +(last term in A.3a) as 2κ∆κp +he +→ κBC. +The resulting formulation can then be retrieved from our scheme (3.5) – ignoring +the time derivative and isotropic heat flux terms – by setting ψ = √κ∆b · ∇φ in +(3.5b). Note that this is merely a motivation relating the split-transport setup to +an interior penalty method, since √κ∆b · ∇φ will generally not be in VDG +k +and we +therefore cannot set ψ accordingly after discretization. +Importantly, however, the +scheme (3.5) still preserves the correct diffusive and total energy behavior weakly, as +shown in Remark 1. +Appendix B. Base mesh. +Here, we describe the 2D base mesh used for +Fig. B.1. Base mesh. +the base square of side length Lx in the numerical re- +sults section. It is generated starting from a regular +triangular mesh with resolution ∆x = Lx/7, whose +vertices are then perturbed by a factor of 0.06∆x in +each interior coordinate in order to avoid spurious er- +ror cancellations due to symmetries between the mesh +and initial conditions. The resulting mesh is depicted +in Figure B.1. In general, for large anisotropy ratios, +we found the error values in the convergence study to be very sensitive to the under- +lying mesh; we therefore decided to rely on a constructed pseudo-irregular base mesh +rather than a fully irregular one for better reproducibility. +REFERENCES +[1] Patrick R Amestoy, Iain S Duff, and J-Y L’Excellent. Multifrontal parallel distributed symmet- +ric and unsymmetric solvers. Computer Methods in Applied Mechanics and Engineering, +184(2-4):501–520, 2000. +[2] Satish Balay, Shrirang Abhyankar, Mark Adams, Jed Brown, Peter Brune, Kris Buschelman, +Lisandro Dalcin, Alp Dener, Victor Eijkhout, W Gropp, et al. PETSc users manual. 2019. +[3] Nathan Bell, Luke N Olson, and Jacob Schroder. PyAMG: algebraic multigrid solvers in python. +Journal of Open Source Software, 7(72):4142, 2022. +[4] James J Brannick, Yao Chen, and Ludmil T Zikatanov. An algebraic multilevel method for +anisotropic elliptic equations based on subgraph matching. 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Mesh refinement for anisotropic diffusion in +magnetized plasmas. arXiv preprint arXiv:2210.16442, 2022. +24 + diff --git a/c9FQT4oBgHgl3EQfizaw/content/tmp_files/load_file.txt b/c9FQT4oBgHgl3EQfizaw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ddab70b8c862192b0773fd66a497506de823a969 --- /dev/null +++ b/c9FQT4oBgHgl3EQfizaw/content/tmp_files/load_file.txt @@ -0,0 +1,948 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf,len=947 +page_content='A FAST ALGEBRAIC MULTIGRID SOLVER AND ACCURATE DISCRETIZATION FOR HIGHLY ANISOTROPIC HEAT FLUX I: OPEN FIELD LINES ∗ GOLO A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' WIMMER†, BEN S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' SOUTHWORTH†, THOMAS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' GREGORY‡, AND XIANZHU TANG† Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We present a novel solver technique for the anisotropic heat flux equation, aimed at the high level of anisotropy seen in magnetic confinement fusion plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Such problems pose two major challenges: (i) discretization accuracy and (ii) efficient implicit linear solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We si- multaneously address each of these challenges by constructing a new finite element discretization with excellent accuracy properties, tailored to a novel solver approach based on algebraic multigrid (AMG) methods designed for advective operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We pose the problem in a mixed formulation, introducing the heat flux as an auxiliary variable and discretizing the temperature and auxiliary fields in a discontinuous Galerkin space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The resulting block matrix system is then reordered and solved using an approach in which two advection operators are inverted using AMG solvers based on approximate ideal restriction (AIR), which is particularly efficient for upwind discontinuous Galerkin discretizations of advection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' To ensure that the advection operators are non-singular, in this paper we restrict ourselves to considering open (acyclic) magnetic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We demonstrate the proposed discretization’s superior accuracy over other discretizations of anisotropic heat flux, achieving error 1000× smaller for anisotropy ratio of 109, while also demonstrating fast convergence of the proposed iterative solver in highly anisotropic regimes where other diffusion-based AMG methods fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Anisotropic heat flux, anisotropic diffusion, auxiliary operator, algebraic multigrid, AIR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The heat flux of plasmas in magnetic confinement fusion ex- hibits strong anisotropy [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Heat can be conducted at a rate many orders of magni- tude greater parallel to magnetic field lines than it is perpendicular to the magnetic field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' this ratio κ∥ κ⊥ can exceed 1010 in realistic experiments [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' If a numerical sim- ulation does not adequately resolve the anisotropy of the heat flux, the resulting simulated plasma confinement times may be spuriously short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Unfortunately, many numerical approaches suffer from the large parallel component of heat flux polluting the perpendicular direction, leading to non-physically large perpendicular heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In addition, anisotropic heat flux in a magnetized plasma has a very restrictive CFL condition because of the large κ∥, so for realistic applications an implicit solver is typically necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In the isotropic limit with κ∥ = κ⊥, the problem reduces to the heat equation, and classical geometric and algebraic multigrid (AMG) solvers [28] are known to be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Conversely, the highly anisotropic regime, κ∥ ≫ κ⊥, is notoriously difficult for multilevel and iterative solvers, and efficient parallel solvers for realistic applications remain a largely open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' One common approach to address pollution of flux in the perpendicular direction ∗Corresponding author email address: gwimmer@lanl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='gov Funding: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=', and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' have been supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Department of Energy Office of Fusion Energy Sciences and Office of Advanced Scientific Computing Research under the Tokamak Disruption Simulation (TDS) Scientific Discovery through Advanced Computing (SciDAC) project, as well as the Base Theory Program, both at Los Alamos National Laboratory (LANL) under contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 89233218CNA000001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' was supported by the Laboratory Directed Research and Development program of Los Alamos National Laboratory as a Nicholas C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Metropolis Fellow and under project number 20220174ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The computations have been performed using resources of the National Energy Research Scientific Computing Center (NERSC), a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Department of Energy Office of Science User Facility operated under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' DE-AC02-05CH11231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' †Theoretical Division, Los Alamos National Laboratory, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' ‡Imperial College London 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='13351v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='NA] 31 Jan 2023 is to make use of meshes aligned with the magnetic field lines [8, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' However, this can only apply for configurations in which the magnetic field lines form nested flux surfaces, which is not the case, for instance, for tokamak disruptions that can have volume-filling stochastic magnetic field lines [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Mesh-alignment also places limitations on software and algorithm flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Several other avenues have been explored focusing on increased accuracy of heat flux discretizations, thereby allowing for general, non-field-aligned meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Recently, authors have considered higher order discretizations [12] and adaptive mesh refinement [29], where the mesh is refined specifically in areas where large error due to anisotropy is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' One can also make use of an auxiliary variable, explicitly resolving the heat flux along magnetic field lines as a variable in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' This formulation has been shown to better resolve the heat flux after discretization, reducing the aforementioned parallel-to-perpendicular cross-contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' One such auxiliary variable implementation, provided by Gunter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' [13], uses a mixed finite element method, with the temperature field defined on a continuous Lagrange element of order n, and the auxiliary heat flux variable on a discontinuous Lagrange element of order n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For large sparse matrices that arise in finite element discretizations, almost all fast and parallelizable solvers are iterative and multilevel in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The basic idea is to combine a relatively cheap procedure to attenuate error on one “level” in the hierarchy (“relaxation”), with a coarse-representation that represents the complementary error in a smaller solution space (“coarse-grid correction”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Relaxation is typically a local procedure, such as Jacobi, Gauss-Seidel, or (overlapping) Schwarz, which attenuates high-frequency error in a geometric sense, and error associated with large eigenvalues in a spectral sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The fundamental difficulty with developing fast solvers for strongly anisotropic diffusion is that there exists a large set of eigenmodes that are smooth in the parallel direction and high frequency in the perpendicular direction, which have small eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Such modes cannot be effectively attenuated by standard local relaxation procedures due to the small associated eigenvalues, and are equally difficult to represent in a coarse solution space due to the high frequency character in the perpendicular direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' One solution to the aforementioned problems is line-relaxation, where one does a block relaxation aligned with one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=', in 2D, for each discrete point in the x-dimension, one solves the fully coupled problem in the y-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' If the mesh is aligned with the anisotropy, one can perform line relaxation in the direction of anisotropy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' for non-grid aligned anisotropy, one can use alternating line relax in grid-aligned directions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=', in x and y in 2D, or attempt to find an ordering of mesh points that is approximately aligned with the anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In 3D, things become more complicated, typically requiring plane relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Line relaxation was a key compo- nent of the multilevel solvers recently used in [12] for anisotropic heat flux in plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The downsides of line relaxation are its high cost, need for structured grids to define lines/planes on, preferably aligned with anisotropy, and in particular its poor perfor- mance in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' A number of papers have also tried to address anisotropic diffusion without line relaxation through algebraic solvers, primarily variations in AMG (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=', see [4, 6, 11, 20, 25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Although there has been some success in this regard, there is a disconnect between “hard” anisotropic problems considered in solvers papers, and accurate discretizations of the anisotropic heat flux that actually arises in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=', magnetic confinement fusion problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' This paper conjointly addresses the challenges in discretization accuracy and solver efficiency for strongly anisotropic heat flux with open field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Following [13], we pose the anisotropic heat flux in mixed form using an auxiliary variable, which 2 offers superior accuracy in the strongly anisotropic regime compared with direct dis- cretizations of the second-order diffusion operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' This formulation also uncovers the purely anisotropic diffusion operator as a combination of two linear advection opera- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The 2 × 2 block system resulting from a mixed discretization typically has mass matrices and the isotropic diffusion term on the diagonal, with the advection oper- ators in the off-diagonal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Although robust multilevel methods for (physically realistic) anisotropic diffusion remains a largely open question, significant progress has been recently made developing fast and robust AMG methods for linear advec- tion based on an approximate ideal restriction (AIR) [19, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Thus, rather than follow the traditional approach approximating a Schur complement that represents the anisotropic diffusion operator, we instead swap the rows of the block system so that the advection operators are on the diagonal, and apply a block triangular pre- conditioning, with the diagonal advection blocks inverted using AIR-AMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Given the transport point of view of this solver scheme, it is well-constructed to deal with the anisotropy of the problem: the higher the anisotropy, the more efficient this approach is compared to classical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The remainder of this work is structured as follows: in Section 2 we introduce the problem and motivate our solver strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The proposed solver strategy has several implications on the discretization that must be taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In particular, the advection blocks need to be square and invertible, which requires care in choice of finite element space, enforcement of boundary conditions, and constraints on the anisotropic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In addition, we must choose a discretization that AIR is an effective solver for – as for many problems, some FEM formulations are much easier to solve than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Section 3 describes a new DG-upwind-based spatial discretization of the auxiliary formulation that conforms with each of these constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The full solver strategy is then introduced in Section 4, including a spectral analysis of the (advection-)preconditioned Schur complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Numerical results are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' From an accuracy perspective, the new mixed DG discretization is shown to significantly outperform a non-mixed CG and DG formulation, as well as the mixed CG system developed in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In addition, in the regime of large anisotropy, the proposed solver strategy is shown to be fast in terms of iteration counts and wall- clock times, whereas other classical diffusion-based AMG schemes demonstrate very poor scaling or simply fail to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Finally, in Section 6, we review our results and discuss future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Anisotropic Heat Transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Following Gunter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' [13], we formulate the evolution of temperature T within a plasma through use of a heat flux q, with forcing S: ∂T ∂t − ∇ · q = S, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1a) q = κ∥∇∥T + κ⊥∇⊥T, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1b) over some bounded domain Ω, where κ∥ denotes the conductivity parallel to magnetic field lines and κ∥ denotes the conductivity perpendicular to field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In this work, we will consider pure Dirichlet boundary conditions for T: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2) T|∂Ω = TBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In magnetic confinement fusion applications, the conductivities may depend on the magnetic field and the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For implicit discretizations, the resulting non- 3 linear problem can be solved, for example, with a quasi-Newton iteration scheme, using the given iteration’s known value for T in the conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For simplicity, in what follows, we assume constant conductivities and note that the framework fol- lows equally for non-constant ones (up to the required nonlinear iterative procedure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Similarly, ∇∥ and ∇⊥ are the components of the gradient along and orthogonal to magnetic field lines, respectively: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3) ∇∥(·) := b · ∇(·))b, ∇⊥ := ∇ − ∇∥, for b = B/|B|, where B is the magnetic field along which the anisotropy is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Heat flux in the context of magnetic confinement fusion is highly anisotropic, diffusing significantly faster along magnetic field lines as opposed to perpendicular to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' As mentioned in the introduction, we can expect to see κ∥ κ⊥ ∼ 1010 and larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Next to a split of the heat flux into a parallel and perpendicular component, it is also possible to do a split into an isotropic part, and a remaining purely anisotropic (parallel) part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For this purpose, we define a purely anisotropic conductivity as κ∆ := κ∥ − κ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Diffusion in the plasma is isotropic when κ∆ = 0, and is predominately anisotropic if κ∆ ≫ κ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Defining κ∆ in this way allows us to represent the heat flux as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4) q = κ∆∇∥T + κ⊥∇T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Considering the directional derivative b · ∇T as an auxiliary variable, we may rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1) as ∂T ∂t − √κ∆∇ · (bζ) − ∇ · (κ⊥∇T) = S, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5a) ζ = √κ∆b · ∇T, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5b) where we split up the factor of κ∆ evenly between equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5a) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5b) in view of the solver strategy to be presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In this form, the PDE system has a more conventional advection-diffusion character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Gunter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' [13] demonstrated that this mixed system – solved for T and ζ – may lead to higher accuracy than a single-variable model which solves for T only, by resolving the parallel heat flux term ζ more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Finally, we note the equation’s two limiting cases in order to motivate the solver strategy presented in this work: for κ∥ = κ⊥, we find that κ∆ = 0 and the system reduces to a (forced) heat equation given by ∂T ∂t = ∇ · (κ⊥∇T) + S, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6) and we can expect classical multigrid methods to perform well when solving implicit discretizations of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' However, for κ⊥ = 0, we instead have a purely anisotropic system of equations ∂T ∂t − √κ∆∇ · (bζ) = S, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='7a) ζ = √κ∆b · ∇T, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='7b) in which the dynamics are governed by two transport operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Solver considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Written in block operator form, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5) can be for- mulated as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='8) � ∂ ∂t + κ⊥∆ −√κ∆∇ · � b(·) � −√κ∆b · ∇ I � �T ζ � = �S 0 � , where ∆ denotes the Laplacian, and I the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Upon discretizing in space with a suitable finite element method and applying implicit integration in time (including diagonally implicit Runge Kutta or linear multistep methods), the resulting discrete linear systems may take the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9) � 1 ∆tMT + κ⊥L √κ∆GT b −√κ∆Gb Mζ � �T n+1 ζn+1 � = �FT Fζ � , where MT and Mζ are mass matrices, L denotes the discrete Laplacian, Gb the discrete scalar-form transport operator, and FT , Fζ correspond to forcing terms as well as terms on the known time level n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Consider the limiting case of a steady-state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' no time derivative) and purely anisotropic problem (κ⊥ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Plugging this value of κ⊥ into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9) and swapping the block rows in the corresponding equations yields the equivalent block-triangular system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='10) �−√κ∆Gb Mζ 0 √κ∆GT b � �T n+1 ζn+1 � = �Fζ FT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Here, we have decoupled the implicit purely anisotropic diffusion equation into the solution of two successive linear advection equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Although it may seem unusual to take a parabolic problem and reformulate in terms of hyperbolic operators, in this case there is a fundamental lack of efficient, parallelizable iterative solvers for the parabolic purely anisotropic setting, while there do exist fast AMG methods for linear advection [19, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Note that we opted for a multiplicatively even split of κ∆ between Gb and GT b in view of future work considering temperature dependent conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' While the split is not necessary for the constant values of κ∆ considered here, it may avoid that one solve is more challenging than the other when such a temperature dependency is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The above limiting case motivates the more general solver strategy proposed in this paper (and detailed in Section 4), focused on inverting transport operators rather than directly trying to solve the highly anisotropic diffusion equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For this frame- work to be viable, we must construct the transport operator to be invertible, which can be achieved by (i) choosing the same finite element space for T and ζ, and (ii) assuming open magnetic field lines throughout the domain, that is, the magnetic field is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The former ensures that Gb is square, and the latter ensures that the operator is nonsingular – in contrast, if the magnetic field has closed field lines, then constant values of T along any closed field line constitute a non-trivial kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In addi- tion, we must choose a transport discretization that is amenable to the fast reduction based AMG solver, AIR, developed in [19, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Because AIR is reduction based, it is most effective on discretizations with low matrix connectivity, a property particuarly enjoyed by upwind discontinuous Galerkin discretizations, where matrix connectivity only occurs to immediately adjacent elements that are upwind with respect to the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The following section constructs a suitable discretization, and the solver strategy is expanded in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Finite element discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' To solve the anisotropic diffusion system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5), we discretize the transport term (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1) ∇ · �√κ∆b(·) � , using the classical DG-upwind method (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For this purpose, we start with an operator of the form Lb(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' φ) := − ⟨√κ∆θb, ∇φ⟩ + � Γ �√κ∆φb · n�˜θ dS + � ∂Ωout √κ∆φb · nθ dS, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2) for L2-inner product ⟨·, ·⟩, and any functions θ, φ in the given kth polynomial order DG space VDG k (Ω), which we will specify further below in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The set ∂Ωout denotes the outflow subset of the boundary ∂Ω relative to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Further, Γ denotes the set of all interior facets of the mesh, and we applied jump and upwind facet operations defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3) �ψ� := ψ+ − ψ−, ˜ψ := � ψ+ if b+ · n+ < 0, ψ− otherwise, for any scalar field ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' n denotes the facet normal vector, and the two sides of each mesh facet are arbitrarily denoted by + and − (and hence n+ = −n−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Given (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2), we then define two transport operators according to Lb,T (ζh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' φ) := Lb(ζh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' φ) + � ∂Ωin √κ∆φ(b · n)ζin dS ∀φ ∈ VDG k , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4a) Lb,ζ(ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Th) := Lb(ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Th) − � ∂Ωout √κ∆TBC(b · n)ψ dS ∀ψ ∈ VDG k , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4b) where ∂Ωin denotes the inflow part of the boundary, defined analogously to ∂Ωout, and ζin denotes a known value to be specified at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Note that this setup of known inflow boundary integrals ensures that Lb,T and Lb,ζ are non-singular for problems where all magnetic field lines are open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' An intuition for this choice of operators and boundary integrals is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Given these operators, we can formulate the system to be solved as finding (Th, ζh) ∈ (VDG k × VDG k ) such that � φ, ∂Th ∂t � − Lb,T (ζh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' φ) − IP(Th;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' φ) = − � ∂Ω κBCφ(Th − TBC) dS + ⟨φ, S⟩ ∀φ ∈ VDG k , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5a) ⟨ψ, ζh⟩ + Lb,ζ(ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Th) = 0 ∀ψ ∈ VDG k , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5b) for penalty parameter κBC, and where IP(Th;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' φ) is the discretized Laplacian using an interior penalty method (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' [5]): IP(Th;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' φ) = − ⟨∇φ, κ⊥∇Th⟩ + � Γ �Th� {κ⊥∇φ} dS + � Γ �φ� {κ⊥∇Th} dS + � ∂Ω κ⊥n · ∇φ(Th − TBC) dS+ � ∂Ω κ⊥n · ∇Thφ dS− � Γ κ⊥κp he �φ� �Th� dS, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6) 6 for any test function φ ∈ VDG k , penalty parameter κp, and where he denotes the given facet’s length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For the numerical results below, we used as a length scale |K+|+|K−| 2|∂K| , where |K±| denote the facet’s adjacent cell’s volumes, and |∂K| denotes the facet area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Further, the average operation is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='7) {u} := 1 2 � u+ + u−� n+, for any vector field u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Finally, the non-dimensional interior penalty parameter in IP(Th;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' φ) is set to κp = 2 in the numerical results section below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The discussion for the penalty parameter κBC for exterior facets (first term on right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5a)) is more intricate and we postpone it to the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In the numerical results section below, we consider 3D extruded meshes based on prism cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Such cells are of interest in magnetic confinement fusion applications, where tokamak meshes may be created starting from a triangular mesh for the poloidal plane (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' a vertical cross section), which is then extruded to create a 3D mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' While the extrusion would be curved in the context of tokamak meshes, here we restrict ourselves to straight extrusions for simplicity1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We then set VDG k (Ω) = VDG 2 (Ω) to the corresponding second polynomial order discontinuous Galerkin space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For prisms, the latter space can be constructed via a finite element which is given by the tensor product of the standard DG finite element dP2 for triangles, together with the analogous one for intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Further, we discretize B in the second order div- conforming Nedelec space Nc2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' There is no specific reason to this choice of finite element space other than related work on magnetic confinement fusion discretizations by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' With regards to the choice of magnetic field space, we generally found the scheme’s accuracy to be only significantly affected by the latter space’s polynomial degree, which should be at least equal to the temperature space’s polynomial degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Structural properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The non-discretized anisotropic diffusion operator in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5) acts to diffuse temperature and may create a heat flux across the domain’s boundary in the direction of the magnetic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' This can easily be seen by taking the L2-inner product of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5) with T, and setting κ⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' After integrating by parts, this yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='8) 1 2 d dt∥T∥2 2 = −∥ζ∥2 2 + � ∂Ω √κ∆ζb · nTBC dS, up to contributions due to the forcing S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Further, the anisotropic transport operator does not impact the system’s total energy other than at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' This can by seen by integrating equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5) (with κ⊥ = 0) and applying integration by parts again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We then have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9) d dt � Ω T dx = � ∂Ω √κ∆ζb · n dS, where in this context, the integral over T plays the role of total energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Similarly, for the newly introduced DG discretization (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5) with the anisotropic part only (with κ⊥ = 0), we can set φ = Th in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5a) and further ψ = ζh in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5b), 1In exploratory numerical tests, we found our framework to also work in curved meshes, albeit with an additional error that depends on the finite element space and mesh degrees in the direction of the curved extrusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 7 which leads to 1 2 d dt∥Th∥2 2 = − ∥ζh∥2 2 − � ∂Ω κBCTh(Th − TBC) dS + � ∂Ωin √κ∆ζin(b · n)Th dS + � ∂Ωout √κ∆ζh(b · n)TBC dS, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='10) where we substituted Lb(ζh, Th) from Lb,T (ζh, Th) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5a) by the corresponding term from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' This behavior is consistent in the sense that when Th is equal to the strong solution T, and ζin is set to the strong directional gradient √κ∆b · ∇T, these terms are equal to the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Note that the correct discrete diffusive behavior (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='10) may initially seem counterintuitive, since we obtained it by combining two transport operators that apply upwinding in the same direction (along b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' This can be resolved by observing that while we upwind the trial function for Lb,T in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4a), we instead upwind the test function for Lb,ζ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The latter in turn can be considered as downwinding the trial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Altogether, the upwinded and downwinded trial functions then combine to the above diffusive behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Finally, to obtain a discrete version of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9), we set φ = 1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5a), which leads to d dt � Ω Th dx = � ∂Ωout √κ∆ζh(b · n) dS + � ∂Ωin √κ∆ζin(b · n) dS (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='11) − � ∂Ω κBC(Th − TBC) dS + � Γ �√κ∆b · n�˜ζh dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For the boundary integrals, as before this is consistent with the non-discretized version (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9) when Th is equal to the strong solution T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' However, this time, there is an additional interior facet integral term (last term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='11)), which need not vanish since b = B/|B| need not have continuous normal components, even if B is a div- conforming finite element field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' has continuous normal components across facets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' However, as for the boundary integrals, the term is weakly consistent in the sense that for a strong solution with a continuous field B, the term vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We end this section with a brief description of the time discretization to be used in the numerical results section, as well as terminology for the next section, the choice of ζin, and a discussion on the penalty terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The space discretization (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5) is coupled to an implicit midpoint rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Given the fully discretized scheme, we denote the operator Lb in matrix form as −√κ∆Gb, where the change in sign and factorization of √κ∆ are to aid the transport solver discussion below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Similarly, we denote the discrete Laplacian operator IP in matrix form as −κ⊥L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Further, we denote the VDG 2 mass matrix by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Finally, for the implicit system, the boundary penalty term (first term on righ-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5a)) leads to a mass matrix defined along the boundary only, and we denote the latter by MBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The need for specifying ζin as a value independent of the unknown ζh to be solved for is a shortcoming of the proposed space discretized formulation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5), which arises from employing DG-upwinded transport operators for both Th and ζh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We therefore require known inflow boundary values for both fields for the operators to be invertible, while only values for either Th (Dirichlet) or ζh (Neumann) are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' One possible way to circumvent this is to set ζin = √κ∆b · ∇ ¯Th, where ¯Th is the midpoint-in-time discretization of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In particular, this is an expression independent 8 of ζh, and therefore does not change the resulting transport block −√κ∆Gb which we are required to invert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' While we found this approach to work well in terms of solution accuracy, it adds a contribution with overall factor κ∆ to the top-left block of the matrix system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9), thereby rendering our transport operator based solver motivation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='10) invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Instead, we initially set the inflow flux ζin equal to the projection of √κ∆b · ∇T|t=0 into VDG 2 , and subsequently set it equal to the most recent solution for ζh of the resulting mixed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In other words, ζin can be seen as an explicit time discretization of ζh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Generally, we found this setup to be stable, likely due to the interior penalty term that weakly enforces the Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We further found the resulting heat flux error at the inflow boundary generally to be small, including test cases where the heat flux varies in time at the latter boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In practice, this shortcoming can likely further be remedied at no additional cost by incorporating updates to the value of ζin in an outer nonlinear iterative procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' As mentioned in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1, the latter procedure is needed in magnetic confinement fusion simulations where the conductivities κ⊥, κ∥ depend on the temperature Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Given this choice of ζin, both additional boundary integrals in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4) are absorbed into the right-hand side vectors FT and Fζ, respectively, leading to an overall discrete 2 × 2 block system of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='12) � 1 ∆tM + κ⊥L + MBC √κ∆GT b −√κ∆Gb M � �T n+1 h ζn+1 h � = �FT Fζ � , whose structure is equal to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9) described in our motivation section Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2, up to the additional boundary penalty term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The latter term can be considered as part of the discrete Laplacian L, in which case the correct penalty parameter formulation is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='13) κBC = ˜κBC κ⊥ he , for some non-dimensional parameter ˜κBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The top-left block of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='12) can then be grouped according to 1 ∆tM + κ⊥(L + ˜κBCMBC,h−1 e ), where we factored ˜κBCκ⊥ out of MBC to highlight the matrix grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Further, we denote the remaining operator as MBC,h−1 e , noting that it contains a factor of h−1 e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' However, this may be impractical for the anisotropic limit in which κ⊥ may be negligible, thereby not ensuring a strong enough weak enforcement of the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In particular, the transport operators Lb,T and Lb,ζ alone are not sufficient to enforce the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We therefore instead consider MBC independently of L, and use a penalty parameter of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='14) κBC = ˜κBC he ∆t, where again ˜κBC is non-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In the numerical results section below, we set ˜κBC = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Given the factor of 1 ∆t, we group MBC together with the mass matrix, leading to an overall mixed system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='15) � 1 ∆t(M + ˜κBCMBC,he) + κ⊥L √κ∆GT b −√κ∆Gb M � �T n+1 h ζn+1 h � = �FT Fζ � , where again we factored ˜κBC/∆t out of MBC to highlight the matrix grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' This time, we denote the remaining matrix as MBC,he, since it contains a factor of he.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Transport solvers for anisotropic diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For standard block diagonal, triangular, and LDU preconditioners applied to block 2 × 2 linear systems, conver- gence of Krylov and fixed-point iterations are fully defined by the preconditioning of the Schur complement [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In the standard form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9) of an auxiliary-variable discretization of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1) or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5), the natural (1,1) Schur complement is effectively a discrete representation of the continuous anisotropic diffusion equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' as discussed in Section 1, it is very difficult to construct effective preconditioners for such equa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Building on the discussion in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2, we reorder the discrete 2×2 block system in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='15) to take the equivalent form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1) � −√κ∆Gb M 1 ∆t(M + ˜κBCMBC,he) + κ⊥L √κ∆GT b � �T n+1 ζn+1 � = �Fζ FT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In doing this reordering, the (2,2) Schur complement defining convergence takes the form (note, the (1,1) Schur complement is the negative transpose of the (2,2)) S22 :=√κ∆GT b + � 1 ∆t(M + ˜κBCMBC,he) � (√κ∆Gb)−1M + κ⊥L(√κ∆Gb)−1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2) Note that MBC,he can be seen as a mass matrix over boundary facets, which is scaled by a factor of he.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Altogether, we expect its eigenspectrum to be akin to that of a mass matrix over cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For simplicity, we further assume ˜κBC to be O(1), so that from the point of view of the spectral analysis to follow, M + ˜κBCMBC,he can be replaced by M without a qualitative change in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We then have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3) S22 ≈ √κ∆GT b + � 1 ∆tM + κ⊥L � (√κ∆Gb)−1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' This section provides a spectral analysis on preconditioning (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2) by inverting the diagonal transport operator to support the proposed strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The analysis is based on the observation that the preconditioned Schur complement is similar via a simple mass-matrix similarity transform to the identity plus two symmetric positive definite (SPD) operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' By nature of this similarity, we expect spectral analysis of the result- ing SPD operators to provide excellent measures of preconditioning (a property which is not always the case with advective operators and inverses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The purely anisotropic case is discussed in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1 and the general case in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We then use a Lanczos algorithm to compute eigenvalue bounds for the preconditioned Schur complement for realistic problems in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3 and demonstrate consistency with the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Finally, we detail the solver in practice in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The purely anisotropic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Recall in this paper we are particularly interested in the strongly anisotropic regime, where κ∆ ≫ κ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' If we consider the limit of κ⊥ = 0, the approximate Schur complement (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3) simplifies to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4) S22 := √κ∆GT b + 1 ∆tM(√κ∆Gb)−1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Now consider preconditioning S22 with the diagonal transport block: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5) (√κ∆GT b )−1S22 = I + 1 ∆tκ∆ G−T b MG−1 b M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Observe that our preconditioned Schur complement is a perturbation of the identity, and (formally) similar to the SPD operator (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6) M 1/2(√κ∆GT b )−1S22M −1/2 = I + 1 ∆tκ∆ M 1/2G−T b MG−1 b M 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 10 Next, we uncover the eigenvalue structure of the second term on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Observe that M −1GT b M −1Gb corresponds to a discrete anisotropic diffusion operator, which is similar to M −1GbM −1GT b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Moreover, recall that b is a unit vector, and the eigenvalues of a discrete diffusion operator span a range of [1/c2, 1/c1h2], for mesh spacing h and O(1) constants c1, c2, independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Thus, the inverse G−T b MG−1 b M that appears in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5), and by similarity the second term of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6), is an SPD operator with eigenvalues in the range [c1h2, c2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Altogether, this implies that the preconditioned Schur complement (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5) has real-valued eigenvalues that are bounded below by one and lie in the range (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='7) σ � (√κ∆GT b )−1S22 � ⊂ 1 + 1 ∆tκ∆ [c1h2, c2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In particular, note that for reasonable values of ∆t and particularly for large κ∆, we achieve excellent preconditioning of the Schur complement by simply inverting the diagonal advection blocks, largely independent of any additional constants scalings that depend on discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Of course, in the limit of a steady-state problem, ∆t → ∞, we have exactness, as used to motivate the solver strategy in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For κ⊥ > 0, the analysis above largely holds, but we have additional perturbations in our preconditioned Schur complement, where (√κ∆GT b )−1S22 = I + 1 ∆tκ∆ G−T b MG−1 b M + κ⊥ κ∆ G−T b LG−1 b M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='8) As before, the above operator is similar to an SPD operator, this time given by M 1/2(√κ∆GT b )−1S22M −1/2 = I + 1 ∆tκ∆ M 1/2G−T b MG−1 b M 1/2 + κ⊥ κ∆ M 1/2G−T b LG−1 b M 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9) Note that both of the perturbation terms in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9) are SPD, and thus the precondi- tioned Schur complement again has positive real-valued eigenvalues, bounded below by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' As done already for the second-to-last term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='8), we aim to establish a connection of the last term with a diffusion operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Observe that by similarity, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='10) G−T b LG−1 b M ∼ G−1 b MG−T b L = � G−1 b MG−T b M �� M −1L � , where the latter term, M −1L, corresponds to a discrete representation of the isotropic diffusion operator, and the first term, G−1 b MG−T b M, corresponds to the inverse of the anisotropic diffusion operator as discussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' To understand the product of these operators, suppose we are in 2D and b is grid-aligned, say in the x-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='10) is a discrete representation of ∂−1 xx (∂xx +∂yy) = 1+∂−1 xx ∂yy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' On a unit domain with zero Dirichlet boundary conditions, the eigenfunctions and eigenvalues of the Laplacian are {ujk, π2(j2 + k2)}, where ujk = 2 sin(jπx) sin(kπy), and similarly for ∂xx only in the (j, x) dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Thus, if we consider the high- est frequency mode represented by our grid in the y-dimension, k = 1/h, and the smoothest mode in the x-dimension, j = 1, we have the eigenpair (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='11) (1 + ∂−1 xx ∂yy)2 sin(πx) sin( 1 hπy) = � 1 + 1 h2 � 2 sin(πx) sin( 1 hπy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' A complete analysis generalized to specific FEM discretizations, non-unit-square do- mains, more complex boundary conditions, or non-grid aligned magnetic field b is 11 a more complex endeavor and outside the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' However, a similar result will still hold, namely that modes that are highly oscillatory orthogonal to b and smooth in the direction of b will typically scale like 1 + c3/h2, with the c3/h2 arising due to oscillatory functions that are not damped by the anisotropic inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Altogether, we then have an estimate for the spectrum contained in the real-valued subset (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='12) σ � κ⊥ κ∆ M 1/2G−T b LG−1 b M 1/2� ⊂ � 0, κ⊥c3 κ∆h2 � , for some constant c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' To conclude, the analysis in the previous two subsections indicates that for suffi- ciently small ratios of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='13) c1 ∆tκ∆ ⪅ 1, κ⊥c3 κ∆h2 ⪅ 1, for constants c1, c3 independent of ∆t, κ∆, h, we expect excellent preconditioning of the Schur complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Numerical results in Section 5 confirm these observations in practice, with rapid convergence obtained in solving (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' On the other hand, for more isotropic regimes O(κ∥) ≈ O(κ⊥), and for setups with smaller time steps or κ∆ such that O(∆t) ≈ O(κ−1 ∆ ), we expect our solver performance to deteriorate since the transport operator alone will no longer be a good preconditioner for the Schur complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Future work will consider more general methods that are also robust in the isotropic or mixed regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' However, we note that in strongly magnetized plasmas, we are primarily concerned with extreme anisotropies for which the methods proposed here are robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Finally, in terms of the time step and therefore the first ratio in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='13), we set ∆t = 10−3 for the numerical tests in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In comparison, the local diffusive time scale h2/κ∥ – which is an important quantity in tokamak applications – lies in an ap- proximate range h2/κ∥ ∈ [10−4, 10−13] for the resolutions and parallel conductivities considered below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' However, the latter scale is not the only factor determining the choice of time step and in extension the scheme’s temporal accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Other factors include the problem’s nonlinearity in the case of T-dependent conductivities and the temperature profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Overall, we found our fixed choice of time step to work well in terms of accuracy for the test cases considered below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Eigenvalue bounds in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' To further investigate the parameter regimes (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='13), we include a brief numerical study to analyze the largest eigenvalue of the components of the preconditioned Schur complement (√κ∆GT b )−1S22, with S22 given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In the following, we denote the second-to-last term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2) by S22,M, and the last one by S22,L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We have tested each operator using the mesh described in Appendix B, as well as a periodically extruded 3D version thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We consider 3 levels of mesh refinement, with 2 extruded cells in 3D for the lowest refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In each refinement, the resolution is doubled in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For each resolution, we tested the behavior for differing choices of time step ∆t ∈ [10−3, 10−2] and κ∥ κ⊥ ∈ [103, 106, 109], with κ⊥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Finally, the magnetic field B is defined as the curved and slanted one used in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The eigenvalues are computed by implementing the preconditioned Schur complement’s components as actions applied to an input vector xin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The actions are then passed to an implicitly restarted Lanczos method [17] as implemented in SciPy2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The resulting largest eigenvalues for the 2D 2The number of Lanczos vectors is capped at 10 for the highest resolution 3D, together with a relative tolerance of 10% for computational speedup, noting that the latter resolution leads to 12 ∆t κ∥/κ⊥ (√κ∆GT b )−1S22,M (√κ∆GT b )−1S22,L (√κ∆GT b )−1S22 − I 103 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='8e0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6e0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9e0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6e-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4e0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='7e0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='8e0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='7e0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3e0 10−3 106 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='8e-3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6e-3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9e-3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6e-4 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='7e-3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='0e-4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5e-3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='7e-3 109 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='8e-7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6e-7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9e-7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6e-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4e-6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='7e-6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='0e-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5e-6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='7e-6 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1 Largest eigenvalues of preconditioned Schur complement components for 2D mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' S22,M and S22,L correspond to second and third terms on right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The three values for each given anisotropy ratio correspond to the three spatial refinement levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' ∆t κ∥/κ⊥ (√κ∆GT b )−1S22,M (√κ∆GT b )−1S22,L (√κ∆GT b )−1S22 − I 103 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6e2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9e2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4e2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6e1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='0e2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='7e2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6e2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='0e2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4e2 10−3 106 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6e-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9e-1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4e-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6e-2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='0e-1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='7e-1 6.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='0e-4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='7e-4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6e4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='0e-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4e-4 103 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6e1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9e1 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1e-1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='8e-1 109 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6e-5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9e-5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4e-5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6e-5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='0e-4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='7e-4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6e-5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1e-4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='8e-4 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2 Largest eigenvalues of preconditioned Schur complement components for 3D mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' S22,M and S22,L correspond to second and third terms on right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The three values for each given anisotropy ratio correspond to the three spatial refinement levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' and 3D setups are given in Tables 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' a larger leading eigenvalue will lead to a less efficient solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The leading eigenvalues λmax SM of (√κ∆GT b )−1S22,M and λmax SL of (√κ∆GT b )−1S22,L evolve according to their leading factors in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='13), except increasing the resolution additionally decreases λmax SM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Upon further investigation, we found this to be related to the inclusion of the boundary penalty term MBC in the Schur complement S22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Without it, the value λmax SM is independent of the spatial resolution as expected – for instance, in 2D and for ∆t = 10−3, κ∥/κ⊥ = 103, we get λmax SM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='84 for all three resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Finally, we found that as we increase the spatial resolution, the largest eigenvalues that are obtained when including MBC converge to the one when the latter operator is excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We also observe that either λmax SM or λmax SL may be larger, depending on the choice of h and ∆t (for fixed conductivities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Additionally, the observed preconditioned Schur complement’s largest eigenvalues (ignoring the additive factor of 1 from the identity) are slightly less than the sum λmax M + λmax L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Finally, we find both (√κ∆GT b )−1S22,L and (√κ∆GT b )−1S22,M to have considerably larger largest eigenvalues in 3D than in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In further studies including different discretization and initial condition setups, we found this likely to be due to the degree of the magnetic field’s alignment with respect to the 3D mesh, rather than discretization factors such as the choice of finite approximately 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5 × 105 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 13 element, type of cell and cell aspect ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The solver in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In practice, we apply a block lower triangular preconditioner of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='14) � −√κ∆Gb 0 1 ∆t(M + ˜κBCMBC) + κ⊥L √κ∆GT b �−1 , as a preconditioner to an outer flexible GMRES Krylov iteration solving the system in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1), with an outer tolerance set to 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We use lower triangular as opposed to upper triangular so as to directly include the isotropic operator in our preconditioning, even if is it not being inverted (and experimental results have supported this, with lower triangular preconditioning converging 10-20% faster than upper triangular).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Following [26], we do not use block LDU preconditioning, noting that it typically provides minimal improvement in convergence compared with block triangular, at the expense of an additional (approximate) solve of a diagonal block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Inner solves of (√κ∆Gb)−1 and (√κ∆Gb)−T are approximated using AIR-AMG [19, 21] as a preconditioner for right-preconditioned GMRES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We choose an inner relative residual stopping tolerance of min{10−3, 10−3/∥b∥}, where b is the current right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' This is effectively enforcing both an absolute and relative residual tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We do this because when using block preconditioning with inexact Schur complements, it is almost always more computationally efficient to use approximate rather than exact inner solves (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=', Navier Stokes results in [26] suggest a tolerance of about 10−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' However, when (∆tκ∆)−1 or κ⊥/κ∆ is not trivially small (in our experience ⪆ 10−4), the right-hand side provided to the lower triangular solve can be very large, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=', O(104).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In such cases, we find that a relative stopping tolerance of 10−3 doesn’t even lead to a residual < 1 in norm, and the outer iteration fails to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Thus, for each inner solver we enforce achieving at least three digits in both absolute and relative residual tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Block AIR is applied using the PyAMG library [3], with the discretization ma- trices grouped by DG element blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Ruge-Stuben coarsening [24] with second pass is applied on a block level using a classical strength of connection with tolerance θC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='01 based on a hard minimum, {−aij ≥ θC maxk |aik|}, and calculated on a block basis as implemented in PyAMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' No pre-relaxation is used and FFC-block- Jacobi post relaxation is applied with no damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We use one-point interpolation and distance-one block AIR [19] with strong connections determined through a classical strength of connection as above with tolerance θR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Having described the scheme and solver strategy, we next test their properties in terms of convergence and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' To set the scene, we first describe other discretizations and a standard solver strategy to compare against, as well as implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Comparison to other discretizations and solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For the test cases below, we will consider three additional discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' This includes the CG-primal system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' without the auxiliary heat flux term) for T in the standard second polynomial order continuous Galerkin space VCG 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For prisms, the latter is constructed analogously to the DG case, using the tensor product of the standard CG element P2 for triangles and the analogous 1D element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In this case, all gradient operations are well-defined, and the discretization is given by � η, ∂Th ∂t � +⟨b · ∇η, κ∆b · ∇Th⟩+⟨∇η, κ⊥, ∇Th⟩ = ⟨η, S⟩ ∀η ∈ VCG 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1) 14 Additionally, we consider the VCG 2 VDG 1 mixed finite element formulation based on Gunter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' [13], which reads � η, ∂Th ∂t � + ⟨b · ∇η, √κ∆ζh⟩ + ⟨∇η, κ⊥∇Th⟩ = ⟨η, S⟩ ∀η ∈ VCG 2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2a) ⟨φ, ζh − √κ∆b · ∇Th⟩ = 0 ∀φ ∈ VDG 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2b) Finally, we consider a DG-primal system for T in VDG 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For this purpose, we use a discretization similar to the conductivity tensor approach in [12, 29], except that we use a split isotropic-anisotropic formulation of the continuous equations in order to have a scheme that is equal to the mixed DG formulation up to the treatment of the purely anisotropic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The temperature equation is then given by � φ, ∂Th ∂t � − IPb(Th;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' φ) − IP(Th;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' φ) = ⟨φ, S⟩ ∀φ ∈ VDG 2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3) where the isotropic term IP is discretized according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6), and the anisotropic term IPb is given by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The interior penalty parameter for the resulting isotropic DG-Laplacian is set as for the mixed DG setup, while the one for the anisotropic DG-Laplacian is set to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' As before, the above space discretizations are discretized in time using the implicit midpoint rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The exception for the inflow boundary term involving ζin – as discussed at the end of Section 3 – does not apply here, since the above space discretizations do not contain such a term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Next to alternative discretizations, we also consider an alternative solver strategy for the DG and CG mixed systems, which is given by the standard block matrix form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='9) with the natural (1,1) Schur complement setup as a precondioner to an outer flexible GMRES Krylov iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The lower right block – which corresponds to the DG auxiliary variable mass matrix – is solved using a conjugate gradient Krylov iteration with block Jacobi as a precondioner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The Schur complement is solved for using a preconditioner ˜S−1 = � 1 ∆tMT + κ⊥L) + κ∆GT b [diag(Mζ)]−1Gb �−1 to an inner GMRES Krylov iteration, with the preconditioner solved for using BoomerAMG from hypre [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The multigrid options are left to the default ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The outer and inner tolerances are set to 10−8 and 10−3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The discretization and solver were implemented using Firedrake [23], which heavily relies on PETSc [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' A parallel version of AIR is im- plemented in hypre, which can be called in Firedrake through the PETSc interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' However, the interface currently does not contain the full set of options available to tune AIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' As mentioned in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4, we therefore instead call AIR through PyAMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The corresponding Python functions are then used in the Firedrake frame- work via a custom preconditioner interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The resulting setup can only be used in serial, and the solver efficiency results below therefore show wall-clock times for serial runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The diffusion-based multigrid based solver strategy described in the previous sub- section uses a classical AMG solver as implemented in hypre’s BoomerAMG [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The hypre implementation allows for more options and is more geared towards efficiency than the PyAMG version of classical multigrid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' indeed we found the BoomerAMG implementation to generally require slightly fewer iterations, as well as less wall-clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Note, we also tested several AMG methods implemented in PyAMG specifically designed for anisotropic diffusion, including [20, 22, 25], but again found the hypre 15 classical AMG implementation to consistently win in terms of iteration count and wall-clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For the convergence results, all systems were solved with a direct solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' DG based discretizations are solved using mumps [1], as this is less taxing on the memory (DG spaces have more degrees of freedom than the corresponding CG spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For CG, mumps did not consistently converge at higher anisotropy ratios and higher resolutions, and so superlu-dist [18] was used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For problems where both of these choices worked, the results were the same up to round-off;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' the choice of solver is for computational speedup and does not affect the solution accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Next to the DG scheme described in Section 3, as well as the above schemes used for comparison purposes, we also attempted an AIR based strategy for a mixed CG discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' However, we found such a scheme difficult to implement in view of the strong Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In typical finite element codes, it is non-trivial to extract the resulting symmetric transport operators, noting that we need the latter operators for our block-row swapped solver strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Other than the implementation difficulty, we further thought it justified to skip the mixed CG discretization together with the AIR framework, since the mixed DG formulation leads to a lower connectivity than the CG one, which is advantageous for the latter framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Convergence Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Here, we test the convergence order of our novel space discretization, using a variation of a test case considered in [13] and originally presented in the paper introducing the magnetohydrodynamic model NIMROD [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The original test case from the literature is posed in 2 dimensions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' for the purposes of our problem, we extrude this into 3D in order to avoid having to deal with the magnetic field singularity (with |B| = 0) present in the 2D problem, which we found to be problematic for the upwinding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Since such singularities do not occur in physically realistic scenarios, we do not consider this as a weakness of our scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The domain is described by Ω = [0, Lx] × [0, Lx] × [0, Lz], which is periodic in the z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We take Lx = 1, Lz = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The initial temperature field is given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4) T0 = sin �πx Lx � sin �πy Lx � , with TBC = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The magnetic field B is chosen to align with the contours of T0 in the two base dimensions and be constant in the third;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' namely that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5) B = (Bx, By, Bz) = (−∂yT, ∂xT, 5), which ensures that (Bx, By) = ∇⊥T0 for 2D curl ∇⊥ = (−∂y, ∂x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Finally, as in the original test case, we add a counter forcing S given by −κ⊥∆T0, which is included in order to ensure a steady state test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The error can then be measured relative to the initial condition expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4) as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6) eT (t) = ∥Th(t) − T0∥2 ∥T0∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' As described in Section 3, in meshing the domain, we use a triangular mesh for [0, Lx] × [0, Lx], which is then extruded in the third dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Since the initial con- ditions are highly symmetric, we use a perturbed version of a regular triangular mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' This breaks any symmetry in the problem due to possible alignments of the initial conditions with the mesh, to ensure that our convergence study really is analyzing the effect of the discretization on the error, as opposed to symmetries in the particular problem chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The base mesh is depicted in Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 16 Three resolutions are tested, starting from resolution ∆x = Lx/7 for the base mesh, which is then twice refined by splitting cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For the resolution in the extruded direction, we note that due to the setup of the initial conditions, we find that 2 cells are sufficient to resolve the latter direction, and increasing the number of cells will not decrease the overall error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' To keep the overall computational cost low, we therefore use 2 cells in the extruded direction for all three base resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Finally, we use a time step of ∆t = 10−3 for 100 time steps, giving a final time of tmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We test anisotropy ratios of 103, 106 and 109 – implementationally, we set κ⊥ = 1, such that the anisotropy ratio is equivalent to κ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The resulting convergence plots are presented Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1, where we consider eT (t) averaged over the last two time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' This is done since for the lower anisotropy ratio 103, we found all four discretizations to exhibit a small degree of oscillatory behavior in the error across time steps, likely due to an interplay between the isotropic diffusion and counter-forcing terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Relative L2 error (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6) convergence plots for the two non-mixed and mixed discretiza- tions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The anisotropy ratios are given from left to right by 103, 106, and 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Orange circles denote the novel mixed DG scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5), green diamonds the mixed CG scheme (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2), red triangles the primal DG scheme (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3), and purple triangles the primal CG scheme (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The black lines indicate first to third order convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For an anisotropy ratio of 103, all four schemes show similar errors and conver- gence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For 106, all schemes still display second to third order convergence rates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' however, the errors for the newly introduced scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5) are approximately two or- ders of magnitude smaller than for the other three schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Finally, for the high degree of anisotropy 109, scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5) still displays favorable error values and conver- gence rates, while the remaining schemes exhibit relative errors of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Inspecting the runs’ field development, we found these large errors likely to be due to a large spurious contribution of the heat flux’ parallel component in the perpendicular direc- tion, thereby rendering the latter direction overly diffusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The mixed DG scheme, on the other hand, is able to avoid such an excessive spurious diffusion likely due to a better representation of cross-cell fluxes through the upwind formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Note that the results for the mixed CG scheme are considerably worse than in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We found this to be unrelated to using a 3D version of the original test case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' instead, the differ- ence seems to be related to the use of triangles in the base mesh in this work, versus quadrilaterals in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Switching to quadrilaterals, in additional tests not shown here, we found the performance of the mixed CG scheme to be drastically improved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' albeit with an error that was still larger than the one for the mixed DG scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 17 100 10-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 10-1 10-1 10-2 10-4 Th E P2 Th E dP2 10-2, 10-3 (Th, h)E P2 × dP1 (Th, Sh)E dP2 X dP2 α △x1 10-4 10-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' α △x2 10-3, α △x3 10-1 6×10-2 4×10-2 10-1 6 × 10-2 4 × 10-2 6×10-24×10-2 10-1 Horizontal cell side lengthAxFinally, we again stress a caveat of the newly introduced scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5), given by the heat flux guess ζin at the inflow boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' While non-dynamical test cases – such as the one considered here – allow for an analytic solution given by the initial condition, they may conceal problems related to dynamics that vary in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' That said, in further tests with quadrilateral meshes, a non-steady dynamical evolution, and an anisotropy ratio of 103, we found little difference between the mixed CG and the mixed DG discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Solver Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' While we used a modification of the test case from [27] for the convergence results, here we consider a new test case, which in particular has open field lines only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We consider the same base mesh and domain as in the previous test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Again, the base mesh starts with resolution ∆x = Lx/7, and we use 2 cells in the extruded direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' As before, we refine twice by splitting cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' However, this time, we also refine the number of cells in the extruded direction in order to ensure a truly three-dimensional solver study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For the highest refinement level, this leads to approximately 105 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5 × 105 degrees of freedom for VCG 2 and VDG 2 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The initial temperature field is given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='7) T0 = 1 + 1 20 � 1 − cos �2πy Lx �� sin �πx Lx � + x + y 10, with TBC defined according to the values of T0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' As in the convergence study, the magnetic field B is chosen to align with the contours of T0 in the two base dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' This leads to field lines which are slightly slanted with respect to the domain, as well as curved towards the domain’s center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Further, B is again chosen to be constant in the third dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' this time, we set Bz = 15 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Finally, since here we only consider the solver performance and not convergence rates, there is no need for an analytic solution and the associated counter-forcing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' hence we set S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In the following, we consider results for a) the mixed DG-scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5) together with the AIR solver strategy described in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4, b) the mixed DG-scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5) together with the classical AMG based solver strategy described in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1, as well as c) the mixed CG scheme (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2) together with the aforementioned classical AMG based solver strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' As before, the time step is set to ∆t = 10−3, and we consider iteration counts as well as wall-clock time from the 2nd to the 5th time step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' a total of four time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Note that we disregard the first time step as a simple way of avoiding to count wall-clock time associated with form assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Further, runs which either exceed a specified average wall-clock time or outer iteration count per time step (approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 1500 seconds and 104 iterations, respectively) are discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The resulting averages per time step for the inner iteration counts, as well as wall-clock times, for anisotropy ratio κ∥/κ⊥ = 10k, k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=', 10, are depicted in Figures 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' As expected for the mixed CG scheme solved using classical AMG, we find a steady increase in solver iteration counts and according wall-clock times as the ratio of anisotropy increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For the highest refinement level, runs with anisotropy ratio 107 or higher did not meet the imposed cutoff criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Passing from the first to the second refinement level, runs at the highest three anisotropy ratios experience a wall-clock time increase by a factor of approximately 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Surprisingly, for the mixed DG scheme solved using classical AMG, we find the increase in solver iteration count and wall-clock times to eventually stall as we increase the ratio of anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For the lowest refinement level, the times stall at approximately 10 seconds, while for the next higher one, they do at about 240 seconds – which indicates a roughly 24-fold increase across the two refinement levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For this setup, most of the highest refinement level 18 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Average total inner iteration counts per time step using setup described in Sub- section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Top row: counts with respect to anisotropy ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Bottom row: counts with respect to refinement level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Left column: mixed CG scheme solved using classical AMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Center column: mixed DG scheme solved using classical AMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Right column: mixed DG scheme solved using AIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' runs did not meet the imposed time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In comparison to the mixed CG scheme solved using classical AMG, this is likely due to the much higher number of degrees of freedom associated with the DG scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For the mixed DG scheme solved using our newly proposed solver strategy in- cluding AIR, we instead find a decrease in iteration counts and wall-clock times as the anisotropy ratio increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' As described in Section 4, this is as expected, as such higher ratios fit better into our solver strategy’s favorable parameter regime (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For the highest refinement level, runs with anisotropy ratio 103 or lower did not meet the imposed time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In particular, the advection-based block preconditioner strug- gles with lower anisotropy ratios, while it performs very well for ratios 108 and higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For instance, at the highest anisotropy ratio 1010 and the medium refinement level, the mixed CG formulation solved with classical AMG requires an average of approx- imately 104 iterations, while the mixed DG formulation solved with our new AIR based strategy only requires approximately 50, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' around 200 times fewer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Given the aforementioned rates of increase for the number of iterations as we increase the 19 (Th, h)E P2 X dP1 (Th,Sh)EdP2XdP2 (Th,h)EdP2XdP2 CLASSICAL CLASSICAL AIR 104 104 103 103 102 102 101 101 103 106 109 103 106 109 103 106 109 Kμ/K 104 104 103 103 102 102 101 101 0 1 2 0 1 2 0 1 2 refinement level ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' level 0 ￥-K|/K11E+2 K/K11E+5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='K/KI1E+8 ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' level 1 --+-K|/K1 1E+3 →-K|/K11E+6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='·K/KI1E+9 ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' level 2 K/KI1E+4 +·K/KI1E+7 -·K//K11E+10Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Average wall-clock times in seconds per time step using setup described in Sub- section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Top row: counts with respect to anisotropy ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Bottom row: counts with respect to refinement level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Left column: mixed CG scheme solved using classical AMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Center column: mixed DG scheme solved using classical AMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Right column: mixed DG scheme solved using AIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' refinement level, we expect this gap to rapidly increase for higher resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Simi- larly, for the same anisotropy ratio and refinement level, we find the mixed DG/AIR based run to be around 45 times faster than the mixed CG/classical AMG based run;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' this holds true even though the DG discretization has far more degrees of freedom than the CG one, and we use a classical AMG based code implementation that is more streamlined for efficiency than our AIR based one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Again, given the rates of increase in wall-clock time as we increase the refinement level, we expect this gap to be larger still for higher resolutions, provided that the AMG based approach still converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In addition, recent work has proposed a modified AIR [7] that reduces total computational time compared with ℓAIR as implemented in hypre [19, 21] by several times, suggesting further multiplicative speedups possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Finally, we find the difference in iteration count for the mixed DG scheme at the three highest anisotropy ratios to not be as pronounced between the AIR- and classical AMG based solver strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Picking again the example of an anisotropy ratio of 1010 at the medium refinement level, we find that the AIR based strategy 20 (Th, Sh)E P2 X dPi (Th,Sh)EdP2XdP2 (Th,h)EdP2XdP2 CLASSICAL CLASSICAL AIR 103 103 102 102 101 101 100 100 103 106 109 103 106 109 103 106 109 Kμ/K1 103 103 102 102 101 101 100 100 0 1 2 0 2 0 1 2 refinement level ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='level 0 ￥K//K11E+2 K//K11E+5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='K/KI1E+8 ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' level 1 --+- K|/KI 1E+3 3-→-K|/K11E+6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='K/KI1E+9 ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='level 2 K/KI1E+4 +·K//K11E+7 +K//K11E+10requires about 10 times fewer iterations than the classical AMG based one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' It is unclear why the classical AMG based strategy performs this well in terms of iteration count for the mixed DG scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' However, considering the wall-clock time, we find that the mixed DG/classical AMG based run is again around 30 times slower than the AIR based one, with significant degradation as the mesh is further refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We end the discussion with observations on the AIR strategy’s solver behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' First, we found the number of inner iterations to remain very confined throughout the range of anisotropy ratios that we tested in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For the highest refine- ment level and an anisotropy ratio of 102, the inner solves for the transport operator and Schur complement take approximately 10 and 22 iterations per outer iteration, re- spectively, with the larger number of Schur complement iterations due to the modified inner tolerance proposed in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For 106, we obtain 10 and 19, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Finally, for 1010, the counts are 10 and 14, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In other words, the inner solver count for the transport operator remains the same, while for the Schur com- plement, it slowly decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' This is because as κ⊥L becomes a more significant off-diagonal contribution to the Schur complement’s right-hand side as κ∆ decreases, we require a larger residual reduction to meat an absolute residual tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Second, considering the relatively low inner iteration counts throughout the range of anisotropy ratios, we note that it is the outer solver that begins to struggle at anisotropy ratios of 105 or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Recall that the Schur complement is preconditioned using the transport operator (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5)), which we expect to work well when we are within the valid parameter regime (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='13) for our solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The latter regime increas- ingly ceases to be satisfied for ratios 105 or less, as can be inferred from the largest eigenvalue Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For a ratio 105, the largest eigenvalues of the preconditioned Schur complement are 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6, 5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='4 for the three spatial refinements, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' They then further increase inversely proportionally to the anisotropy ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Consis- tent with the spectral analysis (see Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1), the proposed method is efficient in 2D for smaller anisotropies, on the order of 103, but results are not presented due to space, and because we are interested in 3D-simulations in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' These considerations aside, we find that altogether, for the very high anisotropy ratios > 106 typically considered in magnetic confinement fusion, our novel mixed DG/AIR based scheme performs very well, and clearly outperforms the classical AMG based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Confined plasma simulations may exhibit high anisotropy, which needs to be resolved accurately in order to avoid cross-contamination of heat flux from components parallel to the magnetic field into components perpendicular to the mag- netic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' This cross-contamination in turn can lead to spuriously short simulated magnetic confinement times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In addition, such heat flux requires implicit integration, and for large-scale simulation requires efficient iterative solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In this work, we pre- sented a novel DG space discretization based on the heat flux as an auxiliary variable, and two upwind transport operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' It is coupled to a novel solver strategy based on the AIR AMG method, where the block matrix system resulting from an implicit time discretization is block-row swapped, and is solved for using a Schur complement approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For high ratios of anisotropy, the corresponding A11 block and Schur com- plement are then both akin to DG upwind transport operators, which can be solved for efficiently using AIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We compared the proposed spatial discretization to a CG and a DG direct dis- cretization, as well as another CG mixed discretization which also uses the heat flux as an auxiliary variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Further, we compared the AIR based solver strategy with 21 a standard one based on classical AMG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In the first test case based on (extruded) prism elements, we showed that for moderate rates of anisotropy, our novel space dis- cretization has comparable errors and an equal convergence rate when compared to the three other discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Further, for higher rates of anisotropy, our discretiza- tion retains the low errors and favorable convergence rate (third-order convergence for 2nd-order basis functions), while the remaining discretizations’ errors and rates deteriorate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For anisotropy ratios of 109, our proposed discretization achieves error ≈ 1000 times smaller than the other discretizations tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In the second set of tests, we demonstrated the newly proposed AIR based solver strategy to be highly efficient at anisotropy ratios 106 and higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We found the latter strategy to take up to 200 times fewer solver iterations and 45 times less wall-clock time than the classical AMG approach at a moderate space resolution, in spite of using a more streamlined and efficient classical AMG implementation than the one for AIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Additionally, we found the AIR based solver strategy to run within our imposed time and iteration count cut-off criteria at higher resolutions, where this was no longer the case for the AMG approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' A major shortcoming of the proposed solver methodology is that it requires all magnetic field lines to be open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Ongoing work is focused on developing a generalized solver strategy for the proposed discretization that is robust for closed field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Intuition for DG-upwind operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The DG-upwind trans- port operator based discretization of the anisotropic heat flux can be motivated start- ing from an interior penalty formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' For the anisotropic heat flux term (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1) ∇ · � κ∆b(b · ∇T) � , one such interior penalty formulation is given analogously to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='6) by IPb(Th;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' φ) = − ⟨b · ∇φ, κ∆b · ∇Th⟩ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2a) + � Γ �(b · n)Th� {κ∆b · ∇φ} dS + � Γ �(b · n)φ� {κ∆b · ∇Th} dS (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2b) − � Γ κ∆κp he �(b · n)φ� �(b · n)Th� dS − � ∂Ω 2κ∆κp he φ(Th − TBC) dS (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2c) + � ∂Ω κ∆(b · n)(b · ∇φ)(Th − TBC) dS + � ∂Ω κ∆(b · n)(b · ∇Th)φ dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2d) Next, we reformulate the central difference facet integrals (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='2b) to upwinded ones, further appropriately replace occurrences of √κ∆b · ∇Th by ζh, and drop the interior facet penalty term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' We then obtain, after some rewriting, � IPb(Th;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' φ) = − ⟨b · ∇φ, κ∆b · ∇Th⟩ − � ∂Ω 2κ∆κp he φ(Th − TBC) dS (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3a) + � Γ �√κ∆Thb · n� √κ∆ � b · ∇φ dS + � Γ �√κ∆φb · n� ˜ζh dS (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3b) + � ∂Ω √κ∆(Th − TBC)(b · n)√κ∆(b · ∇φ) dS + � ∂Ω √κ∆φ(b · n)ζh dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3c) 22 Finally, we modify the boundary terms (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3c) in accordance with the upwind modifi- cation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' to so so, we set Th = TBC and ζh = ζin along ∂Ωin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In particular, the portion along ∂Ωin of the first integral in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3c) then vanishes, and only the one along ∂Ωout remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Additionally, we reformulate the penalty parameter in the boundary term (last term in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='3a) as 2κ∆κp he → κBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The resulting formulation can then be retrieved from our scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5) – ignoring the time derivative and isotropic heat flux terms – by setting ψ = √κ∆b · ∇φ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Note that this is merely a motivation relating the split-transport setup to an interior penalty method, since √κ∆b · ∇φ will generally not be in VDG k and we therefore cannot set ψ accordingly after discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Importantly, however, the scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='5) still preserves the correct diffusive and total energy behavior weakly, as shown in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Base mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Here, we describe the 2D base mesh used for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' Base mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' the base square of side length Lx in the numerical re- sults section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' It is generated starting from a regular triangular mesh with resolution ∆x = Lx/7, whose vertices are then perturbed by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='06∆x in each interior coordinate in order to avoid spurious er- ror cancellations due to symmetries between the mesh and initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' The resulting mesh is depicted in Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' In general, for large anisotropy ratios, we found the error values in the convergence study to be very sensitive to the under- lying mesh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FQT4oBgHgl3EQfizaw/content/2301.13351v1.pdf'} +page_content=' we therefore decided to rely on a constructed pseudo-irregular base mesh rather than a fully irregular one for better 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100644 index 0000000000000000000000000000000000000000..bdea18375ad7c632c5658875d0849f95d6c61581 --- /dev/null +++ b/ctE0T4oBgHgl3EQf5AKl/content/tmp_files/2301.02746v1.pdf.txt @@ -0,0 +1,2083 @@ +arXiv:2301.02746v1 [math.FA] 6 Jan 2023 +REINHARDT FREE SPECTRAHEDRA +MUNIR BEN JEMAA∗ AND SCOTT MCCULLOUGH +Abstract. The automorphism group of a particular free spectrahedron is determined via +a novel argument involving algebraic methods. +1. Introduction +Fix norm 1 matrices C1, C2 of size s × s. For positive integers n, let Mn(C)2 denote the +set of pairs X = (X1, X2) of n×n matrices and let P[n] denote those X ∈ Mn(C)2 for which +the hermitian block 4 × 4 matrix +L (X) = + + + + + +Is ⊗ In +C1 ⊗ X1 +C2 ⊗ X2 +0 +(C1 ⊗ X1)∗ +Is ⊗ In +0 +C2 ⊗ X2 +(C2 ⊗ X2)∗ +0 +Is ⊗ In +C1 ⊗ X1 +0 +(C2 ⊗ X2)∗ +(C1 ⊗ X1)∗ +Is ⊗ In + + + + + +is positive definite. Here X∗ +j is the adjoint (complex transpose) of Xj and In is the n × n +identity matrix. The sequence of sets P = (P[n])n is an example of a free spectrahedron. +Given X ∈ Mn(C)2 and Y ∈ Mm(C)2, and a unitary matrix U ∈ Mn(C), let +X ⊕ Y = +�� +X1 +0 +0 +Y1 +� +, +� +X2 +0 +0 +Y2 +�� +and +U∗XU = (U∗X1U, U∗X2U). +Observe, if X ∈ P[n] and Y ∈ P[m], then X ⊕ Y ∈ P[n + m] and U∗XU ∈ P[n]; that is P +is closed with respect to direct sums and unitary similarity. +Let M(C) denote the sequence (Mn(C)) and let M(C)2 denote the sequence (Mn(C)2). +A free analytic function f : P → M(C) is a sequence (f[n]) of analytic functions f[n] : +P[n] → Mn(C) that respects direct sums and unitary similarities. That is, given X ∈ P[n] +and Y ∈ P[m] and a unitary matrix U ∈ Mn(C), +f[n + m](X ⊕ Y ) = f[n](X) ⊕ f[m](Y ) +2010 Mathematics Subject Classification. 47L25, 32H02 (Primary); 52A05, 46L07 (Secondary). +Key words and phrases. bianalytic map, birational map, free spectrahedron, free analysis. +∗ Research supported by the University of Florida University Scholars undergraduate research program. + +2 +BEN JEMAA AND MCCULLOUGH +and +f[n](U∗XU) = U∗f[n](X)U. +Typically we write f in place of f[n]. The general definition of a free analytic function appears +in Subsection 1.3 below. While it may not immediately appear so, free analytic functions +are the natural (freely) non-commutative analogs of analytic functions in several complex +variables. +A free analytic mapping ϕ = (ϕ1, ϕ2) : P → P is a pair of free analytic functions +ϕj : P → M(C)2 such that +ϕ(X) = (ϕ1(X), ϕ2(X)) ∈ P +for all X ∈ P. An automorphism ϕ of P is a free analytic mapping ϕ : P → P for which +there exists a free analytic mapping ψ : P → P such that ψ(ϕ(X)) = X = ϕ(ψ(X)) for +X ∈ P. +Given γ = (γ1, γ2) ∈ C2 with |γj| = 1, the function f(x) = (γ1x1, γ2x2) is automorphism +of P. Likewise, f(x) = (γ2x2, γ1x1) is an automorphism. We call these automorphism trivial +automorphisms. Theorem 1.1 below is the main result of this paper. +Theorem 1.1. If +(i) C1 and C2 are invertible; and +(ii) the C-star algebras generated by {C∗ +1C1, C∗ +2C2} and {C1C∗ +1, C2C∗ +2} are all of Ms(C), +then the automorphisms of P are trivial. +We wish to highlight two other contributions of this article. Proposition 1.2 provides an +alternate characterization of free analytic functions tailored to the study of automorphisms +of free domains. Its proof is modeled after arguments found in [A+]. Proposition 2.3 charac- +terizing spectraballs is from [EHKM]. Here we provide an alternate proof. The remainder of +this introduction contains more complete definitions of free spectrahedra and spectraballs, +free analytic functions and maps, as well as background and motivation for studying the +automorphism group of P. Preliminary results are contained in Section 2 and the proof of +Theorem 1.1 appears in Section 3. +1.1. Free polynomials and their evaluations. Fix a positive integer g. Let x = (x1, . . . , xg) +denote g freely non-commuting variables and let denote the semigroup of words in x +with ∅, the enpty word, playing the role of the identity. The length of the empty word is 0 +and otherwise the length of a word +(1.1) +w = xj1xj2 · · · xjm +is m denoted |w| = m. + +REINHARDT SPECTRAHEDRA +3 +For positive integers n, let Mn(C)g denote the set of g-tuples X = (X1, . . . , Xg) of n × n +matrices with entries from C. Let M(C)g denote the sequence (Mn(C)g). Given a tuple +X = (X1, . . . , Xg) ∈ M(C)g, let +Xw = Xj1Xj2 · · · Xjm, +with w ∈ as in equation (1.1). Thus Xw is the evaluation of the word w at the tuple +X. This evaluation extends to the free algebra of free polynomials C equal the C linear +combinations of elements of . Elements p ∈ C have the form +(1.2) +p = +� +w∈ +pww, +where the sum is finite. The polynomial p evaluates at X ∈ M(C)g as +p(X) = +� +w∈ +pwXw. +A matrix-valued free polynomial can be viewed either as a matrix with polynomial entries +or a polynomial with matrix coefficients. In the latter case, given positive integers d, e and +pw ∈ Md,e(C), the finite sum in equation (1.2) is a matrix valued polynomial. To evaluate +this p at a tuple X ∈ M(C)g we will make use of the (Kronecker) tensor product S ⊗ T of +matrices S and T, setting +p(X) = +� +pw ⊗ Xw. +1.2. Free spectrahedra. Given A ∈ Md×e(C)g, let ΛA denote the homogeneous linear pencil +ΛA(x) = � +j Ajxj. It evaluates at X ∈ Mn(C)g as +ΛA(X) = +g +� +j=1 +Aj ⊗ Xj ∈ Md×e(C) ⊗ Mn(C). +In the case A is square (d = e), we let +LA(X) = Id ⊗ In + ΛA(X) + ΛA(X)∗ += I + +� +Aj ⊗ Xj + +� +A∗ +j ⊗ X∗ +j ∈ Md(C) ⊗ Mn(C). +The set DA[1] ⊆ Cg consisting of x ∈ Cg such that LA(x) ≻ 0 is a spectrahedron. Spectra- +hedra are basic objects in a number of areas of mathematics; e.g. semidefinite programming, +convex optimization [WSV12] and real algebraic geometry [BPT]. +The free spectrahedron determined by A ∈ Md(C)g is the sequence of sets DA = (DA[n]), +where +DA[n] = {X ∈ Mn(C)g : LA(X) ≻ 0}, + +4 +BEN JEMAA AND MCCULLOUGH +and T ≻ 0 indicates that the square matrix T is positive definite (hermitian with positive +eigenvalues). Observe that P is the free spectrahedron DR, where +(1.3) +R1 = + + + + + +0 +C1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +C1 +0 +0 +0 +0 + + + + + , +R2 = + + + + + +0 +0 +C2 +0 +0 +0 +0 +C2 +0 +0 +0 +0 +0 +0 +0 +0 + + + + + . +A free spectrahedron DA is not determined by the spectrahedron DA[1]. See Proposi- +tion 2.5. Free spectrahedra are canonical objects in the theories of operator systems and +spaces and completely positive maps. They are related to quantum channels from quan- +tum information theory. That they arise naturally in certain systems engineering problems +governed by a signal flow diagram [dOH06, dOHMP, SIG97] also provides motivation for +studying free spectrahedra. +1.2.1. Spectraballs. Given a tuple G = (G1, . . . , Gg) of d × e matrices, the sequence BG = +(BG[n])n defined by +BG[n] = {X ∈ Mn(C)g : ∥ +g +� +j=1 +Gj ⊗ Xj∥ < 1} +is a spectraball. The spectraball at level one, BG[1], is a rotationally invariant convex subset +of Cg. The spectraball BG is a spectrahedron since BG = DB for B = ( 0 G +0 0 ). Under the +hypotheses of Theorem 1.1, the spectrahedron P is not a spectraball. See Proposition 2.2. +A spectrahedron DA has its naturally associated spectraball, +(1.4) +BA = {X : ∥ΛB(X)∥ < 1} = {X : +� +0 +X +0 +0 +� +∈ DA}. +The spectraball BR associated to P = DR plays an important role in this article. +1.2.2. Free sets. A free set S ⊆ M(C)g is a sequence S = (S [n])n such that S [n] ⊆ +Mn(C)g for each positive integer n and such that S is closed with respect to direct and +unitary similarity: +(i) if X ∈ S [n] and Y ∈ S [m], then +X ⊕ Y = (X1 ⊕ Y1, . . . , Xg ⊕ Yg) ∈ S [n + m], +where +Xj ⊕ Yj = +� +Xj +0 +0 +Yj +� +; + +REINHARDT SPECTRAHEDRA +5 +(ii) if X ∈ S [n] and U ∈ Mn(C) is unitary, then +U∗XU = (U∗X1U, . . . , U∗XgU) ∈ S [n]. +We say S is open if each S [n] is open and S is bounded if there exists a κ such that for all +n and X ∈ S [n], the n × ng matrix +� +X1 +. . . +Xg +� +has norm at most κ. +A free spectrahedron is an open free set. +1.3. Free analytic functions. Given an open free set S , a free function f : S → M(C) +is a sequence f = (f[n]), where f[n] : S �→ Mn(C), that satisfies the axioms +(i) if X ∈ S [n] and Y ∈ S [m], then +f[n + m](X ⊕ Y ) = f[n](X) ⊕ f[m](Y ); +(ii) if X ∈ S [n] and U is an n × n unitary matrix, then +f[n](U∗XU) = U∗f[n](X)U. +Thus free functions respect direct sums and unitary similarity. The free function f is analytic +if each f[n] is analytic. We typically write f in place of f[n]. +Turning to examples, free polynomials are evidently free analytic functions. +A free +rational function r (regular at 0) is a free analytic function that has a realization formula; +that is, there exists a positive integer e, a tuple A ∈ Me(C)g and vectors c, b ∈ Ce, such that +r(x) = c∗(I − ΛA(x))−1b. +The natural domain of r consists of those tuples X ∈ Mn(C)g for which I − ΛA(X) is +invertible and for such an X, +r(X) = (Id ⊗ c)∗ (Id ⊗ In − ΛA(X))−1 (Id ⊗ b) ∈ Mn(C). +See [KVV14] for further information about free functions. +There are two other formulations of free functions that are equivalent and more common +in the literature. They do not not assume analyticity, but rather have it as a consequence +of mild additional assumptions such as continuity or boundedness, in which case they are +equivalent to the formulation adopted here. In one formulation, item (ii) is replaced by the +hypothesis that f respects similarities: if X ∈ S [n] and T is an invertible matrix such that +T −1XT ∈ S [n], then f(T −1XT) = T −1f(X)T. The other formulation replaces items (i) +and (ii) with the single axiom that f respects intertwinings: if X ∈ S [n] and Y ∈ S [m] + +6 +BEN JEMAA AND MCCULLOUGH +and Γ is an m × n matrix such that ΓX = Y Γ, then Γf(X) = f(Y )Γ. Proposition 1.2 is a +variation on [A+, Lemma 3.5]. A proof appears in Subsection 2.4. +Proposition 1.2. If S is a free open set and f : S → M(C) is a free analytic function, +then f respects intertwinings. +1.4. Automorphisms of free spectrahedra. Given free sets S ⊆ M(C)g and T +⊆ +M(C)g a free map f : S → T is an g-tuple f = (f 1, . . . , f g) of free functions f : S → M(C) +such that f(X) ∈ T for all X ∈ S . A bianalytic map f between free spectrahedra DA and +DB is a free map f : DA → DB for which there exists a free analytic mapping g : DB → DA +such that g(f(X)) = X and f(g(Y )) = Y for all X ∈ DA and Y ∈ DB. We refer to g as +the inverse of f and write g = f −1. A natural problem, from several different perspectives, +is to determine the bianalytic maps between two free spectrahedra. This problem is the free +analysis analog of rigidity phenomena in several complex variables and from this perspective +it is expected that two free spectrahedra are rarely bianalytic. +The paper[HKMV20] determines, under certain generic irreducibility inspried hypotheses +on A and B, the tuples (A, B, f) of bianalytic maps f : DA → DB. It turns out that A and +B are closely linked and f has a highly algebraic description. In that same paper, bianalytic +maps between spectraballs are determined without any additional hypotheses. Again, these +maps have a highly algebraic description. +An automorphism f of a free spectrahedron DA is a bianalytic map f : DA → DA. If +f, g : DA → DB are bianalytic, then the map g−1 ◦ f : DA → DA is an automorphism. Thus, +the automorphism group of DA places constraints on the bianalytic maps f : DA → DB. +A free set S is circularly symmetric if +γX = (γX1, . . . , γXg) ∈ S +whenever X ∈ S and γ ∈ C is unimodular. A natural class of free spectrahedra not covered +by the results in [HKMV20] are those with circular symmetry that are not spectraballs. In +particular, those P satisfying the hypotheses of Theorem 1.1 provide examples of a spectra- +hedron whose automorphism group is not yet classified. +We are now in a position to provide an overview of the proof of Theorem 1.1. A somewhat +routine argument shows if ϕ is a linear automorphism of P, then ϕ is trivial. A consequence +of the free analog of the Caratheodory-Cartan-Kaup-Wu from [HKM11] is the following. If +ϕ : P → P is an automorphism and ϕ(0) = 0, then ϕ is linear. The strategy employed here +to show if ϕ is an automorphism of P, then ϕ(0) = 0 is novel, using algebraic aspects of the +theory of spectraballs. Given an automorphism ϕ : P → P with ϕ(0) ∈ C2 not necessarily +0, we construct a tuple B described solely in terms of ϕ(0) ∈ C2 and ϕ′(0) ∈ M2(C) such +that BR = BB. This equality is analyzed using algebraic results from [HKMV20], ultimately + +REINHARDT SPECTRAHEDRA +7 +concluding s = 1. When s = 1, the spectrahedron P[1] is the generalized complex ellipsiod +{z = (z1, z2) ∈ C2 : |z1| + |z2| < 1}. It is known [K, JP08], using very different techniques +than those here, that the analytic automorphisms of this complex ellipsiod are trivial. In +particular, ϕ(0) = 0. We give a self contained proof of this fact using results developed in +this paper. +By contrast, the proof strategy for classifying the automorphism group of hyper-Reinhardt +domains, defined below and in [MT, M], proceeds via the dynamics of composing automor- +phisms and using results related to the classical Caratheodory interpolation theorem. +2. Preliminary results +This section collects preliminary and ancillary results to Theorem 1.1. Subsection 2.1 +provides alternate characterizations for membership in P and establishes that, under the +hypotheses of Theorem 1.1, P is neither a spectraball nor a hyper-Reinhardt domain. Ad- +ditionally, an alternate proof of a results from [EHKM] is given. Subsection 2.2 gathers +facts about the spectraball BR associated to P = DR. Ball minimality from [HKMV20] is +reviewed in Subsection 2.3, where a ball minimal tuple E such that BE = BR is identified. +Subsection 2.4 discusses the evaluation of a free function defined near 0 on nilpotent tuples. +The proof of Proposition 1.2 appears in Subsection 2.5. +2.1. Membership in P. A matrix T is a strict contraction if ∥T∥ < 1, where ∥T∥ is the +operator norm of T. +Proposition 2.1. For X = (X1, X2) ∈ M(C)2 and Yj = Cj ⊗ Xj, the following are equiva- +lent. +(a) X ∈ P; +(b) the matrix +T(X) := +� +Y ∗ +1 +Y2 +Y ∗ +2 +Y1 +� +is a strict contraction; +(c) the matrix +L ′(X) := +� +I − Y ∗ +1 Y1 − Y2Y ∗ +2 +−Y ∗ +1 Y2 − Y2Y ∗ +1 +−Y ∗ +2 Y ∗ +1 − Y1Y ∗ +2 +I − Y1Y ∗ +1 − Y ∗ +2 Y2 +� += I − T(X)T(X)∗ +is positive definite; + +8 +BEN JEMAA AND MCCULLOUGH +(d) The matrix +L ′ +∗(X) := +� +I − Y1Y ∗ +1 − Y2Y ∗ +2 +−Y1Y2 − Y2Y2 +−Y ∗ +2 Y ∗ +1 − Y ∗ +1 Y ∗ +2 +I − Y ∗ +1 Y1 − Y ∗ +2 Y2 +� += I − T(X)∗T(X) +is positive definite. +Proof. A selfadjoint block square matrix +M = +� +A +B +B∗ +D +� +with A positive definite is positive definite if and only if the Schur complement of its (1, 1) +block, +S = D − B∗A−1B +is positive definite. +By definition, X ∈ P means L (X) ≻ 0. Taking the Schur complement of L (X) first of +the (1, 1) block entry then off the (3, 3) block entry (an identity matrix) shows L (X) ≻ 0 if +and only if L ′(X) ≻ 0 using the observation at the outset of this proof. A direct computation +shows L ′(X) = I − T(X)T(X)∗. +To complete the proof observe, I − T(X)T(X)∗ ≻ 0 if and only if ∥T(X)∥ < 1 if and +only if I − T(X)∗T(X) ≻ 0 if and only if L ′ +∗(X) ≻ 0. +□ +Specializing to the case of two variables, a spectrahedron D ⊆ M(C)2 is hyper-Reinhardt +if there exists a tuple G = (G1, G2) of matrices of compatible sizes such that, with +A1 = + + + +0 +G1 +0 +0 +0 +0 +0 +0 +0 + + + , +A2 = + + + +0 +0 +0 +0 +0 +G2 +0 +0 +0 + + + , +we have D = DA. A hyper-Reinhardt free spectrahedra is circular, but not necessarily a +spectraball. The automorphisms of hyper-Reinhardt free spectrahedra are determined in +[MT, M]. +Proposition 2.2. Under the hypotheses of Theorem 1.1, the spectrahedron P is neither +hyper-Reinhardt nor a spectraball. +The proof of Proposition 2.2 uses one direction of the following result from [EHKM]. +Proposition 2.3. A spectrahedron DA is a spectraball if and only if for each positive integer +n, each X ∈ DA[n] and each unitary matrix U ∈ Mn(C), the tuple UX ∈ DA. + +REINHARDT SPECTRAHEDRA +9 +Proof. If DA = BB is a spectraball, then it is immediate that X ∈ DA[n] and U unitary +implies UX ∈ DA. +Conversely suppose for each positive integer n, each X ∈ DA[n] and each unitary matrix +U ∈ Mn(C), the tuple UX ∈ DA[n]. To prove DA is spectraball, it suffices to show DA = BA. +To this end, let X ∈ Mn(C)g be given. Let +U = +� +0 +In +In +0 +� +. +and observe X ∈ DA if and only if 0n ⊕ X ∈ DA[2n] if and only if +U(0n ⊕ X) = U +� +0 +0 +0 +X +� += +� +0 +X +0 +0 +� +∈ DA[2n] +if and only if X ∈ BA[n] (see Equation 1.4). Thus DA = BA and the proof is complete. +□ +The following lemma is also needed for the proof of Proposition 2.2. +Lemma 2.4. Under the hypotheses of Theorem 1.1, C∗ +1C1+C∗ +2C2 ≻ Is and C1C∗ +1+C2C∗ +2 ≻ Is. +Proof. If C∗ +1C1 + C∗ +2C2 ⪯ Is, then the kernel K of I − C∗ +1C1 is non-trivial and orthogonal +to the range of C∗ +2C2. Hence K reduces both C∗ +1C1 and C∗ +2C2 and hence the C-star algebra +they generate. Thus C∗ +1C1 + C∗ +2C2 ≻ Is. Likewise C1C∗ +1 + C2C∗ +2 ≻ Is. +□ +Proof of Proposition 2.2. A routine computation shows, if D ⊆ M(C)g is a hyper-Reinhardt +free spectrahedron, X ∈ D[n] and W0, W1, W2 ∈ Mn(C) are unitary matrices, then +W · X := (W ∗ +0 XW1, W ∗ +1 XW2) ∈ D[n]. +Another routine computation and an appeal to Proposition 2.1 shows +(2.1) +X = (X1, X2) = +�� +1 +0 +0 +0 +� +, +� +0 +0 +0 +1 +�� +is in the boundary of P. Let +(2.2) +U = +� +0 +1 +1 +0 +� +and (W0, W1, W2) = (I, U, U2). Thus each Wj is unitary and +W · X = +�� +0 +1 +0 +0 +� +, +� +0 +1 +0 +0 +�� +. +By item (d) or Proposition 2.1 and Lemma 2.4, W · X is not in the closure of P. Hence P +is not hyper-Reinhardt. + +10 +BEN JEMAA AND MCCULLOUGH +If P is a spectraball, X ∈ P[n] and U is an n×n unitary matrix, then UX = (UX1, UX2) +is also in P[n] by Proposition 2.3. Choose X1, X2 and U as in equations (2.1) and (2.2) and +note, with L ′ as in Proposition 2.1, +L ′(UX) = I − + + + + + +C1C∗ +1 + C∗ +2C2 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +C∗ +1C1 + C2C∗ +2 + + + + + , +since X∗ +1X2 = 0 = X2X∗ +1. If UX is in the boundary of P, then I − C∗ +1C1 − C2C∗ +2 ⪰ 0. By +assumption, there is a vector x such that C∗ +1C1x = x, from which it follows that C∗ +2x = 0, +contradicting the assumption that C2 is invertible. +□ +2.2. The spectraball BR. Let {e1, e2} denote the standard basis for C2 and let eT +j denote +the transpose of ej. Let Er ∈ M2(C)2 denote the tuple, +Er +1 = eT +1 ⊗ C1 = +� +C1 +0 +� +, +Er +2 = eT +2 ⊗ C2 = +� +0 +C2 +� +and let Ec ∈ M2(C) denote the tuple Ec = (Ec +1, Ec +2) where Ec +j = ej ⊗ Cj. +The spectraball associated to P is BR, where R is defined in equation (1.3). Since the +first row and last column of Rj are zero, BR = BE, where +(2.3) +E = (E1, E2) = + + + + + + +C1 +0 +0 +0 +0 +0 +0 +0 +C1 + + + , + + + +0 +C2 +0 +0 +0 +C2 +0 +0 +0 + + + + + + = +� +Er +0 +0 +Ec +� +. +Proposition 2.5. The equality BR = BE = BEr ∩ BEc holds. In particular, X ∈ BR if and +only if both +� +C1 ⊗ X1 +C2 ⊗ X2 +� +, +� +(C1 ⊗ X1)∗ +(C2 ⊗ X2)∗� +are strict contractions. On the other hand, under the hypotheses of Theorem 1.1, BEj[2] ̸⊆ +BEk[2] for j, k ∈ {r, c} and j ̸= k. +Proof. The equalities BR = BE = BEr ∩ BEc are immediate from equation (2.3). +Using Lemma 2.4, choose ρ < 1 such that ρ2(C∗ +1C1 + C∗ +2C2) ̸⪯ I. The tuple +X = ρ +�� +0 +1 +0 +0 +� +, +� +0 +0 +0 +1 +�� +is in BEr but not BEc. Hence BEr[2] ̸⊆ BEc[2]. A similar argument with +X = ρ +�� +1 +0 +0 +0 +� +, +� +0 +1 +0 +0 +�� +shows BEc[2] ̸⊆ BEr[2]. +□ + +REINHARDT SPECTRAHEDRA +11 +Remark 2.6. In the terminology of [HKMV20], the direct sum Er ⊕ Ec is irredundant. +□ +2.3. Ball minimality. The discussion of ball-minimality here is borrowed from [HKMV20]. +In particular, Lemma 2.7 below is excerpted from [HKMV20, Lemma 3.1]. A g-tuple E = +(G1, . . . , Gg) of d × e matrices is ball-mimimal for BG if F ∈ Mk×ℓ and BF = BG implies +d ≤ k and e ≤ ℓ. As examples, it is immediate that both Er and Ec are ball-minimal (for +BEr and BEc respectively). Tuples G, F ∈ Md,e(C)g are ball-equivalent if there exists unitary +matrices U ∈ Me,e(C) and V ∈ Md,d(C) such that F = V ∗GU. Observe that, in this case, +BF = BG. A tuple A ∈ Md(C)g is minimal for DA if B ∈ Mk(C)g and DB = cDA implies +d ≤ k. +Lemma 2.7. Ball minimal tuples exists; that is, if B ⊆ M(C)g is a spectraball, then there +exists d, e and a ball minimal tuple G ∈ Md,e(C)g such that B = BG. +Let G ∈ Md,e(C)g be given. +(i) If G ball minimal, F ∈ Mk×ℓ(C)g and BG = BF, then there is a tuple J ∈ M(k−d)×(ℓ−e)(C)g +and unitaries U, V of sizes k × k and ℓ × ℓ respectively such that BG ⊆ BJ and +F = U +� +G +0 +0 +J +� +V. +In particular, +(a) d ≤ k and e ≤ ℓ; +(b) if F ∈ Md×e(C)g is ball minimal too, then G and F are ball-equivalent. +(ii) G ∈ Md,e(C)g is ball minimal if and only if +H = +� +0 +G +0 +0 +� +∈ Md+e(C)g +is minimal for DH. +Proposition 2.8. The tuple E from equation (2.3) is ball-minimal. +Proof. Let +H = +� +0 +E +0 +0 +� +∈ M6s,6s. +Thus BE = DH and, by Lemma 2.7, E is ball minimal for BE if and only if H is minimal for +DH. +Let {Ej,k : 1 ≤ j ≤ 6} denote the matrix units for M6(C). Let +F1 = [E1,2 + E5,6] ⊗ C1, +F2 = [E1,3 + E4,6] ⊗ C2. + +12 +BEN JEMAA AND MCCULLOUGH +Since F and H are unitarily equivalent, we have DH = DF and moreover H is minimal +defining for DH if and only if F is minimal for DF if and only if E is ball minimal for BE. +Let F denote the C-star algebra generated by {F1, F2}. Since F1F ∗ +1 = [E1,1+E5,5]⊗C1C∗ +1 +and F2F ∗ +2 = [E1,1 + E4,4] ⊗ C2C∗ +2 and since, by assumption, {C1C∗ +1, C2C∗ +2} generates Ms(C) +as a C-star algebra, for each X ∈ Ms(C) there exist Y1, Y2 ∈ Ms(C) such that F contains +E1,1 ⊗ X + E4,4 ⊗ Y2 + E5,5 ⊗ Y1. Multiplying by F1F ∗ +1 F2F ∗ +2 on the right, it follows that +F contains E1,1 ⊗ XC1C∗ +1C2C∗ +2. Since C1C∗ +1C2C∗ +2 is invertible, it follows that F contains +E1,1 ⊗ Ms(C). By considering F ∗ +j [E1,1 ⊗ Ms(C)] Fk, for j, k = 0, 1, 2 and F0 = I, and arguing +as above, it follows that F contains �3 +j,k=1 Ej,k⊗Ms(C); that is F contains F1 = M3s(C)⊕0 as +a C-star subalgebra. By considering instead F ∗ +j Fj and using {C∗ +1C1, C∗ +2C2} generates Ms(C) +as a C-star algebra, it follows that F contains F2 = 0 ⊕ M3s(C) as a C-star subalgebra. +Hence F = M3s(C) ⊕ M3s(C). In particular, aside from the trivial ones, the only reducing +subspaces for F, equivalently, {F1, F2}, are C3s ⊕ {0} and {0} ⊕ C3s. +Suppose A ∈ Mk(C)2 is minimal defining for DF. Thus k ≤ 6s. By [EHKM, Proposi- +tion 2.2], there exist a tuple J ∈ M(C)g such that, up to unitary equivalence, F = A ⊕ J. +Suppose A ̸= F. It follows that either A = (E1,2 ⊗C1, E1,3 ⊗C2) or A = (E4,6 ⊗C1, E4,6 ⊗C2). +But then, DA is either BEr or BEc, contradicting the conclusion of Proposition 2.8 ({BEr, BEc} +is irredundant). Thus F is minimal for DF and E is ball minimal. +□ +2.4. Nilpotent evaluations. Given δ > 0, let +B(0, δ) = {X ∈ M(C)g : +g +� +j=1 +XjX∗ +j < δ2} +It is straightforward to verify B(0, δ) is an open free set. +A (formal) power series F in is an expression of the form +(2.4) +F(x) = +� +w∈ +Fww. +A tuple X ∈ M(C)g is nilpotent of order at most m if Xw = 0 for all words w of length +m. Theorem 2.9 below can be found in [KVV14, KS, HKM12] among other places. +Theorem 2.9. Suppose S is an open free set and there is a δ > 0 such that B(0, δ) ⊆ S. +If f : S → M(C) is a free analytic function, then there exists a formal power series as in +equation (2.4) such that if X ∈ B(0, δ), then +f(X) = F(X) = +∞ +� +ℓ=0 +� +|w|=ℓ +FwXw, + +REINHARDT SPECTRAHEDRA +13 +with the series converging in norm. Further, f extends analytically to all nilpotent tuples. +In particular, if X is nilpotent of order at most m, then +f(zX) = +m +� +ℓ=0 +( +� +|w|=ℓ +FwXw)zℓ +for z ∈ C. +Let +S = +� +0 +1 +0 +0 +� +. +Thus, given X = (X1, X2) ∈ Mn(C)2, the tuple +S ⊗ X = (S ⊗ X1, S ⊗ X2) = +�� +0 +X1 +0 +0 +� +, +� +0 +X2 +0 +0 +�� +∈ M2n(C)2 +is nilpotent of order two. The evaluation of free map f : P → M(C) on S ⊗ X takes a +particularly simple form. +Proposition 2.10. If f : P → M(C) is a free analytic function, then f extends uniquely to +a function, still denoted f, defined on all tuples of the form S ⊗X for X ∈ M(C)2. Moreover, +there exists ℓ1, ℓ2 ∈ C such that +f(S ⊗ X) = f(0) + S ⊗ ( +2 +� +j=1 +ℓjXj) ∈ M2(C) ⊗ Mn(C) = M2n(C). +We close this section with a proof of Proposition 1.2. The ideas borrow freely from [A+]. +2.5. Proof of Proposition 1.2. We are to show, if S is a free open set and f : S → M(C) +is analytic and a free function in the sense that f respects both direct sums and unitary +similarity, then f respects intertwinings. To this end, suppose X ∈ S [m] and Y ∈ S [n] +and Γ is an m × n matrix such that XΓ = ΓY. To show f(X)Γ = Γf(Y ), first observe we +may replace Γ with tΓ for any non-zero t ∈ C. Since S is closed with respect to direct sums +Z = X ⊕ Y ∈ S [n + m]. Since S [n + m] is open there is an ǫ > 0 such that if T is an +(n + m) × (n + m) matrix and ∥T − I∥ < ǫ, then T is invertible and +T −1 [X ⊕ Y ] T ∈ S [n + m]. +For a matrix M and ρ ∈ C with |ρ| ∥M∥ < 1, let +Dρ(M) = (I − ρ2M∗M) +1 +2 + +14 +BEN JEMAA AND MCCULLOUGH +and let, for 0 ̸= z ∈ C, +Tρ(M)[z] = +� +Dρ(M∗) +ρ +zM +−ρ zM∗ +Dρ(M) +� +. +For |z| = 1 the matrix Tρ[M](z) is a version of the Julia matrix and is unitary. +By choosing ∥Γ∥ sufficiently small, we may assume, for all 0 < ρ < 1, and ρ < |z| < 1 +ρ, +that +∥I − Tρ(Γ)[z]∥ < ǫ. +Hence, +Tρ(Γ)[z]−1 [X ⊕ Y ] Tρ(Γ)[z] ∈ S [n + m]. +The function (for fixed ρ) +F(z) = f +� +Tρ(Γ)[z]−1 [X ⊕ Y ] Tρ(Γ)[z] +� +− Tρ(Γ)[z]−1 f(X ⊕ Y ) Tρ(Γ)[z] +is defined and analytic on the annulus ρ ≤ |z| ≤ 1 +ρ and vanishes for |z| = 1 since f respects +unitary similarities. Thus F vanishes identically. Choosing z = ρ and then letting ρ tend to +0 gives, +f( +� +I +−Γ +0 +I +� � +X +0 +0 +Y +� � +I +Γ +0 +I +� +) = +� +I +−Γ +0 +I +� +f( +� +X +0 +0 +Y +� +) +� +I +Γ +0 +I +� +. +Thus, using f respects direct sums, +� +f(X) +0 +0 +f(Y ) +� += f( +� +X +0 +0 +Y +� +) = f( +� +X +XΓ − ΓY +0 +Y +� +) += f( +� +I +−Γ +0 +I +� � +X +0 +0 +Y +� � +I +Γ +0 +I +� +) = +� +I +−Γ +0 +I +� +f( +� +X +0 +0 +Y +� +) +� +I +Γ +0 +I +� += +� +I +−Γ +0 +I +� � +f(X) +0 +0 +f(Y ) +� � +I +Γ +0 +I +� += +� +f(X) +f(X)Γ − Γf(Y ) +0 +f(Y ) +� +and the desired conclusion follows. +3. A proof of Theorem 1.1 +In this section, we present a proof of Theorem 1.1. In Subsection 3.1, we construct +a spectraballs BB that is canonically associated with an automorphism ϕ : P → P. In +Subsection 3.2, we collect consequences of the equality BB = BR = BE that are then used in +Subsection 3.3 that b = 0. The proof concludes in Subsection 3.4. + +REINHARDT SPECTRAHEDRA +15 +3.1. An affine change of variables. In this section a spectraball BB canonically associated +to an automorphism ϕ : P → P is constructed. +Suppose ϕ : P → P is bianalytic. The mapping ϕ has coordinate functions ϕj so that +ϕ(X) = (ϕ1(X), ϕ2(X)) +for X ∈ P. +Express the series expansions for ϕj, up to the first degree terms as +ϕj(x) = bj + +� +k +ℓj,kxk + · · · . +Since ϕ is bianalytic, the matrix +(3.1) +L = +� +ℓ1,1 +ℓ1,2 +ℓ2,1 +ℓ2,2 +� +is invertible. +Given a tuple X ∈ Mn(C)2, +ϕj(S ⊗ X) = +� +bj +� +k ℓj,kXk +0 +bj +� +. +Let bj = bjCj. Let +B0 = + + + + + +I +b1 +b2 +0 +b∗ +1 +I +0 +b2 +b∗ +2 +0 +I +b1 +0 +b∗ +2 +b∗ +1 +I + + + + + . +Since ϕ(0) = b ∈ P, the matrix B0 = L (ϕ(0)) is positive definite and hence invertible. +Let +(3.2) +Yj = + + + + + +0 +ℓ1,jC1 +ℓ2,jC2 +0 +0 +0 +0 +ℓ2,jC2 +0 +0 +0 +ℓ1,jC1 +0 +0 +0 +0 + + + + + = +� +k +ℓk,jRk, +where R is defined in equation (1.3). Let B = (B1, B2) denote the tuple defined by +Bj = B +− 1 +2 +0 +YjB +− 1 +2 +0 . +Let ΛB(x) = B1x1 + B2x2 and let BB denote the resulting spectraball, +BB = {X : ∥ΛB(X)∥ < 1}. +Proposition 3.1. The equality BR = BB holds. + +16 +BEN JEMAA AND MCCULLOUGH +Before proving Proposition 3.1 we record the following lemma. +Lemma 3.2. There is a permutation matrix Σ on {1, 2, . . ., 8} such that +(3.3) +(Σ∗ ⊗ Is)L (ϕ(S ⊗ x))(Σ ⊗ Is) = +� +B0 +ΛY (x) +ΛY (x)∗ +B0 +� +. +Proof. Let yj = (�2 +k=1 ℓj,kxk) Cj. Identifying 0 with the 0 matrix of size s × s, +L (ϕ(S ⊗ x)) = + + + + + + + + + + + + + + +I +0 +b1 +y1 +b2 +y2 +0 +0 +0 +I +0 +b1 +0 +b2 +0 +0 +b∗ +1 +0 +I +0 +0 +0 +b2 +y2 +y∗ +1 +b∗ +1 +0 +I +0 +0 +0 +b2 +b∗ +2 +0 +0 +0 +I +0 +b1 +y1 +y∗ +2 +b∗ +2 +0 +0 +0 +I +0 +b1 +0 +0 +b∗ +2 +0 +b∗ +1 +0 +I +0 +0 +0 +y∗ +2 +b∗ +2 +y∗ +1 +b∗ +1 +0 +I + + + + + + + + + + + + + + +. +Thus equation (3.3) holds with Σ equal the tensor product of the matrix associated to the +permutation (1, 3, 5, 7, 2, 4, 6, 8) with Is, the s × s identity, after noting that + + + + + +0 +y1 +y2 +0 +0 +0 +0 +y2 +0 +0 +0 +y1 +0 +0 +0 +0 + + + + + = ΛY (x). +□ +Proof of Proposition 3.1. Using the assumption that ϕ is an automorphism of P, Equa- +tion (1.4) and Lemma 3.2, a tuple X is in BR[n] if and only if S ⊗ X ∈ P[2n] if and only if +ϕ(S ⊗ X) ∈ P if and only if L (ϕ(S ⊗ X)) ≻ 0 if and only if +� +B0 ⊗ In +ΛY (X) +ΛY (X)∗ +B0 ⊗ In +� +≻ 0 +if and only if +0 ≺ +� +B +− 1 +2 +0 +⊗ In +0 +0 +B +− 1 +2 +0 +⊗ In +� � +B0 ⊗ In +ΛY (X) +ΛY (X)∗ +B0 ⊗ In +� � +B +− 1 +2 +0 +⊗ In +0 +0 +B +− 1 +2 +0 +⊗ In +� += +� +I +ΛB(X) +ΛB(X)∗ +I +� +if and only if X ∈ BB[n]. +□ +3.2. Analyzing the equality BR = BB. In this section we deduce consequences of the +equality the two representation BE and BB of BR appearing in Propositions 2.5 and 3.1 +respectively using the theory of spectraballs. + +REINHARDT SPECTRAHEDRA +17 +Since, by Proposition 2.8, E is ball-minimal and BE = BB, Lemma 2.7 implies there +exist unitary matrices U, V such that +U +� +E +0 +0 +J +� +V ∗ = B, +where J = (J1, J2) ∈ Ms(C)2 and BE ⊆ BJ. Letting {e1, . . . , e4} denote the standard basis +for C4 and Hj = {ej ⊗ h : h ∈ Cs}, +ker(B) := ∩2 +j=1 ker(Bj) = B +1 +2 +0 H1, +since ∩2 +j=1 ker(Yj) = H1. Thus +� +Ej +0 +0 +Jj +� +V ∗B +1 +2 +0 H1 = 0. +Since E has no kernel, it follows that J = 0 and V ∗B +1 +2 +0 H1 = H4. In particular, +U∗BjV = +� +Ej +0 +0 +0 +� +where the lower left 0 matrix has size s × s. +Summarizing, there exist unitary matrices U, V such that +Bj = U +� +Ej +0 +0 +0 +� +V ∗. +Set P = B +1 +2 +0 U and Q∗ = V ∗B +1 +2 +0 . Thus PP ∗ = B0 = QQ∗ and +Yj = P +� +Ej +0 +0 +0 +� +Q∗. +Express P and Q in terms of their block columns as +P = +� +p1 +p2 +p3 +p4 +� += +� +Pj,k +�4 +j,k=1 +Q = +� +q1 +q2 +q3 +q4 +� += +� +Qj,k +�4 +j,k=1 , +, +where Pj,k and Qj,k are s × s matrices. With these notations, +Y1 =P +� +E1 +0 +0 +0 +� +Q∗ = P[(e1e∗ +1 + e3e∗ +3) ⊗ C1]Q∗ = p1C1q∗ +1 + p3C1q∗ +3 +Y2 =P +� +E2 +0 +0 +0 +� +Q∗ = P[(e1e∗ +2 + e2e∗ +3) ⊗ C2]Q∗ = p1C2q∗ +2 + p2C2q∗ +3, +(3.4) + +18 +BEN JEMAA AND MCCULLOUGH +where {e1, e2, e3, e4} is the standard basis for C4. Let T = Q−∗ +4,3 and S = P −∗ +1,1 . +Lemma 3.3. With notations above, Q1,4 and P4,4 are unitary, Q4,3 and P1,1 are invertible +and +P = + + + + + +P1,1 +P1,2 +P1,3 +0 +0 +ℓ2,2C2TC−1 +2 +ℓ2,1C2TC−1 +1 +b2C2P4,4 +0 +ℓ1,2C1TC−1 +2 +ℓ1,1C1TC−1 +1 +b1C1P4,4 +0 +0 +0 +P4,4 + + + + + +Q = + + + + + +0 +0 +0 +Q1,4 +ℓ∗ +1,1C∗ +1SC−∗ +1 +ℓ∗ +1,2C∗ +1SC−∗ +2 +0 +b∗ +1C∗ +1Q1,4 +ℓ∗ +2,1C∗ +2SC−∗ +1 +ℓ∗ +2,2C∗ +2SC−∗ +2 +0 +b∗ +2C∗ +2Q1,4 +Q4,1 +Q4,2 +Q4,3 +0 + + + + + . +Proof. With +Wj = +� +0 +ℓ1,jC−1 +2 +−ℓ2,jC−1 +1 +0 +� +, +observe +0 = W1Y1 = W1p1C1q∗ +1 + W1p2C1q∗ +3. +Since Q is invertible, there exist r1, . . . , r4 ∈ M4s,s(C) such that +q∗ +j rk = δj,k. +Hence, from the first identity in equation (3.4) and equation (3.2), +0 = W1p1C1q∗ +1r1 + W1p2C1q∗ +3r1 = W1p1C1 = [ℓ1,1C−1 +2 P2,1 − ℓ2,1C−1 +1 P3,1]C1 +and therefore ℓ1,1C−1 +2 P2,1 − ℓ2,1C−1 +1 P3,1 = 0. A similar argument using the second identity in +equation (3.4) shows +0 = [ℓ1,2C−1 +2 P2,1 − ℓ2,2C−1 +1 P3,1]C2 +and thus ℓ1,2C−1 +2 P2,1−ℓ2,2C−1 +1 P3,1 = 0. Since L from equation (3.1) is invertible, we conclude +C−1 +2 P2,1 = C−1 +1 P3,1 = 0 and therefore P2,1 = P3,1 = 0. +Since [e∗ +4 ⊗ Is] Yj = 0, it also follows that P4,1 = P4,2 = P4,3 = 0. In particular P1,1 +is invertible. Examining the (block) (4, 4) entry of PP ∗ = B0 shows P4,4P ∗ +4,4 = Is. From +the last column of PP ∗ = B0 it now follows that P1,4 = 0, and P2,4 = b2C2P4,4 as well as +P3,4 = b1C1P4,4. At this point we have identified, as indicated, the first and last row and +column of P. Similar reasoning applies to the third and forth columns and first and fourth +rows of Q. + +REINHARDT SPECTRAHEDRA +19 +Comparing the descriptions of Yj from equations (3.4). and (3.2) gives, +P1,1C1Q∗ +2,1 = ℓ1,1C1 +P1,1C1Q∗ +3,1 = ℓ2,1C2 +P1,1C2Q∗ +2,2 = ℓ1,2C1 +P1,1C2Q∗ +3,2 = ℓ2,2C2 +P2,3C1Q∗ +4,3 = ℓ2,1C2 +P3,3C1Q∗ +4,3 = ℓ1,1C1 +P2,2C2Q∗ +4,3 = ℓ2,2C2 +P3,2C1Q∗ +4,3 = ℓ1,2C2. +(3.5) +Comparing the identities above with the definitions of T and S completes the proof. +□ +Let +Σ = +� +P2,2 +P2,3 +P3,2 +P3,3 +� +, +Λ = +� +b2C2P4,4 +b1C1P4,4 +� +. +From the middle 2 × 2 block of PP ∗ = B0, +(3.6) +ΣΣ∗ + ΛΛ∗ = I = +� +I +0 +0 +I +� +. +Let +G = +� +ℓ1,1 +−ℓ2,1 +−ℓ1,2 +ℓ2,2 +� � +C−1 +2 +0 +0 +C−1 +1 +� +, +and observe +GΣ = det L +� +TC−1 +2 +0 +0 +TC−1 +1 +� +, +GΛ = +� +ℓ1,1b2 − ℓ2,1b1 +−ℓ1,2b2 + ℓ2,2b1 +� +P4,4. +Thus, applying G on the left and G∗ on the right of equation (3.6) and comparing the (1, 2) +(block) entries gives, +(3.7) +− [ℓ1,1ℓ∗ +1,2C−1 +2 C−∗ +2 ++ ℓ2,1ℓ∗ +2,2C−1 +1 C−∗ +1 ] = (ℓ1,1b2 − ℓ2,1b1)(−ℓ1,2b2 + ℓ2,2b1)∗. +3.3. A dichotomy. With equation (3.7) in place, we are now in position to state and prove +the following lemma. Recall s is the size of C. +Lemma 3.4. If s > 1, then b1b∗ +2 = 0 and +(i) ℓ1,2 = 0 = ℓ2,1 or +(ii) ℓ1,1 = 0 = ℓ2,2. +Proof. If the right hand side of equation (3.7) is not 0, then either C−1 +1 C−∗ +1 +and C−1 +2 C−∗ +2 +commute or C−1 +j C−∗ +j +is a multiple of the identity for either j = 1 or j = 2. In either case +C∗ +1C1 and C∗ +2C2 commute and thus, as {C∗ +1C1, C∗ +2C2} generates Ms(C) as a C-star algebra, +s = 1. Thus, if s > 1, then the right hand side of equation (3.7) is 0. +If ℓ1,1ℓ∗ +1,2 ̸= 0 or ℓ2,1ℓ∗ +2,2 ̸= 0, then they are both not zero and again, C∗ +1C1 and C∗ +2C2 +commute and s = 1. Thus, if s > 1, then ℓ1,1ℓ∗ +1,2 = 0 = ℓ2,1ℓ∗ +2,2. Since L is invertible, there are + +20 +BEN JEMAA AND MCCULLOUGH +two cases, either ℓ2,1 = 0 = ℓ1,2 (and ℓ1,1 ̸= 0 ̸= ℓ2,2) or ℓ1,1 = 0 = ℓ2,2 (and ℓ2,1 ̸= 0 ̸= ℓ1,2). +In either case b1b∗ +2 = 0. +□ +Lemma 3.5. If s > 1, then b = 0. +Proof. Arguing by contradiction, suppose s > 1 and b ̸= 0. Observe that conclusion that +b1b∗ +2 = 0 of Lemma 3.4 applies to any automorphism of P. Thus, without loss of generality, +b1 ̸= 0 and b2 = 0. Another appeal to Lemma 3.4 gives either ℓj,k = 0 for j ̸= k or ℓj,k = 0 +for j = k. Suppose the first case holds. Given θ real, let fθ denote the automorphism of fθ +given by fθ(y1, y2) = eiθ(y2, y1) and let ψ = ϕ ◦ fθ. Thus ψ : P → P is an automorphism and +ψ(0) = ϕ ◦ fθ(b1, 0) = ϕ(0, eiθb1) = (ϕ1(0, eiθb1), ϕ2(0, eiθb1)). +Hence ϕ1(0, eiθb1) ϕ2(0, eiθb1) is identically 0. By analyticity, ϕk(0, eiθb1) = 0 for some k +and all θ and hence, for either k = 1 or k = 2, we have gk(z) = ϕk(0, z) is identically 0. +Since g1(0) = b1 ̸= 0, it follows that g2(z) is identically 0. Thus 0 = g′ +2(0) = ℓ2,2 ̸= 0, a +contradiction. +Now suppose instead that ϕj,j = 0 for j = 1, 2. In this case set fθ(y1, y2) = eiθ(y1, y2) +and ψ = ϕ ◦ fθ ◦ ϕ. Thus, +ψ(0) = ϕ(eiθb1, 0) +and ϕ1(eiθb1, 0) ϕ2(eiθb1, 0) is identically 0. Hence g2(z) = ϕ2(z, 0) is identically 0 and thus +0 = g′ +2(0) = ℓ2,1 ̸= 0, a contradiction which shows b1 = 0 = b2. +□ +Lemma 3.6. If s = 1, then b = 0. +Proof. Since s = 1, we have C1, C2 ∈ C are unimodular and P[1] is the set {z = (z1, z2) ∈ +C2 : |z1| + |z2| < 1}. Indeed, given (z1, z2) ∈ C2, the matrix +� +z∗ +1 +z2 +z∗ +2 +z1 +� +is a contraction if and only if the self-adjoing matrix, +� +r +z2eit +z∗ +2e−it +r +� +is a contraction, where z1 = reit is the polar decomposition of z1. This latter matrix has +eigenvalues r ± |z2| and hence is a contraction if and only if |z1| + |z2| < 1. +The set P[1] is known as a psuedo-ellipse and f = ϕ[1] : P[1] → P[1] is an auto- +morphism (in the classical several complex variables sense). It is known (see [JP08]) that +automorphisms of P[1] are compositions of maps of the form (z1, z2) �→ (γ1z1, γ2z2) and +(z1, z2) �→ (z2, z1), where γj ∈ C are unimodular (the proof uses techniques from several + +REINHARDT SPECTRAHEDRA +21 +complex variables and lie groups). In particular, b = 0. Thus in any case b = 0 and ϕ +is linear by [HKM11, Theorem 4.4], since ϕ(0) = b = 0 and the domain P is circularly +symmetric. +□ +We now give a self contained alternate proof of Lemma 3.6 based upon results in this +article. +Second proof of Lemma 3.6. In addition to the identities of equation (3.5) obtained by com- +paring the equations (3.4) and (3.2), observe +P1,1Q∗ +4,1 + P1,3Q∗ +4,3 = 0 +P1,1Q∗ +4,2 + P1,2Q∗ +4,3 = 0, +which can be summarized as +(3.8) +Q∗ +4,3 +� +P1,3 +P1,2 +� += −P1,1 +� +Q∗ +4,1 +Q∗ +4,2 +� +. +From (1, 2) and (1, 3) entries of PP ∗ = B0 and using Tj = Q−∗ +4,3, +� +P1,2 +P1,3 +� � +ℓ∗ +2,2 +ℓ∗ +1,2 +ℓ∗ +2,1 +ℓ∗ +1,1 +� += Q4,3 +� +b1 +b2 +� +. +Equivalently, +Q−∗ +4,3 +� +P ∗ +1,3 +P ∗ +1,2 +� � +ℓ1,1 +ℓ2,1 +ℓ1,2 +ℓ2,2 +� += +� +b∗ +2 +b∗ +1 +� +. +Similarly from the (4, 2) and (4, 3) entries of QQ∗ = B0, +P −1 +1,1 +� +Q4,1 +Q4,2 +� � +ℓ1,1 +ℓ2,1 +ℓ1,2 +ℓ2,2 +� += +� +b∗ +2 +b∗ +1 +� +. +It follows that +(3.9) +Q−1 +4,3 +� +P1,3 +P1,2 +� += P −∗ +1,1 +� +Q∗ +4,1 +Q4,2 +� +. +From equations (3.8) and (3.9), +(|Q4,3|2 + |P1,1|2) +� +Q∗ +4,1 +Q4,2 +� += 0 +and we conclude Q4,1 = Q4,2 = 0 and thus b = 0. +□ + +22 +BEN JEMAA AND MCCULLOUGH +3.4. Completion of the proof of Theorem 1.1. From Lemmas 3.5 and Lemma 3.6 it +follows that b = 0 and hence, by [HKM11, Theorem 4.4], ϕ is linear. +Since ϕ is linear, +ϕ(x) = (w1, w2), +where +� +w1 +w2 +� += L +� +x1 +x2 +� +. +If s > 1, then, by Lemma 3.4 and composing with the automorphism (x1, x2) �→ (x2, x1), +we assume L is diagonal, in which case the diagonal entries must be unimodular and the +proof is complete. +Now suppose s = 1. From the (first) proof of Lemma 3.6 it can be seen that ϕ is trivial. +Alternately, from Lemma 3.3 and the relation PP ∗ = B0, it follows that the matrix +� +ℓ2,2 +ℓ2,1 +ℓ1,2 +ℓ1,1 +� +T +is unitary. Hence L is a multiple of a unitary. Since ϕ is an automorphism of P, we conclude +that L is unitary. In particular, |ℓ1,1|2 + |ℓ2,1|2 = 1. Since (t, 0) is in P for 0 < t < 1, so +is ϕ(t, 0) = t(ℓ1,1, ℓ2,1). It now follows from item (c) of Proposition 2.1 that ℓ1,1ℓ2,1 = 0. A +similar argument shows ℓ1,2ℓ2,2 = 0. Hence L is unitary and either ℓj,k = 0 for j ̸= k or +ℓj,k = 0 for j = k and the proof is complete. +□ +References +[A+] Free Potential Functions, arXiv:2005.01850. 2, 6, 13 +[BPT] G. Blekherman, P.A. Parrilo, R.R. Thomas (editors), Semidefinite optimization and convex algebraic +geometry, SIAM, 2013. 3 +[dOH06] M.C. De Oliveira and J. W. Helton, Computer algebra tailored to matrix inequalities incontrol +International Journal of Control, (2006) 79:11, 1382-1400, DOI:10.1080/00207170600725529 4 +[dOHMP] Mauricio C. de Oliveira, J. William Helton, Scott McCullough and Mihai Putinar, Engineering +systems and free semi-algebraic geometry in Emerging applications of algebraic geometry, 17–61, IMA +Vol. Math. Appl., 149, Springer, New York, 2009. 4 +[EHKM] Eric Evert, Bill Helton, Scott McCullough and Igor Klep, Circular Free Spectrahedra, J. Math. +Anal. Appl, 445 (2017), no. 1, 1047–1070. 2, 7, 8, 12 +[HKM11] Bill Helton, Igor Klep and Scott McCullough, Proper analytic fre maps, J. Functional Analysis +260 (2011), no. 5, 1476–1490. 6, 21, 22 +[HKM12] Bill Helton, Igor Klep and Scott McCullough, Free analysis, convexity and LMI domains. Math- +ematical methods in systems, optimization, and control, 195–219, Oper. Theory Adv. Appl., 222, +Birkh¨auser/Springer Basel AG, Basel, 2012. 12 + +REINHARDT SPECTRAHEDRA +23 +[HKMV20] Bill Helton, +Igor Klep, +Scott McCullough and Jurij Volˇciˇc, +Bianalytic free maps be- +tween spectrahedra and spectraballs, Journal of Functional Analysis, 278 no. 11, 15 June 2020, +doi.org/10.1016/j.jfa.2020.108472. 6, 7, 11 +[KVV14] Dmitry Kalyuzhnyi-Verbovetski˘i and Victor Vinnikov, Foundations of free noncommutative func- +tion theory, Mathematical Surveys and Monographs 199, Amer. Math. Soc., 2014. 5, 12 +[KS] Igor Klep and ˇSpela ˇSpenko, Free function theory through matrix invariants, Canad. J. Math. 69 (2017), +no. 2, 408–433. 12 +[K] A. Kodama, On the holomorphic automorphism group of a generalized complex ellipsoid, Complex +Variables and Elliptic Equations 59 (2014) 1342–1349. 7 +[M] Scott McCullough, Hyper-Reinhardt Free Spectrahedra, arXiv:2107.11641 7, 8 +[MT] S. McCullough and N. Tuovila, Reinhardt Free Spectrahedra, Linear Algebra and its Applications, 640 +(2022), 91–117. 7, 8 +[SIG97] R.E. Skelton, T. Iwasaki, K.M. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, +Taylor & Francis, 1997. 4 +[JP08] M. Jarnicki and P. Pflug, First Steps in Several Complex Variables: Reinhardt domains EMS Text- +books in Mathematics. European Mathematical Society (EMS), Z¨urich, 2008. viii+359 pp. ISBN: 978- +3-03719-049-4. 7, 20 +[WSV12] H. Wolkowicz, R. Saigal, L. Vandenberghe (eds.), Handbook of semidefinite programming: theory, +algorithms, and applications, International Series in Operations Research & Management Science 27, +Springer, 2012 3 + +Index +B0, 15 +Ec, 10 +Er, 10 +LA, 3 +S, 13 +U ∗XU, 1 +X ⊕ Y , 1 +C, 3 +ΛA(x), 3 +BG, 4 +DA, 3 +DR, 4 +L , 1 +L, 15 +P[n], 1 +f[n], 5 +analytic, 5 +associated spectraball, 4 +automorphism, 2, 6 +ball-equivalent, 11 +ball-mimimal, 11 +bianalytic, 6 +bounded, 5 +circularly symmetric, 6 +closed with respect to direct and unitary +similarity, 4 +closed with respect to direct sums, 1 +closed with respect to unitary similarity, 1 +evaluation, 3 +free algebra, 3 +free analytic function, 1 +free analytic mapping, 2 +free function, 5 +free map, 6 +free polynomials, 3 +free set, 4 +free spectrahedron, 1, 3 +homogeneous linear pencil, 3 +hyper-Reinhardt, 8 +inverse, 6 +length of a word, 2 +level, 4 +minimal defining, 12 +nilpotent of order at most m, 12 +open, 5 +power series, 12 +respect direct sums and unitary similarity, 5 +respects intertwinings, 5 +respects similarities, 5 +spectraball, 4 +spectrahedron, 3 +strict contraction, 7 +trivial automorphisms, 2 +24 + +REINHARDT SPECTRAHEDRA +25 +University of Flordia, Gainesville, FL +Email address: munirbenjemaa@ufl.edu +Department of Mathematics, University of Flordia, Gainesville, FL +Email address: sam@ufl.edu + diff --git a/ctE0T4oBgHgl3EQf5AKl/content/tmp_files/load_file.txt b/ctE0T4oBgHgl3EQf5AKl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d99d117c08d3d221e1b86e39cd488ac255908e77 --- /dev/null +++ b/ctE0T4oBgHgl3EQf5AKl/content/tmp_files/load_file.txt @@ -0,0 +1,738 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf,len=737 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='02746v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='FA] 6 Jan 2023 REINHARDT FREE SPECTRAHEDRA MUNIR BEN JEMAA∗ AND SCOTT MCCULLOUGH Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The automorphism group of a particular free spectrahedron is determined via a novel argument involving algebraic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Introduction Fix norm 1 matrices C1, C2 of size s × s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' For positive integers n, let Mn(C)2 denote the set of pairs X = (X1, X2) of n×n matrices and let P[n] denote those X ∈ Mn(C)2 for which the hermitian block 4 × 4 matrix L (X) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed Is ⊗ In C1 ⊗ X1 C2 ⊗ X2 0 (C1 ⊗ X1)∗ Is ⊗ In 0 C2 ⊗ X2 (C2 ⊗ X2)∗ 0 Is ⊗ In C1 ⊗ X1 0 (C2 ⊗ X2)∗ (C1 ⊗ X1)∗ Is ⊗ In \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Here X∗ j is the adjoint (complex transpose) of Xj and In is the n × n identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The sequence of sets P = (P[n])n is an example of a free spectrahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Given X ∈ Mn(C)2 and Y ∈ Mm(C)2, and a unitary matrix U ∈ Mn(C), let X ⊕ Y = �� X1 0 0 Y1 � , � X2 0 0 Y2 �� and U∗XU = (U∗X1U, U∗X2U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Observe, if X ∈ P[n] and Y ∈ P[m], then X ⊕ Y ∈ P[n + m] and U∗XU ∈ P[n];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' that is P is closed with respect to direct sums and unitary similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let M(C) denote the sequence (Mn(C)) and let M(C)2 denote the sequence (Mn(C)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A free analytic function f : P → M(C) is a sequence (f[n]) of analytic functions f[n] : P[n] → Mn(C) that respects direct sums and unitary similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' That is, given X ∈ P[n] and Y ∈ P[m] and a unitary matrix U ∈ Mn(C), f[n + m](X ⊕ Y ) = f[n](X) ⊕ f[m](Y ) 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 47L25, 32H02 (Primary);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 52A05, 46L07 (Secondary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' bianalytic map, birational map, free spectrahedron, free analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' ∗ Research supported by the University of Florida University Scholars undergraduate research program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 2 BEN JEMAA AND MCCULLOUGH and f[n](U∗XU) = U∗f[n](X)U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Typically we write f in place of f[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The general definition of a free analytic function appears in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' While it may not immediately appear so, free analytic functions are the natural (freely) non-commutative analogs of analytic functions in several complex variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A free analytic mapping ϕ = (ϕ1, ϕ2) : P → P is a pair of free analytic functions ϕj : P → M(C)2 such that ϕ(X) = (ϕ1(X), ϕ2(X)) ∈ P for all X ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' An automorphism ϕ of P is a free analytic mapping ϕ : P → P for which there exists a free analytic mapping ψ : P → P such that ψ(ϕ(X)) = X = ϕ(ψ(X)) for X ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Given γ = (γ1, γ2) ∈ C2 with |γj| = 1, the function f(x) = (γ1x1, γ2x2) is automorphism of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Likewise, f(x) = (γ2x2, γ1x1) is an automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' We call these automorphism trivial automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1 below is the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' If (i) C1 and C2 are invertible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' and (ii) the C-star algebras generated by {C∗ 1C1, C∗ 2C2} and {C1C∗ 1, C2C∗ 2} are all of Ms(C), then the automorphisms of P are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' We wish to highlight two other contributions of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2 provides an alternate characterization of free analytic functions tailored to the study of automorphisms of free domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Its proof is modeled after arguments found in [A+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3 charac- terizing spectraballs is from [EHKM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Here we provide an alternate proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The remainder of this introduction contains more complete definitions of free spectrahedra and spectraballs, free analytic functions and maps, as well as background and motivation for studying the automorphism group of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Preliminary results are contained in Section 2 and the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1 appears in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Free polynomials and their evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Fix a positive integer g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' , xg) denote g freely non-commuting variables and let denote the semigroup of words in x with ∅, the enpty word, playing the role of the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The length of the empty word is 0 and otherwise the length of a word (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1) w = xj1xj2 · · · xjm is m denoted |w| = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' REINHARDT SPECTRAHEDRA 3 For positive integers n, let Mn(C)g denote the set of g-tuples X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' , Xg) of n × n matrices with entries from C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let M(C)g denote the sequence (Mn(C)g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Given a tuple X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' , Xg) ∈ M(C)g, let Xw = Xj1Xj2 · · · Xjm, with w ∈ as in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus Xw is the evaluation of the word w at the tuple X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' This evaluation extends to the free algebra of free polynomials C equal the C linear combinations of elements of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Elements p ∈ C have the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2) p = � w∈ pww, where the sum is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The polynomial p evaluates at X ∈ M(C)g as p(X) = � w∈ pwXw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A matrix-valued free polynomial can be viewed either as a matrix with polynomial entries or a polynomial with matrix coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In the latter case, given positive integers d, e and pw ∈ Md,e(C), the finite sum in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2) is a matrix valued polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' To evaluate this p at a tuple X ∈ M(C)g we will make use of the (Kronecker) tensor product S ⊗ T of matrices S and T, setting p(X) = � pw ⊗ Xw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Free spectrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Given A ∈ Md×e(C)g, let ΛA denote the homogeneous linear pencil ΛA(x) = � j Ajxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' It evaluates at X ∈ Mn(C)g as ΛA(X) = g � j=1 Aj ⊗ Xj ∈ Md×e(C) ⊗ Mn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In the case A is square (d = e), we let LA(X) = Id ⊗ In + ΛA(X) + ΛA(X)∗ = I + � Aj ⊗ Xj + � A∗ j ⊗ X∗ j ∈ Md(C) ⊗ Mn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The set DA[1] ⊆ Cg consisting of x ∈ Cg such that LA(x) ≻ 0 is a spectrahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Spectra- hedra are basic objects in a number of areas of mathematics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' semidefinite programming, convex optimization [WSV12] and real algebraic geometry [BPT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The free spectrahedron determined by A ∈ Md(C)g is the sequence of sets DA = (DA[n]), where DA[n] = {X ∈ Mn(C)g : LA(X) ≻ 0}, 4 BEN JEMAA AND MCCULLOUGH and T ≻ 0 indicates that the square matrix T is positive definite (hermitian with positive eigenvalues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Observe that P is the free spectrahedron DR, where (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3) R1 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 0 C1 0 0 0 0 0 0 0 0 0 C1 0 0 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , R2 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 0 0 C2 0 0 0 0 C2 0 0 0 0 0 0 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A free spectrahedron DA is not determined by the spectrahedron DA[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' See Proposi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Free spectrahedra are canonical objects in the theories of operator systems and spaces and completely positive maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' They are related to quantum channels from quan- tum information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' That they arise naturally in certain systems engineering problems governed by a signal flow diagram [dOH06, dOHMP, SIG97] also provides motivation for studying free spectrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Spectraballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Given a tuple G = (G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' , Gg) of d × e matrices, the sequence BG = (BG[n])n defined by BG[n] = {X ∈ Mn(C)g : ∥ g � j=1 Gj ⊗ Xj∥ < 1} is a spectraball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The spectraball at level one, BG[1], is a rotationally invariant convex subset of Cg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The spectraball BG is a spectrahedron since BG = DB for B = ( 0 G 0 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Under the hypotheses of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1, the spectrahedron P is not a spectraball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' See Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A spectrahedron DA has its naturally associated spectraball, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4) BA = {X : ∥ΛB(X)∥ < 1} = {X : � 0 X 0 0 � ∈ DA}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The spectraball BR associated to P = DR plays an important role in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Free sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A free set S ⊆ M(C)g is a sequence S = (S [n])n such that S [n] ⊆ Mn(C)g for each positive integer n and such that S is closed with respect to direct and unitary similarity: (i) if X ∈ S [n] and Y ∈ S [m], then X ⊕ Y = (X1 ⊕ Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' , Xg ⊕ Yg) ∈ S [n + m], where Xj ⊕ Yj = � Xj 0 0 Yj � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' REINHARDT SPECTRAHEDRA 5 (ii) if X ∈ S [n] and U ∈ Mn(C) is unitary, then U∗XU = (U∗X1U, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' , U∗XgU) ∈ S [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' We say S is open if each S [n] is open and S is bounded if there exists a κ such that for all n and X ∈ S [n], the n × ng matrix � X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Xg � has norm at most κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A free spectrahedron is an open free set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Free analytic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Given an open free set S , a free function f : S → M(C) is a sequence f = (f[n]), where f[n] : S �→ Mn(C), that satisfies the axioms (i) if X ∈ S [n] and Y ∈ S [m], then f[n + m](X ⊕ Y ) = f[n](X) ⊕ f[m](Y );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' (ii) if X ∈ S [n] and U is an n × n unitary matrix, then f[n](U∗XU) = U∗f[n](X)U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus free functions respect direct sums and unitary similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The free function f is analytic if each f[n] is analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' We typically write f in place of f[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Turning to examples, free polynomials are evidently free analytic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A free rational function r (regular at 0) is a free analytic function that has a realization formula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' that is, there exists a positive integer e, a tuple A ∈ Me(C)g and vectors c, b ∈ Ce, such that r(x) = c∗(I − ΛA(x))−1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The natural domain of r consists of those tuples X ∈ Mn(C)g for which I − ΛA(X) is invertible and for such an X, r(X) = (Id ⊗ c)∗ (Id ⊗ In − ΛA(X))−1 (Id ⊗ b) ∈ Mn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' See [KVV14] for further information about free functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' There are two other formulations of free functions that are equivalent and more common in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' They do not not assume analyticity, but rather have it as a consequence of mild additional assumptions such as continuity or boundedness, in which case they are equivalent to the formulation adopted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In one formulation, item (ii) is replaced by the hypothesis that f respects similarities: if X ∈ S [n] and T is an invertible matrix such that T −1XT ∈ S [n], then f(T −1XT) = T −1f(X)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The other formulation replaces items (i) and (ii) with the single axiom that f respects intertwinings: if X ∈ S [n] and Y ∈ S [m] 6 BEN JEMAA AND MCCULLOUGH and Γ is an m × n matrix such that ΓX = Y Γ, then Γf(X) = f(Y )Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2 is a variation on [A+, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A proof appears in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' If S is a free open set and f : S → M(C) is a free analytic function, then f respects intertwinings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Automorphisms of free spectrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Given free sets S ⊆ M(C)g and T ⊆ M(C)g a free map f : S → T is an g-tuple f = (f 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' , f g) of free functions f : S → M(C) such that f(X) ∈ T for all X ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A bianalytic map f between free spectrahedra DA and DB is a free map f : DA → DB for which there exists a free analytic mapping g : DB → DA such that g(f(X)) = X and f(g(Y )) = Y for all X ∈ DA and Y ∈ DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' We refer to g as the inverse of f and write g = f −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A natural problem, from several different perspectives, is to determine the bianalytic maps between two free spectrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' This problem is the free analysis analog of rigidity phenomena in several complex variables and from this perspective it is expected that two free spectrahedra are rarely bianalytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The paper[HKMV20] determines, under certain generic irreducibility inspried hypotheses on A and B, the tuples (A, B, f) of bianalytic maps f : DA → DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' It turns out that A and B are closely linked and f has a highly algebraic description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In that same paper, bianalytic maps between spectraballs are determined without any additional hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Again, these maps have a highly algebraic description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' An automorphism f of a free spectrahedron DA is a bianalytic map f : DA → DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' If f, g : DA → DB are bianalytic, then the map g−1 ◦ f : DA → DA is an automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus, the automorphism group of DA places constraints on the bianalytic maps f : DA → DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A free set S is circularly symmetric if γX = (γX1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' , γXg) ∈ S whenever X ∈ S and γ ∈ C is unimodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A natural class of free spectrahedra not covered by the results in [HKMV20] are those with circular symmetry that are not spectraballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In particular, those P satisfying the hypotheses of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1 provide examples of a spectra- hedron whose automorphism group is not yet classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' We are now in a position to provide an overview of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A somewhat routine argument shows if ϕ is a linear automorphism of P, then ϕ is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A consequence of the free analog of the Caratheodory-Cartan-Kaup-Wu from [HKM11] is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' If ϕ : P → P is an automorphism and ϕ(0) = 0, then ϕ is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The strategy employed here to show if ϕ is an automorphism of P, then ϕ(0) = 0 is novel, using algebraic aspects of the theory of spectraballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Given an automorphism ϕ : P → P with ϕ(0) ∈ C2 not necessarily 0, we construct a tuple B described solely in terms of ϕ(0) ∈ C2 and ϕ′(0) ∈ M2(C) such that BR = BB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' This equality is analyzed using algebraic results from [HKMV20], ultimately REINHARDT SPECTRAHEDRA 7 concluding s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' When s = 1, the spectrahedron P[1] is the generalized complex ellipsiod {z = (z1, z2) ∈ C2 : |z1| + |z2| < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' It is known [K, JP08], using very different techniques than those here, that the analytic automorphisms of this complex ellipsiod are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In particular, ϕ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' We give a self contained proof of this fact using results developed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' By contrast, the proof strategy for classifying the automorphism group of hyper-Reinhardt domains, defined below and in [MT, M], proceeds via the dynamics of composing automor- phisms and using results related to the classical Caratheodory interpolation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Preliminary results This section collects preliminary and ancillary results to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1 provides alternate characterizations for membership in P and establishes that, under the hypotheses of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1, P is neither a spectraball nor a hyper-Reinhardt domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Ad- ditionally, an alternate proof of a results from [EHKM] is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2 gathers facts about the spectraball BR associated to P = DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Ball minimality from [HKMV20] is reviewed in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3, where a ball minimal tuple E such that BE = BR is identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4 discusses the evaluation of a free function defined near 0 on nilpotent tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2 appears in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Membership in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A matrix T is a strict contraction if ∥T∥ < 1, where ∥T∥ is the operator norm of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' For X = (X1, X2) ∈ M(C)2 and Yj = Cj ⊗ Xj, the following are equiva- lent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' (a) X ∈ P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' (b) the matrix T(X) := � Y ∗ 1 Y2 Y ∗ 2 Y1 � is a strict contraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' (c) the matrix L ′(X) := � I − Y ∗ 1 Y1 − Y2Y ∗ 2 −Y ∗ 1 Y2 − Y2Y ∗ 1 −Y ∗ 2 Y ∗ 1 − Y1Y ∗ 2 I − Y1Y ∗ 1 − Y ∗ 2 Y2 � = I − T(X)T(X)∗ is positive definite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 8 BEN JEMAA AND MCCULLOUGH (d) The matrix L ′ ∗(X) := � I − Y1Y ∗ 1 − Y2Y ∗ 2 −Y1Y2 − Y2Y2 −Y ∗ 2 Y ∗ 1 − Y ∗ 1 Y ∗ 2 I − Y ∗ 1 Y1 − Y ∗ 2 Y2 � = I − T(X)∗T(X) is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A selfadjoint block square matrix M = � A B B∗ D � with A positive definite is positive definite if and only if the Schur complement of its (1, 1) block, S = D − B∗A−1B is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' By definition, X ∈ P means L (X) ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Taking the Schur complement of L (X) first of the (1, 1) block entry then off the (3, 3) block entry (an identity matrix) shows L (X) ≻ 0 if and only if L ′(X) ≻ 0 using the observation at the outset of this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A direct computation shows L ′(X) = I − T(X)T(X)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' To complete the proof observe, I − T(X)T(X)∗ ≻ 0 if and only if ∥T(X)∥ < 1 if and only if I − T(X)∗T(X) ≻ 0 if and only if L ′ ∗(X) ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' □ Specializing to the case of two variables, a spectrahedron D ⊆ M(C)2 is hyper-Reinhardt if there exists a tuple G = (G1, G2) of matrices of compatible sizes such that, with A1 = \uf8eb \uf8ec \uf8ed 0 G1 0 0 0 0 0 0 0 \uf8f6 \uf8f7 \uf8f8 , A2 = \uf8eb \uf8ec \uf8ed 0 0 0 0 0 G2 0 0 0 \uf8f6 \uf8f7 \uf8f8 , we have D = DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A hyper-Reinhardt free spectrahedra is circular, but not necessarily a spectraball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The automorphisms of hyper-Reinhardt free spectrahedra are determined in [MT, M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Under the hypotheses of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1, the spectrahedron P is neither hyper-Reinhardt nor a spectraball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2 uses one direction of the following result from [EHKM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A spectrahedron DA is a spectraball if and only if for each positive integer n, each X ∈ DA[n] and each unitary matrix U ∈ Mn(C), the tuple UX ∈ DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' REINHARDT SPECTRAHEDRA 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' If DA = BB is a spectraball, then it is immediate that X ∈ DA[n] and U unitary implies UX ∈ DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Conversely suppose for each positive integer n, each X ∈ DA[n] and each unitary matrix U ∈ Mn(C), the tuple UX ∈ DA[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' To prove DA is spectraball, it suffices to show DA = BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' To this end, let X ∈ Mn(C)g be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let U = � 0 In In 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' and observe X ∈ DA if and only if 0n ⊕ X ∈ DA[2n] if and only if U(0n ⊕ X) = U � 0 0 0 X � = � 0 X 0 0 � ∈ DA[2n] if and only if X ∈ BA[n] (see Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus DA = BA and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' □ The following lemma is also needed for the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Under the hypotheses of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1, C∗ 1C1+C∗ 2C2 ≻ Is and C1C∗ 1+C2C∗ 2 ≻ Is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' If C∗ 1C1 + C∗ 2C2 ⪯ Is, then the kernel K of I − C∗ 1C1 is non-trivial and orthogonal to the range of C∗ 2C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Hence K reduces both C∗ 1C1 and C∗ 2C2 and hence the C-star algebra they generate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus C∗ 1C1 + C∗ 2C2 ≻ Is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Likewise C1C∗ 1 + C2C∗ 2 ≻ Is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' □ Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A routine computation shows, if D ⊆ M(C)g is a hyper-Reinhardt free spectrahedron, X ∈ D[n] and W0, W1, W2 ∈ Mn(C) are unitary matrices, then W · X := (W ∗ 0 XW1, W ∗ 1 XW2) ∈ D[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Another routine computation and an appeal to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1 shows (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1) X = (X1, X2) = �� 1 0 0 0 � , � 0 0 0 1 �� is in the boundary of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2) U = � 0 1 1 0 � and (W0, W1, W2) = (I, U, U2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus each Wj is unitary and W · X = �� 0 1 0 0 � , � 0 1 0 0 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' By item (d) or Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4, W · X is not in the closure of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Hence P is not hyper-Reinhardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 10 BEN JEMAA AND MCCULLOUGH If P is a spectraball, X ∈ P[n] and U is an n×n unitary matrix, then UX = (UX1, UX2) is also in P[n] by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Choose X1, X2 and U as in equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2) and note, with L ′ as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1, L ′(UX) = I − \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed C1C∗ 1 + C∗ 2C2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C∗ 1C1 + C2C∗ 2 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , since X∗ 1X2 = 0 = X2X∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' If UX is in the boundary of P, then I − C∗ 1C1 − C2C∗ 2 ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' By assumption, there is a vector x such that C∗ 1C1x = x, from which it follows that C∗ 2x = 0, contradicting the assumption that C2 is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The spectraball BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let {e1, e2} denote the standard basis for C2 and let eT j denote the transpose of ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let Er ∈ M2(C)2 denote the tuple, Er 1 = eT 1 ⊗ C1 = � C1 0 � , Er 2 = eT 2 ⊗ C2 = � 0 C2 � and let Ec ∈ M2(C) denote the tuple Ec = (Ec 1, Ec 2) where Ec j = ej ⊗ Cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The spectraball associated to P is BR, where R is defined in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Since the first row and last column of Rj are zero, BR = BE, where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3) E = (E1, E2) = \uf8eb \uf8ec \uf8ed \uf8eb \uf8ec \uf8ed C1 0 0 0 0 0 0 0 C1 \uf8f6 \uf8f7 \uf8f8 , \uf8eb \uf8ec \uf8ed 0 C2 0 0 0 C2 0 0 0 \uf8f6 \uf8f7 \uf8f8 \uf8f6 \uf8f7 \uf8f8 = � Er 0 0 Ec � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The equality BR = BE = BEr ∩ BEc holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In particular, X ∈ BR if and only if both � C1 ⊗ X1 C2 ⊗ X2 � , � (C1 ⊗ X1)∗ (C2 ⊗ X2)∗� are strict contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' On the other hand, under the hypotheses of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1, BEj[2] ̸⊆ BEk[2] for j, k ∈ {r, c} and j ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The equalities BR = BE = BEr ∩ BEc are immediate from equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4, choose ρ < 1 such that ρ2(C∗ 1C1 + C∗ 2C2) ̸⪯ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The tuple X = ρ �� 0 1 0 0 � , � 0 0 0 1 �� is in BEr but not BEc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Hence BEr[2] ̸⊆ BEc[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A similar argument with X = ρ �� 1 0 0 0 � , � 0 1 0 0 �� shows BEc[2] ̸⊆ BEr[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' □ REINHARDT SPECTRAHEDRA 11 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In the terminology of [HKMV20], the direct sum Er ⊕ Ec is irredundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Ball minimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The discussion of ball-minimality here is borrowed from [HKMV20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In particular, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='7 below is excerpted from [HKMV20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A g-tuple E = (G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' , Gg) of d × e matrices is ball-mimimal for BG if F ∈ Mk×ℓ and BF = BG implies d ≤ k and e ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' As examples, it is immediate that both Er and Ec are ball-minimal (for BEr and BEc respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Tuples G, F ∈ Md,e(C)g are ball-equivalent if there exists unitary matrices U ∈ Me,e(C) and V ∈ Md,d(C) such that F = V ∗GU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Observe that, in this case, BF = BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A tuple A ∈ Md(C)g is minimal for DA if B ∈ Mk(C)g and DB = cDA implies d ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Ball minimal tuples exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' that is, if B ⊆ M(C)g is a spectraball, then there exists d, e and a ball minimal tuple G ∈ Md,e(C)g such that B = BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let G ∈ Md,e(C)g be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' (i) If G ball minimal, F ∈ Mk×ℓ(C)g and BG = BF, then there is a tuple J ∈ M(k−d)×(ℓ−e)(C)g and unitaries U, V of sizes k × k and ℓ × ℓ respectively such that BG ⊆ BJ and F = U � G 0 0 J � V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In particular, (a) d ≤ k and e ≤ ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' (b) if F ∈ Md×e(C)g is ball minimal too, then G and F are ball-equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' (ii) G ∈ Md,e(C)g is ball minimal if and only if H = � 0 G 0 0 � ∈ Md+e(C)g is minimal for DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The tuple E from equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3) is ball-minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let H = � 0 E 0 0 � ∈ M6s,6s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus BE = DH and, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='7, E is ball minimal for BE if and only if H is minimal for DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let {Ej,k : 1 ≤ j ≤ 6} denote the matrix units for M6(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let F1 = [E1,2 + E5,6] ⊗ C1, F2 = [E1,3 + E4,6] ⊗ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 12 BEN JEMAA AND MCCULLOUGH Since F and H are unitarily equivalent, we have DH = DF and moreover H is minimal defining for DH if and only if F is minimal for DF if and only if E is ball minimal for BE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let F denote the C-star algebra generated by {F1, F2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Since F1F ∗ 1 = [E1,1+E5,5]⊗C1C∗ 1 and F2F ∗ 2 = [E1,1 + E4,4] ⊗ C2C∗ 2 and since, by assumption, {C1C∗ 1, C2C∗ 2} generates Ms(C) as a C-star algebra, for each X ∈ Ms(C) there exist Y1, Y2 ∈ Ms(C) such that F contains E1,1 ⊗ X + E4,4 ⊗ Y2 + E5,5 ⊗ Y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Multiplying by F1F ∗ 1 F2F ∗ 2 on the right, it follows that F contains E1,1 ⊗ XC1C∗ 1C2C∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Since C1C∗ 1C2C∗ 2 is invertible, it follows that F contains E1,1 ⊗ Ms(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' By considering F ∗ j [E1,1 ⊗ Ms(C)] Fk, for j, k = 0, 1, 2 and F0 = I, and arguing as above, it follows that F contains �3 j,k=1 Ej,k⊗Ms(C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' that is F contains F1 = M3s(C)⊕0 as a C-star subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' By considering instead F ∗ j Fj and using {C∗ 1C1, C∗ 2C2} generates Ms(C) as a C-star algebra, it follows that F contains F2 = 0 ⊕ M3s(C) as a C-star subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Hence F = M3s(C) ⊕ M3s(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In particular, aside from the trivial ones, the only reducing subspaces for F, equivalently, {F1, F2}, are C3s ⊕ {0} and {0} ⊕ C3s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Suppose A ∈ Mk(C)2 is minimal defining for DF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus k ≤ 6s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' By [EHKM, Proposi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2], there exist a tuple J ∈ M(C)g such that, up to unitary equivalence, F = A ⊕ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Suppose A ̸= F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' It follows that either A = (E1,2 ⊗C1, E1,3 ⊗C2) or A = (E4,6 ⊗C1, E4,6 ⊗C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' But then, DA is either BEr or BEc, contradicting the conclusion of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='8 ({BEr, BEc} is irredundant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus F is minimal for DF and E is ball minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Nilpotent evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Given δ > 0, let B(0, δ) = {X ∈ M(C)g : g � j=1 XjX∗ j < δ2} It is straightforward to verify B(0, δ) is an open free set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A (formal) power series F in is an expression of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4) F(x) = � w∈ Fww.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A tuple X ∈ M(C)g is nilpotent of order at most m if Xw = 0 for all words w of length m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='9 below can be found in [KVV14, KS, HKM12] among other places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Suppose S is an open free set and there is a δ > 0 such that B(0, δ) ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' If f : S → M(C) is a free analytic function, then there exists a formal power series as in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4) such that if X ∈ B(0, δ), then f(X) = F(X) = ∞ � ℓ=0 � |w|=ℓ FwXw, REINHARDT SPECTRAHEDRA 13 with the series converging in norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Further, f extends analytically to all nilpotent tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In particular, if X is nilpotent of order at most m, then f(zX) = m � ℓ=0 ( � |w|=ℓ FwXw)zℓ for z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let S = � 0 1 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus, given X = (X1, X2) ∈ Mn(C)2, the tuple S ⊗ X = (S ⊗ X1, S ⊗ X2) = �� 0 X1 0 0 � , � 0 X2 0 0 �� ∈ M2n(C)2 is nilpotent of order two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The evaluation of free map f : P → M(C) on S ⊗ X takes a particularly simple form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' If f : P → M(C) is a free analytic function, then f extends uniquely to a function, still denoted f, defined on all tuples of the form S ⊗X for X ∈ M(C)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Moreover, there exists ℓ1, ℓ2 ∈ C such that f(S ⊗ X) = f(0) + S ⊗ ( 2 � j=1 ℓjXj) ∈ M2(C) ⊗ Mn(C) = M2n(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' We close this section with a proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The ideas borrow freely from [A+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' We are to show, if S is a free open set and f : S → M(C) is analytic and a free function in the sense that f respects both direct sums and unitary similarity, then f respects intertwinings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' To this end, suppose X ∈ S [m] and Y ∈ S [n] and Γ is an m × n matrix such that XΓ = ΓY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' To show f(X)Γ = Γf(Y ), first observe we may replace Γ with tΓ for any non-zero t ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Since S is closed with respect to direct sums Z = X ⊕ Y ∈ S [n + m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Since S [n + m] is open there is an ǫ > 0 such that if T is an (n + m) × (n + m) matrix and ∥T − I∥ < ǫ, then T is invertible and T −1 [X ⊕ Y ] T ∈ S [n + m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' For a matrix M and ρ ∈ C with |ρ| ∥M∥ < 1, let Dρ(M) = (I − ρ2M∗M) 1 2 14 BEN JEMAA AND MCCULLOUGH and let, for 0 ̸= z ∈ C, Tρ(M)[z] = � Dρ(M∗) ρ zM −ρ zM∗ Dρ(M) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' For |z| = 1 the matrix Tρ[M](z) is a version of the Julia matrix and is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' By choosing ∥Γ∥ sufficiently small, we may assume, for all 0 < ρ < 1, and ρ < |z| < 1 ρ, that ∥I − Tρ(Γ)[z]∥ < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Hence, Tρ(Γ)[z]−1 [X ⊕ Y ] Tρ(Γ)[z] ∈ S [n + m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The function (for fixed ρ) F(z) = f � Tρ(Γ)[z]−1 [X ⊕ Y ] Tρ(Γ)[z] � − Tρ(Γ)[z]−1 f(X ⊕ Y ) Tρ(Γ)[z] is defined and analytic on the annulus ρ ≤ |z| ≤ 1 ρ and vanishes for |z| = 1 since f respects unitary similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus F vanishes identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Choosing z = ρ and then letting ρ tend to 0 gives, f( � I −Γ 0 I � � X 0 0 Y � � I Γ 0 I � ) = � I −Γ 0 I � f( � X 0 0 Y � ) � I Γ 0 I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus, using f respects direct sums, � f(X) 0 0 f(Y ) � = f( � X 0 0 Y � ) = f( � X XΓ − ΓY 0 Y � ) = f( � I −Γ 0 I � � X 0 0 Y � � I Γ 0 I � ) = � I −Γ 0 I � f( � X 0 0 Y � ) � I Γ 0 I � = � I −Γ 0 I � � f(X) 0 0 f(Y ) � � I Γ 0 I � = � f(X) f(X)Γ − Γf(Y ) 0 f(Y ) � and the desired conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1 In this section, we present a proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1, we construct a spectraballs BB that is canonically associated with an automorphism ϕ : P → P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2, we collect consequences of the equality BB = BR = BE that are then used in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3 that b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The proof concludes in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' REINHARDT SPECTRAHEDRA 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' An affine change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In this section a spectraball BB canonically associated to an automorphism ϕ : P → P is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Suppose ϕ : P → P is bianalytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The mapping ϕ has coordinate functions ϕj so that ϕ(X) = (ϕ1(X), ϕ2(X)) for X ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Express the series expansions for ϕj, up to the first degree terms as ϕj(x) = bj + � k ℓj,kxk + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Since ϕ is bianalytic, the matrix (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1) L = � ℓ1,1 ℓ1,2 ℓ2,1 ℓ2,2 � is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Given a tuple X ∈ Mn(C)2, ϕj(S ⊗ X) = � bj � k ℓj,kXk 0 bj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let bj = bjCj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let B0 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed I b1 b2 0 b∗ 1 I 0 b2 b∗ 2 0 I b1 0 b∗ 2 b∗ 1 I \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Since ϕ(0) = b ∈ P, the matrix B0 = L (ϕ(0)) is positive definite and hence invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2) Yj = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 0 ℓ1,jC1 ℓ2,jC2 0 0 0 0 ℓ2,jC2 0 0 0 ℓ1,jC1 0 0 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = � k ℓk,jRk, where R is defined in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let B = (B1, B2) denote the tuple defined by Bj = B − 1 2 0 YjB − 1 2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let ΛB(x) = B1x1 + B2x2 and let BB denote the resulting spectraball, BB = {X : ∥ΛB(X)∥ < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The equality BR = BB holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 16 BEN JEMAA AND MCCULLOUGH Before proving Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1 we record the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' There is a permutation matrix Σ on {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=', 8} such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3) (Σ∗ ⊗ Is)L (ϕ(S ⊗ x))(Σ ⊗ Is) = � B0 ΛY (x) ΛY (x)∗ B0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let yj = (�2 k=1 ℓj,kxk) Cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Identifying 0 with the 0 matrix of size s × s, L (ϕ(S ⊗ x)) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed I 0 b1 y1 b2 y2 0 0 0 I 0 b1 0 b2 0 0 b∗ 1 0 I 0 0 0 b2 y2 y∗ 1 b∗ 1 0 I 0 0 0 b2 b∗ 2 0 0 0 I 0 b1 y1 y∗ 2 b∗ 2 0 0 0 I 0 b1 0 0 b∗ 2 0 b∗ 1 0 I 0 0 0 y∗ 2 b∗ 2 y∗ 1 b∗ 1 0 I \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3) holds with Σ equal the tensor product of the matrix associated to the permutation (1, 3, 5, 7, 2, 4, 6, 8) with Is, the s × s identity, after noting that \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 0 y1 y2 0 0 0 0 y2 0 0 0 y1 0 0 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = ΛY (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' □ Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Using the assumption that ϕ is an automorphism of P, Equa- tion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2, a tuple X is in BR[n] if and only if S ⊗ X ∈ P[2n] if and only if ϕ(S ⊗ X) ∈ P if and only if L (ϕ(S ⊗ X)) ≻ 0 if and only if � B0 ⊗ In ΛY (X) ΛY (X)∗ B0 ⊗ In � ≻ 0 if and only if 0 ≺ � B − 1 2 0 ⊗ In 0 0 B − 1 2 0 ⊗ In � � B0 ⊗ In ΛY (X) ΛY (X)∗ B0 ⊗ In � � B − 1 2 0 ⊗ In 0 0 B − 1 2 0 ⊗ In � = � I ΛB(X) ΛB(X)∗ I � if and only if X ∈ BB[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Analyzing the equality BR = BB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In this section we deduce consequences of the equality the two representation BE and BB of BR appearing in Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1 respectively using the theory of spectraballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' REINHARDT SPECTRAHEDRA 17 Since, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='8, E is ball-minimal and BE = BB, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='7 implies there exist unitary matrices U, V such that U � E 0 0 J � V ∗ = B, where J = (J1, J2) ∈ Ms(C)2 and BE ⊆ BJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Letting {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' , e4} denote the standard basis for C4 and Hj = {ej ⊗ h : h ∈ Cs}, ker(B) := ∩2 j=1 ker(Bj) = B 1 2 0 H1, since ∩2 j=1 ker(Yj) = H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus � Ej 0 0 Jj � V ∗B 1 2 0 H1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Since E has no kernel, it follows that J = 0 and V ∗B 1 2 0 H1 = H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In particular, U∗BjV = � Ej 0 0 0 � where the lower left 0 matrix has size s × s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Summarizing, there exist unitary matrices U, V such that Bj = U � Ej 0 0 0 � V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Set P = B 1 2 0 U and Q∗ = V ∗B 1 2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus PP ∗ = B0 = QQ∗ and Yj = P � Ej 0 0 0 � Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Express P and Q in terms of their block columns as P = � p1 p2 p3 p4 � = � Pj,k �4 j,k=1 Q = � q1 q2 q3 q4 � = � Qj,k �4 j,k=1 , , where Pj,k and Qj,k are s × s matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' With these notations, Y1 =P � E1 0 0 0 � Q∗ = P[(e1e∗ 1 + e3e∗ 3) ⊗ C1]Q∗ = p1C1q∗ 1 + p3C1q∗ 3 Y2 =P � E2 0 0 0 � Q∗ = P[(e1e∗ 2 + e2e∗ 3) ⊗ C2]Q∗ = p1C2q∗ 2 + p2C2q∗ 3, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4) 18 BEN JEMAA AND MCCULLOUGH where {e1, e2, e3, e4} is the standard basis for C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let T = Q−∗ 4,3 and S = P −∗ 1,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' With notations above, Q1,4 and P4,4 are unitary, Q4,3 and P1,1 are invertible and P = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed P1,1 P1,2 P1,3 0 0 ℓ2,2C2TC−1 2 ℓ2,1C2TC−1 1 b2C2P4,4 0 ℓ1,2C1TC−1 2 ℓ1,1C1TC−1 1 b1C1P4,4 0 0 0 P4,4 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 Q = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 0 0 0 Q1,4 ℓ∗ 1,1C∗ 1SC−∗ 1 ℓ∗ 1,2C∗ 1SC−∗ 2 0 b∗ 1C∗ 1Q1,4 ℓ∗ 2,1C∗ 2SC−∗ 1 ℓ∗ 2,2C∗ 2SC−∗ 2 0 b∗ 2C∗ 2Q1,4 Q4,1 Q4,2 Q4,3 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' With Wj = � 0 ℓ1,jC−1 2 −ℓ2,jC−1 1 0 � , observe 0 = W1Y1 = W1p1C1q∗ 1 + W1p2C1q∗ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Since Q is invertible, there exist r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' , r4 ∈ M4s,s(C) such that q∗ j rk = δj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Hence, from the first identity in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4) and equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2), 0 = W1p1C1q∗ 1r1 + W1p2C1q∗ 3r1 = W1p1C1 = [ℓ1,1C−1 2 P2,1 − ℓ2,1C−1 1 P3,1]C1 and therefore ℓ1,1C−1 2 P2,1 − ℓ2,1C−1 1 P3,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A similar argument using the second identity in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4) shows 0 = [ℓ1,2C−1 2 P2,1 − ℓ2,2C−1 1 P3,1]C2 and thus ℓ1,2C−1 2 P2,1−ℓ2,2C−1 1 P3,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Since L from equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1) is invertible, we conclude C−1 2 P2,1 = C−1 1 P3,1 = 0 and therefore P2,1 = P3,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Since [e∗ 4 ⊗ Is] Yj = 0, it also follows that P4,1 = P4,2 = P4,3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In particular P1,1 is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Examining the (block) (4, 4) entry of PP ∗ = B0 shows P4,4P ∗ 4,4 = Is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' From the last column of PP ∗ = B0 it now follows that P1,4 = 0, and P2,4 = b2C2P4,4 as well as P3,4 = b1C1P4,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' At this point we have identified, as indicated, the first and last row and column of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Similar reasoning applies to the third and forth columns and first and fourth rows of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' REINHARDT SPECTRAHEDRA 19 Comparing the descriptions of Yj from equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2) gives, P1,1C1Q∗ 2,1 = ℓ1,1C1 P1,1C1Q∗ 3,1 = ℓ2,1C2 P1,1C2Q∗ 2,2 = ℓ1,2C1 P1,1C2Q∗ 3,2 = ℓ2,2C2 P2,3C1Q∗ 4,3 = ℓ2,1C2 P3,3C1Q∗ 4,3 = ℓ1,1C1 P2,2C2Q∗ 4,3 = ℓ2,2C2 P3,2C1Q∗ 4,3 = ℓ1,2C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='5) Comparing the identities above with the definitions of T and S completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' □ Let Σ = � P2,2 P2,3 P3,2 P3,3 � , Λ = � b2C2P4,4 b1C1P4,4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' From the middle 2 × 2 block of PP ∗ = B0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='6) ΣΣ∗ + ΛΛ∗ = I = � I 0 0 I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Let G = � ℓ1,1 −ℓ2,1 −ℓ1,2 ℓ2,2 � � C−1 2 0 0 C−1 1 � , and observe GΣ = det L � TC−1 2 0 0 TC−1 1 � , GΛ = � ℓ1,1b2 − ℓ2,1b1 −ℓ1,2b2 + ℓ2,2b1 � P4,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus, applying G on the left and G∗ on the right of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='6) and comparing the (1, 2) (block) entries gives, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='7) − [ℓ1,1ℓ∗ 1,2C−1 2 C−∗ 2 + ℓ2,1ℓ∗ 2,2C−1 1 C−∗ 1 ] = (ℓ1,1b2 − ℓ2,1b1)(−ℓ1,2b2 + ℓ2,2b1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A dichotomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' With equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='7) in place, we are now in position to state and prove the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Recall s is the size of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' If s > 1, then b1b∗ 2 = 0 and (i) ℓ1,2 = 0 = ℓ2,1 or (ii) ℓ1,1 = 0 = ℓ2,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' If the right hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='7) is not 0, then either C−1 1 C−∗ 1 and C−1 2 C−∗ 2 commute or C−1 j C−∗ j is a multiple of the identity for either j = 1 or j = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In either case C∗ 1C1 and C∗ 2C2 commute and thus, as {C∗ 1C1, C∗ 2C2} generates Ms(C) as a C-star algebra, s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus, if s > 1, then the right hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='7) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' If ℓ1,1ℓ∗ 1,2 ̸= 0 or ℓ2,1ℓ∗ 2,2 ̸= 0, then they are both not zero and again, C∗ 1C1 and C∗ 2C2 commute and s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus, if s > 1, then ℓ1,1ℓ∗ 1,2 = 0 = ℓ2,1ℓ∗ 2,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Since L is invertible, there are 20 BEN JEMAA AND MCCULLOUGH two cases, either ℓ2,1 = 0 = ℓ1,2 (and ℓ1,1 ̸= 0 ̸= ℓ2,2) or ℓ1,1 = 0 = ℓ2,2 (and ℓ2,1 ̸= 0 ̸= ℓ1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In either case b1b∗ 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' If s > 1, then b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Arguing by contradiction, suppose s > 1 and b ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Observe that conclusion that b1b∗ 2 = 0 of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4 applies to any automorphism of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus, without loss of generality, b1 ̸= 0 and b2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Another appeal to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4 gives either ℓj,k = 0 for j ̸= k or ℓj,k = 0 for j = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Suppose the first case holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Given θ real, let fθ denote the automorphism of fθ given by fθ(y1, y2) = eiθ(y2, y1) and let ψ = ϕ ◦ fθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus ψ : P → P is an automorphism and ψ(0) = ϕ ◦ fθ(b1, 0) = ϕ(0, eiθb1) = (ϕ1(0, eiθb1), ϕ2(0, eiθb1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Hence ϕ1(0, eiθb1) ϕ2(0, eiθb1) is identically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' By analyticity, ϕk(0, eiθb1) = 0 for some k and all θ and hence, for either k = 1 or k = 2, we have gk(z) = ϕk(0, z) is identically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Since g1(0) = b1 ̸= 0, it follows that g2(z) is identically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus 0 = g′ 2(0) = ℓ2,2 ̸= 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Now suppose instead that ϕj,j = 0 for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In this case set fθ(y1, y2) = eiθ(y1, y2) and ψ = ϕ ◦ fθ ◦ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus, ψ(0) = ϕ(eiθb1, 0) and ϕ1(eiθb1, 0) ϕ2(eiθb1, 0) is identically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Hence g2(z) = ϕ2(z, 0) is identically 0 and thus 0 = g′ 2(0) = ℓ2,1 ̸= 0, a contradiction which shows b1 = 0 = b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' If s = 1, then b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Since s = 1, we have C1, C2 ∈ C are unimodular and P[1] is the set {z = (z1, z2) ∈ C2 : |z1| + |z2| < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Indeed, given (z1, z2) ∈ C2, the matrix � z∗ 1 z2 z∗ 2 z1 � is a contraction if and only if the self-adjoing matrix, � r z2eit z∗ 2e−it r � is a contraction, where z1 = reit is the polar decomposition of z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' This latter matrix has eigenvalues r ± |z2| and hence is a contraction if and only if |z1| + |z2| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' The set P[1] is known as a psuedo-ellipse and f = ϕ[1] : P[1] → P[1] is an auto- morphism (in the classical several complex variables sense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' It is known (see [JP08]) that automorphisms of P[1] are compositions of maps of the form (z1, z2) �→ (γ1z1, γ2z2) and (z1, z2) �→ (z2, z1), where γj ∈ C are unimodular (the proof uses techniques from several REINHARDT SPECTRAHEDRA 21 complex variables and lie groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In particular, b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Thus in any case b = 0 and ϕ is linear by [HKM11, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4], since ϕ(0) = b = 0 and the domain P is circularly symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' □ We now give a self contained alternate proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='6 based upon results in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Second proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In addition to the identities of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='5) obtained by com- paring the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='2), observe P1,1Q∗ 4,1 + P1,3Q∗ 4,3 = 0 P1,1Q∗ 4,2 + P1,2Q∗ 4,3 = 0, which can be summarized as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='8) Q∗ 4,3 � P1,3 P1,2 � = −P1,1 � Q∗ 4,1 Q∗ 4,2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' From (1, 2) and (1, 3) entries of PP ∗ = B0 and using Tj = Q−∗ 4,3, � P1,2 P1,3 � � ℓ∗ 2,2 ℓ∗ 1,2 ℓ∗ 2,1 ℓ∗ 1,1 � = Q4,3 � b1 b2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Equivalently, Q−∗ 4,3 � P ∗ 1,3 P ∗ 1,2 � � ℓ1,1 ℓ2,1 ℓ1,2 ℓ2,2 � = � b∗ 2 b∗ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Similarly from the (4, 2) and (4, 3) entries of QQ∗ = B0, P −1 1,1 � Q4,1 Q4,2 � � ℓ1,1 ℓ2,1 ℓ1,2 ℓ2,2 � = � b∗ 2 b∗ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' It follows that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='9) Q−1 4,3 � P1,3 P1,2 � = P −∗ 1,1 � Q∗ 4,1 Q4,2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' From equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='9), (|Q4,3|2 + |P1,1|2) � Q∗ 4,1 Q4,2 � = 0 and we conclude Q4,1 = Q4,2 = 0 and thus b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' □ 22 BEN JEMAA AND MCCULLOUGH 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Completion of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' From Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='5 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='6 it follows that b = 0 and hence, by [HKM11, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4], ϕ is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Since ϕ is linear, ϕ(x) = (w1, w2), where � w1 w2 � = L � x1 x2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' If s > 1, then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='4 and composing with the automorphism (x1, x2) �→ (x2, x1), we assume L is diagonal, in which case the diagonal entries must be unimodular and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Now suppose s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' From the (first) proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='6 it can be seen that ϕ is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Alternately, from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='3 and the relation PP ∗ = B0, it follows that the matrix � ℓ2,2 ℓ2,1 ℓ1,2 ℓ1,1 � T is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Hence L is a multiple of a unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Since ϕ is an automorphism of P, we conclude that L is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' In particular, |ℓ1,1|2 + |ℓ2,1|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Since (t, 0) is in P for 0 < t < 1, so is ϕ(t, 0) = t(ℓ1,1, ℓ2,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' It now follows from item (c) of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content='1 that ℓ1,1ℓ2,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' A similar argument shows ℓ1,2ℓ2,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Hence 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Vandenberghe (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' Handbook of semidefinite programming: theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' algorithms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' and applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE0T4oBgHgl3EQf5AKl/content/2301.02746v1.pdf'} +page_content=' International Series in Operations Research & Management Science 27,' metadata={'source': 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+Interpretation of flat energy spectra upstream of fast interplanetary shocks +Silvia Perri,1 Giuseppe Prete,1 Gaetano Zimbardo,1 Domenico Trotta,2 +Lynn B. Wilson III,3 David Lario,3 Sergio Servidio,1 Francesco Valentini,1 and +Joe Giacalone4 +1Dipartimento di Fisica, University of Calabria, Rende, Italy +2The Blackett Laboratory Imperial College London, London SW7 2AZ, UK +3Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA +4Lunar and Planetary Laboratory, University of Arizona, Tucson, USA +(Received; Revised; Accepted) +Submitted to ApJ +ABSTRACT +Interplanetary shocks are large-scale heliospheric structures often caused by eruptive +phenomena at the Sun, and represent one of the main sources of energetic particles. +Several interplanetary shock crossings by spacecraft at 1 AU have revealed enhanced +energetic-ion fluxes that extend far upstream of the shock. Surprisingly, in some shock +events, ion fluxes with energies between 100 keV and about 2 MeV acquire similar +values (which we refer to as “overlapped” fluxes), corresponding to flat energy spectra +in that range. In contrast, closer to the shock, the fluxes are observed to depend on +energy. In this work, we analyze three interplanetary shock-related energetic particle +events observed by the Advanced Composition Explorer spacecraft where flat ion energy +spectra were observed upstream of the shock. We interpret these observations via a +Corresponding author: Silvia Perri +silvia.perri@fis.unical.it + +2 +Perri et al. +velocity filter mechanism for particles in a given energy range. This reveals that low +energy particles tend to be confined to the shock front and cannot easily propagate +upstream, while high energy particles can. +The velocity filter mechanism has been +corroborated from observations of particle flux anisotropy by the Solid-State Telescope +of Wind/3DP. +Keywords: shock waves — energetic particles — turbulence — heliosphere +1. INTRODUCTION +Collisionless shock waves are observed to be one of the main sources of energetic particles and +cosmic rays in astrophysical environments. Efficient particle acceleration at shocks can result when +particles remain confined near the shock, either by scattering in turbulent magnetic fields, or as a +result of the geometry, and gain energy due to the compression in the plasma velocity at the shock +front (e.g., Drury 1983). Shock acceleration is supported by in-situ measurements in the heliosphere, +where energetic particle fluxes are clearly peaking at the time of shock passage (Giacalone 2012), +and by remote observations of supernova remnants (e.g., Bamba et al. 2003; Morlino et al. 2010; +Reynoso et al. 2013). On the other hand, the quantitative agreement with the predictions of diffusive +shock acceleration (DSA), like spectral slope, acceleration times, and maximum energies, remains +elusive (Lagage & Cesarsky 1983; Giacalone 2012; Vainio et al. 2014). +For instance, to reach an +energy of ∼ 1015 eV (i.e., the so-called knee of the cosmic ray energy spectrum), a substantial +amplification of the pre-existing (upstream) magnetic field by means of self-generated turbulence +is required (e.g., Drury 1983; Blasi 2013; Amato 2014). In a similar way, the particle mean free +paths in the heliosphere are estimated to be in the range of 0.01–1.0 AU, but this is far too large +to explain the rapid acceleration of particles by interplanetary (IP) shocks (Perri & Zimbardo 2015). +Thus, turbulence enhancement, related to the backstreaming of energetic particles upstream of IP +shocks, is required (e.g., Ng et al. 2003; Afanasiev et al. 2018). Similar electromagnetic fluctuation +amplification is required to explain the efficient acceleration of solar energetic particles (SEPs) by +CME-driven shocks (e.g., Lee et al. 2012). + +Flat energy spectra +3 +In collisionless plasmas, because of particle reflection at supercritical shocks (Burgess & Scholer +2015), shock properties also depend on the (acute) angle, θBn, between the average upstream magnetic +field, B, and the unit-normal vector to the shock. If θBn < 45◦ the shock is termed quasi-parallel +and reflected ions can efficiently propagate upstream, thereby forming the ion foreshock. +Here, +reflected particles form a beam in velocity space which can excite ion cyclotron, Alfv´en waves, or +fast-magnetosonic, and whistlers, making the ion foreshock a very turbulent region, as often observed +by spacecraft in the terrestrial ion foreshock (e.g., Schwartz & Burgess 1991; Perri et al. 2009; Wilson +2016; Karimabadi et al. 2014). These disturbances can give the required enhanced level of magnetic +fluctuations to efficiently scatter both electrons (Wilson et al. 2016) and ions (Turner et al. 2018; +Wilson et al. 2013), thereby trapping them near the shock for further acceleration to high energies +according to the mechanism of DSA. Conversely, if θBn > 45◦ the shock is termed quasi-perpendicular +and most reflected ions re-enter the shock, contributing to enhanced level of fluctuations downstream, +as usually observed (Greenstadt et al. 1975). +In the absence of rapid scattering upstream, DSA +can be slow, as suggested by some observations (Reynoso et al. 2013) and numerical simulations +(e.g., Caprioli et al. 2015; Sundberg et al. 2016; Trotta et al. 2020; Preisser et al. 2020). However, it +should be noticed that enhanced levels of magnetic fluctuations upstream of quasi-parallel IP shocks +are not always observed (Blanco-Cano et al. 2016); for instance, in a recent study Zimbardo et al. +(2020) found that the magnetic power at the scales corresponding to energetic particle resonance was +approximately constant from upstream to downstream for three IP shocks with different values of +θBn. Therefore, the conditions under which self-generated turbulence is actually found need to be +better understood. +Observations of fluxes of supra-thermal particles can be highly influenced by the upstream trans- +port conditions, which are established by the level of magnetic field fluctuations. +For example, +Lario et al. (2022) investigated the formation of an anisotropic field-aligned beam of protons up- +stream of an oblique shock with energies ≤ 30 keV together with a population of protons at higher +energy propagating at small pitch-angle. The unusually long duration (and therefore spatial extent) + +4 +Perri et al. +of the field-aligned beam was interpreted as due to the absence of magnetic field fluctuations over a +large distance upstream of the shock wave — a scenario where efficient scattering is not favoured. +Another fundamental phenomenon is the formation of a precursor of energetic particles in the +upstream: this precursor is distinct from the ion foreshock (which is formed mostly by reflected +thermal particles), or from the fast/magnetosonic whistler wave precursor (Wilson et al. 2017), and +is due to those energetic particles satisfying +vµ > V sh +1 +sec(θBn), +(1) +i.e., those particles that have velocity parallel to the local magnetic field (being v the particle speed +and µ the cosine of the pitch angle) larger than the speed of the intersection point of a field line with +the shock front along the direction of the shock front (e.g., le Roux & Webb 2012; Lario et al. 2019). +Notice that V sh +1 +represents the upstream solar wind speed in the frame of reference of the shock. +This velocity filter can lead to a spectral flattening upstream of the shock, since only higher energy +particles, can easily propagate back upstream, while lower energy particles tend to be confined +within the shock region, reducing their fluxes far upstream. Nonetheless, the presence of turbulent +fluctuations causes pitch angle scattering and a meandering of magnetic field lines, which can influence +the velocity filter effect on particles. +Such spectral flattening has been predicted in earlier works by Ng et al. (2003), as a consequence of +the differential growth of Alfv´en waves upstream of the shock, amplified by high energy ions streaming +away from the shock. This creates a sort of delay in the lower energy particle transport upstream, +favouring the formation of a flat energy spectrum. That model, however, suffers the limitation of +predicting very short intervals for spectral flattening. More recently, overlapping particle fluxes have +been modelled by Prinsloo et al. (2019), and explained as a balance between shock acceleration and +adiabatic cooling; however, at 1 AU they obtained a very narrow (in energy) spectral flattening. +Thus, mismatches with observations are found. +In this paper, we analyze three IP shock crossings and associated energetic ion intensities observed +by the Advanced Composition Explorer (ACE) and the Wind spacecraft on 2005 May 15, on 2012 + +Flat energy spectra +5 +July 14, and on 2003 November 4. These events were chosen because continuous data measurements +from various instruments are available, no particular IP disturbances affect energetic-ion fluxes, +and the IP shocks have similar compression ratios. All these events are characterized by energetic +particle profiles that are overlapping far upstream and up to about 1-3 hours before the shock +passage (i.e., the upstream energy spectrum is flat for a broad range of particle energies). Closer +to the shocks, the fluxes at each energy separate and reach values more typical of those predicted +by DSA theory. This peculiar feature of upstream overlapping particle fluxes was first reported by +Lario et al. (2018), looking at particles upstream of IP shocks using ACE measurements at energies +between approximately 50 keV and 4 MeV. They noted that the region of spectral flattening for these +energetic protons happens prior of the shock arrival, as the energy spectrum steepens close to the +shock, and the observed spectral shapes are closely related to the particle transport processes from +the evolving shock to the observer. +Here, we analyze energetic particle fluxes and magnetic fluctuations close to and far upstream of +the shocks. We interpret observations of flat upstream energy spectra in terms of the velocity filter +mechanism described above. Importantly, we derive how particle fluxes are modified from such a +condition for different values of the angle θBn and of the upstream plasma speed as measured in the +shock frame. +The paper is organized as follows: Section 2 contains the data description and the data analysis; +Section 3 presents the interpretation of the data according to the velocity filter condition; in Section +4 concluding remarks are given. +2. DATA COLLECTION AND ANALYSIS +2.1. Shock crossings +We first analyze the shock event of 2005 May 15, which exhibits overlapping energetic particle +intensity profiles over a large distance upstream of the shock crossing. +Energetic particle fluxes +separate at about 50 min before the shock crossing and downstream of the shock, thus resulting +in a steeper energy spectrum (Lario et al. 2018). The solar origin of this relatively strong event + +6 +Perri et al. +was associated with a halo, fast (plane-of-sky speed ∼1689 km s−1) coronal mass ejection (CME) +observed at 17:12 UT on 2005 May 13, as reported in the Coordinated Data Analysis Workshops +(CDAW) Data Center SOHO LASCO CME catalog1 and temporary associated with a M8.0 GOES +class X-ray flare from the NOAA active region 10759 at N12E12 with onset at 16:13 UT on 2005 +May 13 (Lario et al. 2018). The transit time for the shock to travel from the Sun to 1 AU was about +2038 minutes corresponding to an average transit speed of ∼1223 km s−1 (see Table 1 in Lario et al. +2018). Using data from GOES-11, Lario et al. (2018) show that energetic particles peak at the shock +even in the high energy channel of 32.5–56.4 MeV, confirming that it is a strong particle accelerator. +Figure 1 shows an overview of the 2005 May 15 event. From top to bottom we report (a) the 12 s +resolution particles fluxes detected by the Electron, Proton, and Alpha Monitor (EPAM) instrument +(Gold et al. 1998) onboard ACE from the LEMS120 sensor in the energy range 68 keV-1.90 MeV; (b) +the magnetic field components in Radial-Tangential-Normal (RTN) coordinates as sampled by the +MAG instrument (Smith et al. 1998) onboard ACE at a time cadence of 60 s; (c) and (d) the plasma +proton density and the solar wind bulk speed, respectively, as detected by the Solar Wind Electron, +Proton, and Alpha Monitor (SWEPAM) experiment (McComas et al. 1998) onboard ACE (black +line) at 64 s resolution. Because of a data gap in the ACE time series around the shock crossing, we +add the proton solar wind density measured by the Solar Wind Experiment (SWE) (Ogilvie et al. +1995) on board Wind at 92 s resolution (red line). The panel (e) shows the angle ψ between the mean +field and the radial direction (e.g., Sonnerup 1969). The latter gives us insights into the magnetic +connectivity of the observer with the shock surface (the horizontal dashed red line indicates a 90◦ +angle). The angle ψ has been computed using 5-minutes running average windows. +The vertical black dashed line indicates the shock crossing time. We display a time window ex- +tending towards ≃ 24 hours before the ACE shock crossing. For a period of about 50 min up to 16 +hours prior to the shock crossing (indicated by a shaded region in Figure 1) the particle fluxes at the +displayed energies overlap. For earlier times (i.e. more than 17 hours prior to the shock crossing), +1 available at cdaw.gsfc.nasa.gov/CME list/ + +Flat energy spectra +7 +Figure 1. Overview of the shock crossing by the ACE spacecraft on 2005 May 15. From top to bottom: +particle fluxes from the EPAM-LEMS120 instrument in the energy range 68-1890 keV measured at 12 s +cadence, the 60 s resolution magnetic field components in the Radial-Tangential-Normal (RTN) reference +frame along with the magnetic field magnitude, the proton density and the solar wind bulk speed both from +ACE (at 64 s resolution) and Wind (at 92 s resolution), and the angle between the radial direction and the +mean field direction computed over a running window of 5 min. The vertical dashed line indicates the shock +position, the yellow shaded area shows the region where energetic particle fluxes overlap indicating a period +with a relatively flat energy spectrum. +particle fluxes at different energies are well separated. It can be seen in panel (b) that within the +region upstream where the energetic particle fluxes overlap, the magnetic field tends to be approxi- +mately radial (see also panel (e) in Figure 1 where ψ < 30◦ for most of the time interval indicated +by the shaded region), while it largely deviates from the radial direction close to the shock. This + +68 - 115keV +(a) +115- 195keV +105 - +195- 321keV +321 - 580keV +0.580 - 1.06 MeV +104 +1.06 - 1.90MeV +103 +-2 +(cm +102 +101 +60 +B +(b) +40 - +BR +BT +(lu) +20 +BN +0 +B +-20 +-40 +30 +ACE +(c) +3 +WIND +20 - +cm +10 +v(km/s) +(d) +1000 +750 +500 +150 +(e) +100 +50 +0 +-1250 +-1000 +-750 +-500 +-250 +250 +500 +t(min)8 +Perri et al. +Table 1. Main parameters for the three shock crossings analyzed, as deduced from ACE and Wind measure- +ments (see text for details). The θBn estimates using both the minimum variance (MVA) and the magnetic +coplanarity (MC) methods are reported, the Alfv´enic and the sonic mach numbers, the compression ratio of +the shock, the plasma beta, and the shock speed in the satellite frame. Associated errors are also reported. +date +time (ACE) +θBn(◦)-MVA +θBn(◦)-MC +MA +MS +r +βplasma +V s/c +sh (km/s) +15/05/05 +02:05 +78.7 ± 12.9 +51.3 ± 13.6 +11.6 +17.5 +3.0 ± 0.6 +1.57 +926.9 +14/07/12 +17:26 +68.6 ± 22.9 +67.6 ± 7.3 +42.7 +81.4 +3.0 ± 0.5 +0.99 +710.6 +04/11/03 +05:59 +43.4 ± 12.5 +9.9 ± 5.7 +6.01 +10.5 +3.6 ± 1.1 +1.2 +803.7 +suggests that for the entire period when the upstream flat spectrum was observed, the spacecraft was +well connected to the shock front. However, the fluxes remain overlapped even closer to the shock +when the field starts changing direction. Indeed, fluxes separate only very close to the shock (−40 +min), while the magnetic field deviates from the radial direction at about −85 min. +In the downstream region, ∼3 hours after the shock crossing, a sudden decrease of about two orders +of magnitude in the particle fluxes is observed, which is due to the arrival of the magnetic obstacle +of the CME that generated the SEP event (Lario et al. 2018; Perri et al. 2022). +Fundamental shock parameters have been computed and are shown in Table 1, although single- +spacecraft shock parameter estimation is a process that is known to be subject to many sources of +uncertainty (Paschmann & Daly 2000; Koval & Szabo 2008). For θBn, we used both the Minimum +Variance Analysis (MVA) over a 30 minutes time interval before the shock crossing, similarly to what +has been done in van Nes et al. (1984), although this technique can be highly unstable because of +rapid variations of the magnetic field, and the magnetic coplanarity (MC) method (Colburn & Sonett +1966). The MC technique with a systematic variation of upstream and downstream averaging win- +dows from ∼ 2 to 30 minutes (as done in Trotta et al. 2022) yields a large spread of values, indicating +high levels of upstream/downstream disturbances. The Alfv´en and the sonic Mach numbers, the com- +pression ratio of the shock, the plasma beta β = nkBT/(B2/2µ0) (being kB the Boltzmann constant +and µ0 the vacuum magnetic permeability), and the shock speed in the spacecraft frame of reference +(Perri et al. 2015) are also reported (values averaged over a time window of 30 min). It should be also + +Flat energy spectra +9 +noted that the parameter estimations assume a stationary, planar, and infinite shock, a hypothesis +that is often not satisfied. Actually, the shock structure can be further complicated by the interaction +with pre-existing structures that modify the local geometry of the shock front (as recently addressed +for IP shocks (Blanco-Cano et al. 2019; Giacalone et al. 2021) as well as for the Earth’s bow shock +(Trotta et al. 2022)). Moreover, the shocks analysed here are supercritical (see Table 1), indicating +that they may be affected by many self-induced spatial and temporal irregularities happening at a +variety of scales (e.g. Kajdiˇc et al. 2019, 2021). +From Table 1, the 2005 May 15 event is a quasi-perpendicular shock, with sonic and Alfv´enic Mach +numbers Ms ∼ MA ∼ 16, which is able to accelerate ions to tens of MeV at 1 AU (i.e., at the +spacecraft position). +The second analyzed event is a quasi-perpendicular shock at 17:26 UT on 2012 July 14. +The +solar origin of this event was associated with a fast (plane-of-sky speed 885 km s−1) halo CME +observed at 16:48 UT on 2012 July 12 as reported by the CDAW SOHO LASCO CME Catalog and +temporally associated with a X.4 flare from NOAA Active Region 11520 at S15W01 at 15:37 UT +(Wijsen et al. 2022). The transit time of the shock to travel from the Sun to 1 AU is about 2989 +minutes, corresponding to an average transit speed of ∼ 834 km s−1. Figure 2 shows data for the +event on 2012 July 14 (with the same format as Figure 1). This event displayed a flat particle energy +spectrum from more than 12 hrs to about 2 hrs prior to the shock passage (shaded region in Figure +2). In this case the magnetic field was almost radial (ψ < 50◦) but gradually started to change +direction about 4.5 hrs prior to the shock crossing, while the fluxes remained overlapped (at least +the ones in the lower energy channels, from 68 keV to 580 keV). +The third shock crossing analyzed occurred at 05:59 UT on 2003 November 4 and corresponds +to a quasi-parallel shock (see Table 1). The solar origin of this event was associated with a fast +(plane-of-sky speed 2598 km s−1) halo CME at 17:30 UT on 2003 November 2 as reported by the +CDAW SOHO LASCO CME Catalog and temporally associated with a X8.3 flare at 17:03 UT from +NOAA Active Region 10486 at S14W56 (Lario et al. 2005). The transit time of the shock to travel +from the Sun to 1 AU is about 2216 minutes, corresponding to an average transit speed of ∼1125 + +10 +Perri et al. +Figure 2. Same as Figure 1 but for the shock crossing by the ACE spacecraft on 2012 July 14. +km s−1. Figure 3 shows, with the same format as Figure 1, data for the event on 2003 November 4. +Energetic ion fluxes overlapping for several hours before the shock passage at the spacecraft position +within the energy channels between 115 keV and 1.06 MeV. The ion fluxes prior to the shaded region +in Figure 3 were already elevated due to prior SEP events. We note that the fluxes in the lowest +energy channel (i.e., 68–115 keV) and in the highest energy channel (1.06–1.90 MeV) overlap partly +within the shaded box in Figure 3. The magnetic field is oriented sunward in this period, oscillating +around ψ ∼ 135◦. + +1 +68 - 115keV +(a) +115 - 195keV +s +105 +195 - 321keV +321 - 580keV +0.580 - 1.06 MeV +104 +1.06 - 1.90 MeV +103 +102 +101 +[BI +(b) +20 +BR +10 +Bt +BN +n +0 +B +-10 +-20 +30 +ACE +() +WIND +20 +(cm +10 +0 +ACE +V(km/s) +600 +400 + = 90° +150 +100 +B +50 +0 +1250 +-1000 +-750 +500 +250 +0 +250 +500 +t(min)Flat energy spectra +11 +Figure 3. Same as Figure 1 but for the shock crossing by the ACE spacecraft on 2003 November 4. +2.2. Magnetic field turbulence +In order to characterize the environment through which the shocks propagate, we have studied the +properties of magnetic field turbulence within both (i) the upstream region where flat energy spectra +are detected, and (ii) close to the shock where power-law energy spectra are observed. This has been +done by computing the power spectral density (PSD) of the magnetic field fluctuations along each +direction in the RTN reference frame, using the Fast Fourier Transform. + +1 +68 - 115keV +(a) +115 - 195keV +195- 321keV +105 +321 - 580keV +0.580 - 1.06 MeV +1.06 - 1.90MeV +104 +103 +30 +B +(b) +20 +BR +BT +(lu) +10 +B +-10 +-20 +38 +ACE +(c) +-3) +20 +WIND +(cm +10 +0 +800 +V(km/s) +600 +Ww +400 +150 +(e) +100 +B +50 +=90° +0 +1250 +-1000 +-750 +-500 +250 +0 +250 +500 +t(min)12 +Perri et al. +Figure 4. PSD of the magnetic field fluctuations for the 2005 May 15 event, computed in the region far +upstream where the fluxes of energetic particles overlap (black line) and in the region close to the shock +front upstream (blue line). The vertical dashed lines indicates the frequencies corresponding to the Larmor +radius of the energetic particles (see Figure legend). The best power-law fits are also shown by the dashed +lines. +Figure 4 displays the power spectrum of the magnetic field in the two regions described above for the +event on 2005 May 15. In the region where the energetic particle fluxes overlap, from -800 min to -100 +min from the shock, the spectrum follows a Kolmogorov-like power-law, i.e., PSD(f) ∝ f −5/3, where +f is the frequency, indicating ambient solar wind turbulence (Bruno & Carbone 2013; Pitˇna et al. +2021). By contrast, in the close upstream region, from -80 min to -10 min, we find PSD(f) ∝ f −1 +for the 2005 May 15 event, while in the other two crossings (where we have used the same time +interval for PSDs computation) the PSD is bumped at low frequencies (see Figures 5 and 6). In +Figure 4 the best power-law fits are displayed with the black and the red dashed lines. The values + +104 +Far upstream +f -1.51±0.01 +103 +Close upstream +f -0.98 ± 0.01 +102, +PSD(nT2/Hz) +101 +100 +68- 115keV +10-1 +115- 195keV +195-321keV +321-580keV +10-2 +0.580 - 1.06 MeV +1.06 - 1.90 MeV +10-3. +10-5 +10-4 +10-3 +10-2 +f(Hz)Flat energy spectra +13 +Figure 5. Same as Figure 4 but for the shock on 2012 July 14. +found for the slopes, along with their associated errors, are also reported in the legend. Notice that +we have computed the power-law fits within a range of frequencies corresponding to the Larmor +radius of the energetic particles, namely, fE = Vup/(2πρE) where Vup is the proton bulk speed in +the relevant upstream region in the spacecraft frame and ρE is the Larmor radius of protons with +energy E. Such frequencies are indicated by the vertical dashed lines in Figure 4. The presence of +a broad PSD(f) ∝ f −1 range suggests that fluctuations are freshly-injected in that region and that +turbulence has not become fully-developed yet. In addition, the presence of a bumped spectrum close +upstream for the 2012 July 14 and 2003 November 4 crossings can be related to fluctuations driven +by the shock-reflected energetic ions. + +104 +Far upstream +f -1.36 ±0.01 +Close upstream +103 +f -0.89 ± 0.03 +102 +PSD(nT2/Hz) +101 +100 +68 - 115keV +115 - 195keV +195 -321keV +10-1 +321 - 580keV +0.580 - 1.06 MeV +1.06 - 1.90 MeV +10-2 +10-5 +10-4 +10-3 +10-2 +f(Hz)14 +Perri et al. +Figure 6. Same as Figure 4 but for the shock on 2003 November 4. +Indeed, the PSD close upstream decays as a f −1 power-law with a bend over towards frequencies +higher than the ones corresponding to energetic particle scales (see Figs. 5 and 6). Notice that in +these two events, the power in the magnetic field fluctuations at low frequencies is smaller (or almost +comparable) in the close upstream region than far upstream (see Figs. 5 and 6 and the discussion +below). +Further, we have explored how turbulent fluctuations are distributed both in frequency and as +a function of the distance from the shock via a wavelet analysis of the magnetic field vector. +The square of the wavelet coefficients of the magnetic field components have been computed as + +104 +Far upstream +f -1.34 ±0.01 +Close upstream +103 +f -0.81 ± 0.03 +PSD (nT2/Hz) +102. +101 +68 - 115keV +M1 +115 - 195keV +195-321keV +100 +321 - 580keV +0.580 - 1.06 MeV +1.06 - 1.90MeV +10-1 +10-5 +10-4 +10-3 +10-2 +f(Hz)Flat energy spectra +15 +(Alexandrova et al. 2008) +|WB(τ, t)|2 = +� +i +|Wi(τ, t)|2, +(2) +where τ = 1/f represents a time scale, and the sum is over the magnetic field components, +i = R, T, N. Wi(τ, t) represent the Morlet wavelet coefficients computed over different τ and time +t (Torrence & Compo 1998), i.e., Wi(τ, t) = �N +j=1 Bi(tj)ψ∗ [(tj − t) /τ], with ψ∗ being the conjugate +of the wavelet function. This allows us to assess the magnetic energy content in frequency and in +time and localize, within the time series, the regions with high magnetic energy. Eq. (2) has been +reported for the 2005 May 15 shock crossing in Figure 7. It is indeed evident how the magnetic field +fluctuation power increases close to the shock front, over a broad frequency range, which includes also +the frequencies corresponding to the Larmor radius of the energetic particles, indicated in Figure 7 by +the horizontal white dashed lines. The yellow box surrounds the flat energy spectrum region. Thus, +the magnetic energy stored in the fluctuations tends to increase very close to the shock (indicated by +the vertical dashed line) and is found to be high downstream of the shock within the turbulent sheath +region. Here fluctuations are highly compressed and enhanced. The increase of magnetic power close +to the shock reconciles with the observation of an extended ∼ f −1 range in the PSD computed in +the close upstream region, which is also in agreement with recent observations of turbulence close to +IP shocks (Zhao et al. 2021). The local increase of the magnetic field power and the detection of a +bumped PSD close upstream might be ascribed to a self-generated, freshly-injected turbulence, due +to the presence of the energetic particle fluxes. +In addition, from the scalograms in Figs. 8 and 9, large amplitude pre-existing magnetic fluctuations +can be observed over a broad range of frequencies and times. This evidence is in agreement with +the comparable power found in the spectra upstream and close upstream in the 2012 July 14 and +in the 2003 November 4 events. Such larger amplitude fluctuations far upstream can favour particle +scattering, especially at lower frequencies where more power is stored. This can explain the tendency +of high energy ion fluxes to separate from the fluxes in the other energy channels in the 2012 July +14 and in the 2003 November 4 events. + +16 +Perri et al. +2.3. Energetic Particle Anisotropy +At this point is crucial to investigate the propagation of energetic particles with respect to the +magnetic field direction using the Solid State Telescope (SST) of Wind 3DP (Lin et al. 1995), in +order to better characterize their motion. Ion fluxes are organized in 9 energy channels with average +energies of 76 keV, 130 keV, 200 keV, 336 keV, 554 keV, 1.0 MeV, 2.0 MeV, 4.0 MeV, 6.8 MeV. +They are also binned in 8 pitch-angle values with respect to the local magnetic field direction; such +fluxes, normalized to their maximum intensity, are displayed in nine panels in Figure 10 for the 2005 +May 15 event, as a function of the pitch-angle cosine µ. Fluxes are in the solar wind frame, in order +to get rid of any anisotropic feature due to the Compton-Getting effect (Compton & Getting 1935; +Forman 1970). The three sets of symbols in Fig. 10 refer to the mean fluxes calculated over three +different regions around the shock: far upstream where flat spectra are detected (from -750 min to +-200 min upstream of the shock), close upstream of the shock (from -80 min to -10 min), and from 10 +min up to 70 min downstream of the shock. In all energy channels there is a tendency to isotropize +the distributions of the fluxes in the downstream region, caused by the presence of the augmented +magnetic field fluctuations which efficiently scatter energetic particles. Far upstream, within the +region of overlapped fluxes, we observed for the lowest energy channel that the flux is higher at large +pitch-angles, namely particles move sunward towards the shock, while higher energy particles travel +mostly anti-sunward at small pitch-angles. Here the amplitude of magnetic field fluctuation decreases +but the field is almost radial and a good connection between the shock and the spacecraft exists. +This permits to detect in the far upstream region those particles that have been isotropized close +upstream, considering that parallel diffusion is larger than perpendicular diffusion. This anisotropy +in pitch angle is somewhat reduced for all the energy channels in the close upstream region. This is +again due to the amplified magnetic fluctuations, since ions can easily be scattered in all directions +by interacting with turbulent fluctuations (Giacalone & Jokipii 1999; Trotta et al. 2021). Here, lower +energy particles tend to be isotropized in µ, while there is still anisotropy in favour of field-aligned +particles at higher energies. A similar behaviour has been observed for the energetic particle fluxes +associated to the 2012 July 14 event (see Fig. 11). + +Flat energy spectra +17 +Figure 7. Power spectral density computed using the wavelet coefficients over the three magnetic field +components (see text for further details). The yellow box delimits the region upstream where a flat energy +spectrum has been detected, while the vertical dashed line remarks the shock crossing time. The horizontal +dashed lines indicates the frequency range corresponding to the Larmor radius of energetic particles from +67 keV to 2 MeV. Typical cascade patches (Greco et al. 2016) can be recognized over the entire upstream +region, with an intensity increase close to the shock front. Just behind the shock the CME sheath region is +characterized by a very high level of turbulence over a broad range of frequencies. +Figure 8. Same as Figure 7 but for the shock crossing by the ACE spacecraft on 2012 July 14. +On the other hand, the mean ion flux values displayed in Fig. 12 for the 2003 November 4 shock +exhibit a high degree of isotropy in the three regions for almost all the energy channels, except for the + +2012/07/14 +2.36608e-05 +9.4643e-05 +25 +0.000378572 +20 +(ZH) +0.00151429 +15 +0.00605715 +10 +0.0242286 +0.0969145 +-1250 +-1000 +-750 +-500 +-250 +0 +250 +500 +t (min)2005/05/15 +2.38434e-05 +25 +9.53736e-05 +0.000381495 +20 +N +0.00152598 +15 +0.00610391 +10 +0.0244157 +5 +0.0976626 +-1250 +-1000 +-750 +-500 +-250 +0 +250 +t (min)18 +Perri et al. +Figure 9. Same as Figure 7 but for the shock crossing by the ACE spacecraft on 2003 November 4. +-1 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +normalized intensity +76 keV +-1 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +130 keV +far upstream +close upstream +downstream +-1 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +200 keV +0 +0.2 +0.4 +0.6 +0.8 +1 +normalized intensity +336 keV +0 +0.2 +0.4 +0.6 +0.8 +1 +554 keV +0 +0.2 +0.4 +0.6 +0.8 +1 +1.0 MeV +0 +0.2 +0.4 +0.6 +0.8 +1 +normalized intensity +2.0 MeV +0 +0.2 +0.4 +0.6 +0.8 +1 +2005-05-15 +4.4 MeV +0 +0.2 +0.4 +0.6 +0.8 +1 +6.8 MeV +Figure 10. Mean ion fluxes, normalized to the maximum intensity in each region (see text for further +details), as measured from the Wind/3DP/SST instrument in the solar wind frame within each pitch-angle +bin during the 2005 May 15 event. Each panel refers to a given energy channel. The black circles indicate +the ion fluxes measured far upstream of the shock wave within the shaded region indicated in Figure 1, the +red diamonds refer to the close upstream region, where ion fluxes start separating, and the blue stars to the +downstream sheath region. +76 keV channel, where particles are almost anti-aligned to the mean field (which is sunward), going + +2003/11/04 +2.51109e-05 +0.000100444 +25 +0.000401774 +20 +(ZH) +0.0016071 +15 +0.00642839 +10 +0.0257135 +5 +0.102854 +-1250 +-1000 +-750 +-500 +250 +0 +250 +500 +t (min)Flat energy spectra +19 +-1 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +normalized intensity +76 keV +-1 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +130 keV +far upstream +close upstream +downstream +-1 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +200 keV +0 +0.2 +0.4 +0.6 +0.8 +1 +normalized intensity +336 keV +0 +0.2 +0.4 +0.6 +0.8 +1 +554 keV +0 +0.2 +0.4 +0.6 +0.8 +1 +1.0 MeV +0 +0.2 +0.4 +0.6 +0.8 +1 +normalized intensity +2.0 MeV +0 +0.2 +0.4 +0.6 +0.8 +1 +2012-07-14 +4.4 MeV +0 +0.2 +0.4 +0.6 +0.8 +1 +6.8 MeV +Figure 11. Same as Figure 10 but for the shock crossing by the ACE spacecraft on 2012 July 14th. +along the anti-sunward direction. Notice from Fig. 3 that the flux of energetic ions in this channel +is not completely overlapped to the other fluxes in the shaded region. Observing Fig. 6 and Fig. 9, +the power stored in the magnetic field fluctuations is a bit higher than in the other two events. This +will induce more particle scattering and therefore more isotropy. +3. DISCUSSION ON THE VELOCITY FILTER CONDITION +Following the interpretation proposed by le Roux & Webb (2012) (see also Lario et al. 2019), we +have derived the energetic particle flux by imposing the velocity filter condition, namely low energy +particles cannot easily escape far upstream. Since the number of particles within a volume in phase +space is defined as (e.g., Moraal 2013) +dN = F(r, p, t)d3rd3p = F(r, p, t)d3rp2dpdΩ, +(3) +where F(r, p, t) is the particle distribution function and dΩ = sin αdαdφ is the element of solid +angle in momentum space, we can define a differential density in momentum, namely the number of +particles in a given volume d3r and with momentum within p and p + dp as + +20 +Perri et al. +-1 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +normalized intensity +76 keV +-1 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +130 keV +far upstream +close upstream +downstream +-1 +-0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +200 keV +0 +0.2 +0.4 +0.6 +0.8 +1 +normalized intensity +336 keV +0 +0.2 +0.4 +0.6 +0.8 +1 +554 keV +0 +0.2 +0.4 +0.6 +0.8 +1 +1.0 MeV +0 +0.2 +0.4 +0.6 +0.8 +1 +normalized intensity +2.0 MeV +0 +0.2 +0.4 +0.6 +0.8 +1 +2003-11-04 +4.4 MeV +0 +0.2 +0.4 +0.6 +0.8 +1 +6.8 MeV +Figure 12. Same as Figure 10 but for the shock crossing by the ACE spacecraft on 2003 November 4. +Up = p2 +� +Ω +F(r, p, t)dΩ. +(4) +This implies that the number of particles becomes dN = Upd3rdp. We align the polar coordinate axis +with the upstream magnetic field, so that α is the pitch angle; considering that only particles with +a sufficiently large parallel velocity can escape from the moving shock, we find that α can vary over +a limited range of values, namely the particle velocity belongs to a limited spherical sector in phase +space (see the cartoon in Figure 13). Then, we can calculate the differential density of upstream +propagating particles as +U′ +p = 2πp2 +� αmax +0 +F(r, p, t) sin αdα. +(5) +Here, we assume that F(r, p, t) is isotropic, i.e., that the particle distribution function depends on +the momentum magnitude p only. On the other hand, the velocity filter involves a condition on µ and +therefore generates anisotropic distributions, as those which are commonly observed far upstream, +see for examples Fig.s 10-12. In other words, near isotropy is assumed close upstream of the shock but +the escaping particles will have an anisotropic distribution function. We also notice that the effect + +Flat energy spectra +21 +Figure 13. Cartoon of the spherical sector around the magnetic field direction with radius given by the +particle velocity v. +of the velocity filter is easily quantified assuming an isotropic F; conversely, if some dependence on +µ would be given, the integration in Eq. (5) would be less straightforward but a velocity filter would +still be acting. With the isotropy assumption, we can insert the omnidirectional distribution function +f(r, p, t) in Eq.(5); using the pitch-angle cosine µ = cos α, defining a minimum pitch-angle cosine +µmin corresponding to αmax, and being dµ = − sin αdα, one can readily find that +U′ +p = 2πp2f(r, p, t)(1 − µmin) = 2πp2f(r, p, t) +� +1 − V sh +1 +sec θBn +v +� +. +(6) +Eq.(6) introduces a correction to the isotropic (in pitch-angle) flux of particles and the correcting +factor comes from the condition for particle escape from the shock µv > V sh +1 +sec θBn (Lario et al. +2019). Thus, the differential density is related to the flux J = U′ +p ∝ p2f(p) (Moraal 2013); according +to the DSA prediction, we have f(p) ∝ p−3r/(r−1), where r = n2/n1 is the compression ratio of the +shock. + +V +uv + αmax +Velocity space +OBn +x +Shock +-22 +Perri et al. +In order to understand the effect of such a velocity filter on the particle energy spectrum observed +upstream of shocks, we have computed the flux J over a broad energy range (from about 70 keV +to 40 MeV) using Eq.(6). First, we have fixed the plasma speed in the shock rest frame and have +varied θBn from quasi-parallel to quasi-perpendicular values. The left panel in Figure 14 shows a +flattening at low energies for quasi-perpendicular shocks. This means that the flux of those particles +is substantially reduced under those conditions and this promotes the observations of flat spectra +upstream of the shocks. Notice that the observed reduction of the flux involves energy channels +below about 400-600 keV, which are the channels where the upstream energetic particle fluxes are +frequently seen to be overlapped (see also Zimbardo et al. 2020, for the θBn dependence). +10 2 +10 3 +10 4 +Energy (keV) +10 39 +10 40 +10 41 +Flux (arb. units) +V1 +sh=800 km/s +Bn=15° +Bn=55° +Bn=75° +Bn=78° +10 2 +10 3 +10 4 +Energy (keV) +10 39 +10 40 +10 41 +Flux (arb. units) +Bn=75° +V1 +sh=100 km/s +V1 +sh=300 km/s +V1 +sh=500 km/s +V1 +sh=700 km/s +V1 +sh=900 km/s +Figure 14. +Left panel: Particle energy spectrum computed fixing the upstream plasma speed in the +shock reference frame at V sh +1 += 800 km/s and by varying the θBn angle. The velocity filter acts for quasi- +perpendicular shocks at low energies, causing a flattening in the spectrum. Right panel: Particle energy +spectrum computed fixing θBn = 75◦ and by varying V sh +1 . The velocity filter acts for high speed plasma +flows at low energies, causing a flattening in the spectrum. +We have further plotted the energy spectrum by fixing θBn = 75◦ and by varying the upstream +plasma speed in the shock frame of reference. Thus, the right panel in Figure 14 displays a reduction +of the low energy flux for high values of the plasma speed. +This is because the particle escape +condition tends to be marginally fulfilled for high V sh +1 +values. +Further, from the above condition it is possible to derive a limit value of θBn for particle upstream +propagation. Namely, since the cosine of the pitch-angle is a bounded quantity, i.e., −1 ≤ µ ≤ 1, the + +Flat energy spectra +23 +0 +500 +1000 +1500 +Ek (keV) +76 +78 +80 +82 +84 +86 +88 +90 +c +Bn +V1 +sh=100 km/s +V1 +sh=300 km/s +V1 +sh=500 km/s +V1 +sh=700 km/s +V1 +sh=900 km/s +Figure 15. θc +Bn as a function of particle energy for several values of the upstream plasma speed (in the +shock frame). +particle escape condition when |µ| = 1 becomes v = V sh +1 +sec θc +Bn, so that a critical value for the angle +between the magnetic field direction and the normal-to-the-shock direction can be easily found, +θc +Bn = arccos +�V sh +1 +v +� +. +(7) +Figure 15 displays the critical θc +Bn as a function of particle energy for several values of the plasma speed +in the shock frame: when θBn > θc +Bn within a given energy range the particle escape condition cannot +be satisfied and particles can be either confined close to the shock front or transmitted downstream. +As V sh +1 +increases, the escape condition is marginally satisfied, especially at low ion energies. +4. CONCLUSIONS +In this paper we have reported peculiar observations of flat particle energy spectra upstream of three +interplanetary shock waves, as detected by ACE and Wind. In all of these events the spacecraft +was well connected to the shock front, so that it has been possible to observe fluxes of particles +previously accelerated at the shock. The analysis performed on the magnetic field measurements, +by means of the computation of the PSD of the field components, has revealed that close upstream +the PSD at the time scales in resonance with energetic particles tends to be bumped, and then + +24 +Perri et al. +bends over when going towards smaller time scales. This suggests the presence of newly injected +magnetic field fluctuations, probably due to the presence of streaming energetic particles. This, of +course, favours particle scattering and diffusion near the shock front. Furthermore, Wind/3DP/SST +observations of energetic ion fluxes in different pitch-angle bins have highlighted isotropic (in pitch- +angle) distributions downstream of the shocks and isotropy is also recovered for lower energy particles +close upstream, where magnetic fluctuations are enhanced. Higher energy particles tend to be more +anisotropic close upstream and far upstream, suggesting that they can escape more easily from the +region in which lower energy particles are confined, thus promoting the formation of a flat energy +spectrum. This scenario also supports the numerical results by Ng et al. (2003). +We have explained these observations by elaborating the suggestion by Lario et al. (2019) that +a velocity filter, depending on both the particle speed and the pitch-angle, favours the upstream +propagation of faster particles, leading to the flattening of the energy spectrum and the overlapping +of the fluxes for moderate energies. We have derived the differential density of upstream propagating +particles and we have found that for a given θBn, particles with higher energies/velocities and large µ +tend to fulfill the velocity filter condition and can be easily detected far upstream, while lower energy +particles with small µ cannot match that condition and tend to be confined at the shock front. As a +consequence, their fluxes are depleted far upstream and this leads to flat spectra. Such a mechanism +would lead to particle anisotropy, and the degree of anisotropy should depend on both the velocity +filter and the scattering undergone by energetic particles. In addition, the energetic particle fluxes +have been computed both as a function of the shock normal angle θBn and as a function of the shock +speed. It is found that the effect of the velocity filter is the more relevant, the more perpendicular +the shock is. Such a flattening compares well with the flux observations of the three shocks here +analyzed, since the energy channels where a significant depletion of the flux is observed match the +ones in which overlapped ion fluxes are detected in the observations (i.e., from 70 keV up to ∼ 600 +keV). On the other hand, these flat spectra are observed several hours prior to the shock arrival, with +particles being accelerated by a shock having different properties than those of the shock crossing, +and in particular different shock normal angles θBn. In this regard, we can consider that for a CME + +Flat energy spectra +25 +propagating not too far from the Sun-Earth line, and because of the typical shape of the Parker +spiral magnetic field, spacecraft at L1 are connected with the westward part of the CME driven +shock, which can be considered to be more “perpendicular” than the shock near to Sun-Earth line. +This allows for an efficient active influence of the velocity filter effect. It is also interesting to note +that the spectral flattening is maintained for a large range of distances between the approaching +shock and Earth, something which suggests that the energetic particle transport properties might be +nearly independent of the particle energy. +In summary, here we have reported and interpreted the observational feature of flat energy spectra +upstream of some interplanetary shocks by using magnetic field, energetic particles, and plasma +measurements. +This joint analysis has allowed us to devise the following scenario: close to the +shock front magnetic field fluctuations are injected at the scales of energetic particles, this creates +an environment in which lower energy particles can be efficiently scattered. Indeed, the pitch-angle +distributions for those particles have been found to be almost isotropic. Higher energy particles can +easily go far upstream outward from the shock because of the velocity filter condition. For lower +energy particles, the escaping condition is not satisfied in case of fast, quasi-perpendicular shocks (as +it is almost the case for the three shocks analyzed). Their flux is then depleted and this promotes +the formation of a flat energy spectrum far upstream. In future works we foresee to support such a +scenario with Solar Orbiter and Parker Solar Probe observations closer to the Sun and to investigate +more deeply the role of the shock geometry on the velocity filter condition from in-situ measurements. +D.L. acknowledges support from NASA Living With a Star (LWS) programs NNH17ZDA001N- +LWS and NNH19ZDA001N-LWS, the Goddard Space Flight Center Internal Scientist Funding Model +(competitive work package) program, and the Heliophysics Innovation Fund (HIF) program. The +work was supported by the Geospace Environment Modeling (GEM) Focus Group ”Particle Heating +and Thermalization in Collisionless Shocks in the MMS Era” led by L.B. 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S., et al. 2021, +A&A, 656, A3 +Zimbardo, G., Prete, G., & Perri, S. 2020, +Frontiers in Astronomy and Space Sciences, 7, +16 + diff --git a/ptE5T4oBgHgl3EQfJA6J/content/tmp_files/load_file.txt b/ptE5T4oBgHgl3EQfJA6J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a84b3e72a376a2eb5c8ce144fb058d68b9a05d2b --- /dev/null +++ b/ptE5T4oBgHgl3EQfJA6J/content/tmp_files/load_file.txt @@ -0,0 +1,1006 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf,len=1005 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='05454v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='SR] 13 Jan 2023 Draft version January 16, 2023 Typeset using LATEX manuscript style in AASTeX62 Interpretation of flat energy spectra upstream of fast interplanetary shocks Silvia Perri,1 Giuseppe Prete,1 Gaetano Zimbardo,1 Domenico Trotta,2 Lynn B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Wilson III,3 David Lario,3 Sergio Servidio,1 Francesco Valentini,1 and Joe Giacalone4 1Dipartimento di Fisica, University of Calabria, Rende, Italy 2The Blackett Laboratory Imperial College London, London SW7 2AZ, UK 3Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA 4Lunar and Planetary Laboratory, University of Arizona, Tucson, USA (Received;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Revised;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Accepted) Submitted to ApJ ABSTRACT Interplanetary shocks are large-scale heliospheric structures often caused by eruptive phenomena at the Sun, and represent one of the main sources of energetic particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Several interplanetary shock crossings by spacecraft at 1 AU have revealed enhanced energetic-ion fluxes that extend far upstream of the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Surprisingly, in some shock events, ion fluxes with energies between 100 keV and about 2 MeV acquire similar values (which we refer to as “overlapped” fluxes), corresponding to flat energy spectra in that range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In contrast, closer to the shock, the fluxes are observed to depend on energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In this work, we analyze three interplanetary shock-related energetic particle events observed by the Advanced Composition Explorer spacecraft where flat ion energy spectra were observed upstream of the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' We interpret these observations via a Corresponding author: Silvia Perri silvia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='perri@fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='unical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='it 2 Perri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' velocity filter mechanism for particles in a given energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This reveals that low energy particles tend to be confined to the shock front and cannot easily propagate upstream, while high energy particles can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The velocity filter mechanism has been corroborated from observations of particle flux anisotropy by the Solid-State Telescope of Wind/3DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Keywords: shock waves — energetic particles — turbulence — heliosphere 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' INTRODUCTION Collisionless shock waves are observed to be one of the main sources of energetic particles and cosmic rays in astrophysical environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Efficient particle acceleration at shocks can result when particles remain confined near the shock, either by scattering in turbulent magnetic fields, or as a result of the geometry, and gain energy due to the compression in the plasma velocity at the shock front (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', Drury 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Shock acceleration is supported by in-situ measurements in the heliosphere, where energetic particle fluxes are clearly peaking at the time of shock passage (Giacalone 2012), and by remote observations of supernova remnants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', Bamba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Morlino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Reynoso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' On the other hand, the quantitative agreement with the predictions of diffusive shock acceleration (DSA), like spectral slope, acceleration times, and maximum energies, remains elusive (Lagage & Cesarsky 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Giacalone 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Vainio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' For instance, to reach an energy of ∼ 1015 eV (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', the so-called knee of the cosmic ray energy spectrum), a substantial amplification of the pre-existing (upstream) magnetic field by means of self-generated turbulence is required (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', Drury 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Blasi 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Amato 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In a similar way, the particle mean free paths in the heliosphere are estimated to be in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='01–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0 AU, but this is far too large to explain the rapid acceleration of particles by interplanetary (IP) shocks (Perri & Zimbardo 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Thus, turbulence enhancement, related to the backstreaming of energetic particles upstream of IP shocks, is required (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', Ng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Afanasiev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Similar electromagnetic fluctuation amplification is required to explain the efficient acceleration of solar energetic particles (SEPs) by CME-driven shocks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Flat energy spectra 3 In collisionless plasmas, because of particle reflection at supercritical shocks (Burgess & Scholer 2015), shock properties also depend on the (acute) angle, θBn, between the average upstream magnetic field, B, and the unit-normal vector to the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' If θBn < 45◦ the shock is termed quasi-parallel and reflected ions can efficiently propagate upstream, thereby forming the ion foreshock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Here, reflected particles form a beam in velocity space which can excite ion cyclotron, Alfv´en waves, or fast-magnetosonic, and whistlers, making the ion foreshock a very turbulent region, as often observed by spacecraft in the terrestrial ion foreshock (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', Schwartz & Burgess 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Perri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Wilson 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Karimabadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' These disturbances can give the required enhanced level of magnetic fluctuations to efficiently scatter both electrons (Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2016) and ions (Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2013), thereby trapping them near the shock for further acceleration to high energies according to the mechanism of DSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Conversely, if θBn > 45◦ the shock is termed quasi-perpendicular and most reflected ions re-enter the shock, contributing to enhanced level of fluctuations downstream, as usually observed (Greenstadt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In the absence of rapid scattering upstream, DSA can be slow, as suggested by some observations (Reynoso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2013) and numerical simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', Caprioli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Sundberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Trotta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Preisser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' However, it should be noticed that enhanced levels of magnetic fluctuations upstream of quasi-parallel IP shocks are not always observed (Blanco-Cano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' for instance, in a recent study Zimbardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (2020) found that the magnetic power at the scales corresponding to energetic particle resonance was approximately constant from upstream to downstream for three IP shocks with different values of θBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Therefore, the conditions under which self-generated turbulence is actually found need to be better understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Observations of fluxes of supra-thermal particles can be highly influenced by the upstream trans- port conditions, which are established by the level of magnetic field fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' For example, Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (2022) investigated the formation of an anisotropic field-aligned beam of protons up- stream of an oblique shock with energies ≤ 30 keV together with a population of protons at higher energy propagating at small pitch-angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The unusually long duration (and therefore spatial extent) 4 Perri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' of the field-aligned beam was interpreted as due to the absence of magnetic field fluctuations over a large distance upstream of the shock wave — a scenario where efficient scattering is not favoured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Another fundamental phenomenon is the formation of a precursor of energetic particles in the upstream: this precursor is distinct from the ion foreshock (which is formed mostly by reflected thermal particles), or from the fast/magnetosonic whistler wave precursor (Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2017), and is due to those energetic particles satisfying vµ > V sh 1 sec(θBn), (1) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', those particles that have velocity parallel to the local magnetic field (being v the particle speed and µ the cosine of the pitch angle) larger than the speed of the intersection point of a field line with the shock front along the direction of the shock front (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', le Roux & Webb 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Notice that V sh 1 represents the upstream solar wind speed in the frame of reference of the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This velocity filter can lead to a spectral flattening upstream of the shock, since only higher energy particles, can easily propagate back upstream, while lower energy particles tend to be confined within the shock region, reducing their fluxes far upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Nonetheless, the presence of turbulent fluctuations causes pitch angle scattering and a meandering of magnetic field lines, which can influence the velocity filter effect on particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Such spectral flattening has been predicted in earlier works by Ng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (2003), as a consequence of the differential growth of Alfv´en waves upstream of the shock, amplified by high energy ions streaming away from the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This creates a sort of delay in the lower energy particle transport upstream, favouring the formation of a flat energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' That model, however, suffers the limitation of predicting very short intervals for spectral flattening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' More recently, overlapping particle fluxes have been modelled by Prinsloo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (2019), and explained as a balance between shock acceleration and adiabatic cooling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' however, at 1 AU they obtained a very narrow (in energy) spectral flattening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Thus, mismatches with observations are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In this paper, we analyze three IP shock crossings and associated energetic ion intensities observed by the Advanced Composition Explorer (ACE) and the Wind spacecraft on 2005 May 15, on 2012 Flat energy spectra 5 July 14, and on 2003 November 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' These events were chosen because continuous data measurements from various instruments are available, no particular IP disturbances affect energetic-ion fluxes, and the IP shocks have similar compression ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' All these events are characterized by energetic particle profiles that are overlapping far upstream and up to about 1-3 hours before the shock passage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', the upstream energy spectrum is flat for a broad range of particle energies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Closer to the shocks, the fluxes at each energy separate and reach values more typical of those predicted by DSA theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This peculiar feature of upstream overlapping particle fluxes was first reported by Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (2018), looking at particles upstream of IP shocks using ACE measurements at energies between approximately 50 keV and 4 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' They noted that the region of spectral flattening for these energetic protons happens prior of the shock arrival, as the energy spectrum steepens close to the shock, and the observed spectral shapes are closely related to the particle transport processes from the evolving shock to the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Here, we analyze energetic particle fluxes and magnetic fluctuations close to and far upstream of the shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' We interpret observations of flat upstream energy spectra in terms of the velocity filter mechanism described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Importantly, we derive how particle fluxes are modified from such a condition for different values of the angle θBn and of the upstream plasma speed as measured in the shock frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The paper is organized as follows: Section 2 contains the data description and the data analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Section 3 presents the interpretation of the data according to the velocity filter condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' in Section 4 concluding remarks are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' DATA COLLECTION AND ANALYSIS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Shock crossings We first analyze the shock event of 2005 May 15, which exhibits overlapping energetic particle intensity profiles over a large distance upstream of the shock crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Energetic particle fluxes separate at about 50 min before the shock crossing and downstream of the shock, thus resulting in a steeper energy spectrum (Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The solar origin of this relatively strong event 6 Perri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' was associated with a halo, fast (plane-of-sky speed ∼1689 km s−1) coronal mass ejection (CME) observed at 17:12 UT on 2005 May 13, as reported in the Coordinated Data Analysis Workshops (CDAW) Data Center SOHO LASCO CME catalog1 and temporary associated with a M8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0 GOES class X-ray flare from the NOAA active region 10759 at N12E12 with onset at 16:13 UT on 2005 May 13 (Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The transit time for the shock to travel from the Sun to 1 AU was about 2038 minutes corresponding to an average transit speed of ∼1223 km s−1 (see Table 1 in Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Using data from GOES-11, Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (2018) show that energetic particles peak at the shock even in the high energy channel of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 MeV, confirming that it is a strong particle accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Figure 1 shows an overview of the 2005 May 15 event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' From top to bottom we report (a) the 12 s resolution particles fluxes detected by the Electron, Proton, and Alpha Monitor (EPAM) instrument (Gold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 1998) onboard ACE from the LEMS120 sensor in the energy range 68 keV-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='90 MeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (b) the magnetic field components in Radial-Tangential-Normal (RTN) coordinates as sampled by the MAG instrument (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 1998) onboard ACE at a time cadence of 60 s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (c) and (d) the plasma proton density and the solar wind bulk speed, respectively, as detected by the Solar Wind Electron, Proton, and Alpha Monitor (SWEPAM) experiment (McComas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 1998) onboard ACE (black line) at 64 s resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Because of a data gap in the ACE time series around the shock crossing, we add the proton solar wind density measured by the Solar Wind Experiment (SWE) (Ogilvie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 1995) on board Wind at 92 s resolution (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The panel (e) shows the angle ψ between the mean field and the radial direction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', Sonnerup 1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The latter gives us insights into the magnetic connectivity of the observer with the shock surface (the horizontal dashed red line indicates a 90◦ angle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The angle ψ has been computed using 5-minutes running average windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The vertical black dashed line indicates the shock crossing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' We display a time window ex- tending towards ≃ 24 hours before the ACE shock crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' For a period of about 50 min up to 16 hours prior to the shock crossing (indicated by a shaded region in Figure 1) the particle fluxes at the displayed energies overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' For earlier times (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' more than 17 hours prior to the shock crossing), 1 available at cdaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='gov/CME list/ Flat energy spectra 7 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Overview of the shock crossing by the ACE spacecraft on 2005 May 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' From top to bottom: particle fluxes from the EPAM-LEMS120 instrument in the energy range 68-1890 keV measured at 12 s cadence, the 60 s resolution magnetic field components in the Radial-Tangential-Normal (RTN) reference frame along with the magnetic field magnitude, the proton density and the solar wind bulk speed both from ACE (at 64 s resolution) and Wind (at 92 s resolution), and the angle between the radial direction and the mean field direction computed over a running window of 5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The vertical dashed line indicates the shock position, the yellow shaded area shows the region where energetic particle fluxes overlap indicating a period with a relatively flat energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' particle fluxes at different energies are well separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' It can be seen in panel (b) that within the region upstream where the energetic particle fluxes overlap, the magnetic field tends to be approxi- mately radial (see also panel (e) in Figure 1 where ψ < 30◦ for most of the time interval indicated by the shaded region), while it largely deviates from the radial direction close to the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This 68 - 115keV (a) 115- 195keV 105 - 195- 321keV 321 - 580keV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='580 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='06 MeV 104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='06 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='90MeV 103 2 (cm 102 101 60 B (b) 40 - BR BT (lu) 20 BN 0 B 20 40 30 ACE (c) 3 WIND 20 - cm 10 v(km/s) (d) 1000 750 500 150 (e) 100 50 0 1250 1000 750 500 250 250 500 t(min)8 Perri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Main parameters for the three shock crossings analyzed, as deduced from ACE and Wind measure- ments (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The θBn estimates using both the minimum variance (MVA) and the magnetic coplanarity (MC) methods are reported, the Alfv´enic and the sonic mach numbers, the compression ratio of the shock, the plasma beta, and the shock speed in the satellite frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Associated errors are also reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' date time (ACE) θBn(◦)-MVA θBn(◦)-MC MA MS r βplasma V s/c sh (km/s) 15/05/05 02:05 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='7 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='9 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='3 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='57 926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='9 14/07/12 17:26 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 ± 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='9 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='3 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='7 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='99 710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 04/11/03 05:59 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='9 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='01 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='7 suggests that for the entire period when the upstream flat spectrum was observed, the spacecraft was well connected to the shock front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' However, the fluxes remain overlapped even closer to the shock when the field starts changing direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Indeed, fluxes separate only very close to the shock (−40 min), while the magnetic field deviates from the radial direction at about −85 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In the downstream region, ∼3 hours after the shock crossing, a sudden decrease of about two orders of magnitude in the particle fluxes is observed, which is due to the arrival of the magnetic obstacle of the CME that generated the SEP event (Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Perri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Fundamental shock parameters have been computed and are shown in Table 1, although single- spacecraft shock parameter estimation is a process that is known to be subject to many sources of uncertainty (Paschmann & Daly 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Koval & Szabo 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' For θBn, we used both the Minimum Variance Analysis (MVA) over a 30 minutes time interval before the shock crossing, similarly to what has been done in van Nes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (1984), although this technique can be highly unstable because of rapid variations of the magnetic field, and the magnetic coplanarity (MC) method (Colburn & Sonett 1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The MC technique with a systematic variation of upstream and downstream averaging win- dows from ∼ 2 to 30 minutes (as done in Trotta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2022) yields a large spread of values, indicating high levels of upstream/downstream disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The Alfv´en and the sonic Mach numbers, the com- pression ratio of the shock, the plasma beta β = nkBT/(B2/2µ0) (being kB the Boltzmann constant and µ0 the vacuum magnetic permeability), and the shock speed in the spacecraft frame of reference (Perri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2015) are also reported (values averaged over a time window of 30 min).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' It should be also Flat energy spectra 9 noted that the parameter estimations assume a stationary, planar, and infinite shock, a hypothesis that is often not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Actually, the shock structure can be further complicated by the interaction with pre-existing structures that modify the local geometry of the shock front (as recently addressed for IP shocks (Blanco-Cano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Giacalone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2021) as well as for the Earth’s bow shock (Trotta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Moreover, the shocks analysed here are supercritical (see Table 1), indicating that they may be affected by many self-induced spatial and temporal irregularities happening at a variety of scales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Kajdiˇc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2019, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' From Table 1, the 2005 May 15 event is a quasi-perpendicular shock, with sonic and Alfv´enic Mach numbers Ms ∼ MA ∼ 16, which is able to accelerate ions to tens of MeV at 1 AU (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', at the spacecraft position).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The second analyzed event is a quasi-perpendicular shock at 17:26 UT on 2012 July 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The solar origin of this event was associated with a fast (plane-of-sky speed 885 km s−1) halo CME observed at 16:48 UT on 2012 July 12 as reported by the CDAW SOHO LASCO CME Catalog and temporally associated with a X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 flare from NOAA Active Region 11520 at S15W01 at 15:37 UT (Wijsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The transit time of the shock to travel from the Sun to 1 AU is about 2989 minutes, corresponding to an average transit speed of ∼ 834 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Figure 2 shows data for the event on 2012 July 14 (with the same format as Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This event displayed a flat particle energy spectrum from more than 12 hrs to about 2 hrs prior to the shock passage (shaded region in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In this case the magnetic field was almost radial (ψ < 50◦) but gradually started to change direction about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 hrs prior to the shock crossing, while the fluxes remained overlapped (at least the ones in the lower energy channels, from 68 keV to 580 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The third shock crossing analyzed occurred at 05:59 UT on 2003 November 4 and corresponds to a quasi-parallel shock (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The solar origin of this event was associated with a fast (plane-of-sky speed 2598 km s−1) halo CME at 17:30 UT on 2003 November 2 as reported by the CDAW SOHO LASCO CME Catalog and temporally associated with a X8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='3 flare at 17:03 UT from NOAA Active Region 10486 at S14W56 (Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The transit time of the shock to travel from the Sun to 1 AU is about 2216 minutes, corresponding to an average transit speed of ∼1125 10 Perri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Same as Figure 1 but for the shock crossing by the ACE spacecraft on 2012 July 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Figure 3 shows, with the same format as Figure 1, data for the event on 2003 November 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Energetic ion fluxes overlapping for several hours before the shock passage at the spacecraft position within the energy channels between 115 keV and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='06 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The ion fluxes prior to the shaded region in Figure 3 were already elevated due to prior SEP events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' We note that the fluxes in the lowest energy channel (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', 68–115 keV) and in the highest energy channel (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='06–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='90 MeV) overlap partly within the shaded box in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The magnetic field is oriented sunward in this period, oscillating around ψ ∼ 135◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 1 68 - 115keV (a) 115 - 195keV s 105 195 - 321keV 321 - 580keV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='580 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='06 MeV 104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='06 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='90 MeV 103 102 101 [BI (b) 20 BR 10 Bt BN n 0 B 10 20 30 ACE () WIND 20 (cm 10 0 ACE V(km/s) 600 400 = 90° 150 100 B 50 0 1250 1000 750 500 250 0 250 500 t(min)Flat energy spectra 11 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Same as Figure 1 but for the shock crossing by the ACE spacecraft on 2003 November 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Magnetic field turbulence In order to characterize the environment through which the shocks propagate, we have studied the properties of magnetic field turbulence within both (i) the upstream region where flat energy spectra are detected, and (ii) close to the shock where power-law energy spectra are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This has been done by computing the power spectral density (PSD) of the magnetic field fluctuations along each direction in the RTN reference frame, using the Fast Fourier Transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 1 68 - 115keV (a) 115 - 195keV 195- 321keV 105 321 - 580keV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='580 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='06 MeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='06 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='90MeV 104 103 30 B (b) 20 BR BT (lu) 10 B 10 20 38 ACE (c) 3) 20 WIND (cm 10 0 800 V(km/s) 600 Ww 400 150 (e) 100 B 50 =90° 0 1250 1000 750 500 250 0 250 500 t(min)12 Perri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' PSD of the magnetic field fluctuations for the 2005 May 15 event, computed in the region far upstream where the fluxes of energetic particles overlap (black line) and in the region close to the shock front upstream (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The vertical dashed lines indicates the frequencies corresponding to the Larmor radius of the energetic particles (see Figure legend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The best power-law fits are also shown by the dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Figure 4 displays the power spectrum of the magnetic field in the two regions described above for the event on 2005 May 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In the region where the energetic particle fluxes overlap, from -800 min to -100 min from the shock, the spectrum follows a Kolmogorov-like power-law, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', PSD(f) ∝ f −5/3, where f is the frequency, indicating ambient solar wind turbulence (Bruno & Carbone 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Pitˇna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' By contrast, in the close upstream region, from -80 min to -10 min, we find PSD(f) ∝ f −1 for the 2005 May 15 event, while in the other two crossings (where we have used the same time interval for PSDs computation) the PSD is bumped at low frequencies (see Figures 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In Figure 4 the best power-law fits are displayed with the black and the red dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The values 104 Far upstream f -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='51±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='01 103 Close upstream f -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='01 102, PSD(nT2/Hz) 101 100 68- 115keV 10-1 115- 195keV 195-321keV 321-580keV 10-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='580 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='06 MeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='06 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='90 MeV 10-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 10-5 10-4 10-3 10-2 f(Hz)Flat energy spectra 13 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Same as Figure 4 but for the shock on 2012 July 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' found for the slopes, along with their associated errors, are also reported in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Notice that we have computed the power-law fits within a range of frequencies corresponding to the Larmor radius of the energetic particles, namely, fE = Vup/(2πρE) where Vup is the proton bulk speed in the relevant upstream region in the spacecraft frame and ρE is the Larmor radius of protons with energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Such frequencies are indicated by the vertical dashed lines in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The presence of a broad PSD(f) ∝ f −1 range suggests that fluctuations are freshly-injected in that region and that turbulence has not become fully-developed yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In addition, the presence of a bumped spectrum close upstream for the 2012 July 14 and 2003 November 4 crossings can be related to fluctuations driven by the shock-reflected energetic ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 104 Far upstream f -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='36 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='01 Close upstream 103 f -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='03 102 PSD(nT2/Hz) 101 100 68 - 115keV 115 - 195keV 195 -321keV 10-1 321 - 580keV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='580 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='06 MeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='06 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='90 MeV 10-2 10-5 10-4 10-3 10-2 f(Hz)14 Perri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Same as Figure 4 but for the shock on 2003 November 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Indeed, the PSD close upstream decays as a f −1 power-law with a bend over towards frequencies higher than the ones corresponding to energetic particle scales (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Notice that in these two events, the power in the magnetic field fluctuations at low frequencies is smaller (or almost comparable) in the close upstream region than far upstream (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 5 and 6 and the discussion below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Further, we have explored how turbulent fluctuations are distributed both in frequency and as a function of the distance from the shock via a wavelet analysis of the magnetic field vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The square of the wavelet coefficients of the magnetic field components have been computed as 104 Far upstream f -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='34 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='01 Close upstream 103 f -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='03 PSD (nT2/Hz) 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 101 68 - 115keV M1 115 - 195keV 195-321keV 100 321 - 580keV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='580 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='06 MeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='06 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='90MeV 10-1 10-5 10-4 10-3 10-2 f(Hz)Flat energy spectra 15 (Alexandrova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2008) |WB(τ, t)|2 = � i |Wi(τ, t)|2, (2) where τ = 1/f represents a time scale, and the sum is over the magnetic field components, i = R, T, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Wi(τ, t) represent the Morlet wavelet coefficients computed over different τ and time t (Torrence & Compo 1998), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', Wi(τ, t) = �N j=1 Bi(tj)ψ∗ [(tj − t) /τ], with ψ∗ being the conjugate of the wavelet function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This allows us to assess the magnetic energy content in frequency and in time and localize, within the time series, the regions with high magnetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (2) has been reported for the 2005 May 15 shock crossing in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' It is indeed evident how the magnetic field fluctuation power increases close to the shock front, over a broad frequency range, which includes also the frequencies corresponding to the Larmor radius of the energetic particles, indicated in Figure 7 by the horizontal white dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The yellow box surrounds the flat energy spectrum region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Thus, the magnetic energy stored in the fluctuations tends to increase very close to the shock (indicated by the vertical dashed line) and is found to be high downstream of the shock within the turbulent sheath region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Here fluctuations are highly compressed and enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The increase of magnetic power close to the shock reconciles with the observation of an extended ∼ f −1 range in the PSD computed in the close upstream region, which is also in agreement with recent observations of turbulence close to IP shocks (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The local increase of the magnetic field power and the detection of a bumped PSD close upstream might be ascribed to a self-generated, freshly-injected turbulence, due to the presence of the energetic particle fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In addition, from the scalograms in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 8 and 9, large amplitude pre-existing magnetic fluctuations can be observed over a broad range of frequencies and times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This evidence is in agreement with the comparable power found in the spectra upstream and close upstream in the 2012 July 14 and in the 2003 November 4 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Such larger amplitude fluctuations far upstream can favour particle scattering, especially at lower frequencies where more power is stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This can explain the tendency of high energy ion fluxes to separate from the fluxes in the other energy channels in the 2012 July 14 and in the 2003 November 4 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 16 Perri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Energetic Particle Anisotropy At this point is crucial to investigate the propagation of energetic particles with respect to the magnetic field direction using the Solid State Telescope (SST) of Wind 3DP (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 1995), in order to better characterize their motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Ion fluxes are organized in 9 energy channels with average energies of 76 keV, 130 keV, 200 keV, 336 keV, 554 keV, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0 MeV, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0 MeV, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0 MeV, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' They are also binned in 8 pitch-angle values with respect to the local magnetic field direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' such fluxes, normalized to their maximum intensity, are displayed in nine panels in Figure 10 for the 2005 May 15 event, as a function of the pitch-angle cosine µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Fluxes are in the solar wind frame, in order to get rid of any anisotropic feature due to the Compton-Getting effect (Compton & Getting 1935;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Forman 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The three sets of symbols in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 10 refer to the mean fluxes calculated over three different regions around the shock: far upstream where flat spectra are detected (from -750 min to 200 min upstream of the shock), close upstream of the shock (from -80 min to -10 min), and from 10 min up to 70 min downstream of the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In all energy channels there is a tendency to isotropize the distributions of the fluxes in the downstream region, caused by the presence of the augmented magnetic field fluctuations which efficiently scatter energetic particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Far upstream, within the region of overlapped fluxes, we observed for the lowest energy channel that the flux is higher at large pitch-angles, namely particles move sunward towards the shock, while higher energy particles travel mostly anti-sunward at small pitch-angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Here the amplitude of magnetic field fluctuation decreases but the field is almost radial and a good connection between the shock and the spacecraft exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This permits to detect in the far upstream region those particles that have been isotropized close upstream, considering that parallel diffusion is larger than perpendicular diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This anisotropy in pitch angle is somewhat reduced for all the energy channels in the close upstream region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This is again due to the amplified magnetic fluctuations, since ions can easily be scattered in all directions by interacting with turbulent fluctuations (Giacalone & Jokipii 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Trotta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Here, lower energy particles tend to be isotropized in µ, while there is still anisotropy in favour of field-aligned particles at higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' A similar behaviour has been observed for the energetic particle fluxes associated to the 2012 July 14 event (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Flat energy spectra 17 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Power spectral density computed using the wavelet coefficients over the three magnetic field components (see text for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The yellow box delimits the region upstream where a flat energy spectrum has been detected, while the vertical dashed line remarks the shock crossing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The horizontal dashed lines indicates the frequency range corresponding to the Larmor radius of energetic particles from 67 keV to 2 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Typical cascade patches (Greco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2016) can be recognized over the entire upstream region, with an intensity increase close to the shock front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Just behind the shock the CME sheath region is characterized by a very high level of turbulence over a broad range of frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Same as Figure 7 but for the shock crossing by the ACE spacecraft on 2012 July 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' On the other hand, the mean ion flux values displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 12 for the 2003 November 4 shock exhibit a high degree of isotropy in the three regions for almost all the energy channels, except for the 2012/07/14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='36608e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4643e-05 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='000378572 20 (ZH) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='00151429 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='00605715 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0242286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0969145 1250 1000 750 500 250 0 250 500 t (min)2005/05/15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='38434e-05 25 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='53736e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='000381495 20 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='00152598 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='00610391 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0244157 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0976626 1250 1000 750 500 250 0 250 t (min)18 Perri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Same as Figure 7 but for the shock crossing by the ACE spacecraft on 2003 November 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 normalized intensity 76 keV 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 130 keV far upstream close upstream downstream 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 200 keV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 normalized intensity 336 keV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 554 keV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0 MeV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 normalized intensity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0 MeV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 2005-05-15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 MeV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 MeV Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Mean ion fluxes, normalized to the maximum intensity in each region (see text for further details), as measured from the Wind/3DP/SST instrument in the solar wind frame within each pitch-angle bin during the 2005 May 15 event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Each panel refers to a given energy channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The black circles indicate the ion fluxes measured far upstream of the shock wave within the shaded region indicated in Figure 1, the red diamonds refer to the close upstream region, where ion fluxes start separating, and the blue stars to the downstream sheath region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 76 keV channel, where particles are almost anti-aligned to the mean field (which is sunward), going 2003/11/04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='51109e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='000100444 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='000401774 20 (ZH) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0016071 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='00642839 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0257135 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='102854 1250 1000 750 500 250 0 250 500 t (min)Flat energy spectra 19 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 normalized intensity 76 keV 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 130 keV far upstream close upstream downstream 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 200 keV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 normalized intensity 336 keV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 554 keV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0 MeV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 normalized intensity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0 MeV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 2012-07-14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 MeV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 MeV Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Same as Figure 10 but for the shock crossing by the ACE spacecraft on 2012 July 14th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' along the anti-sunward direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Notice from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 3 that the flux of energetic ions in this channel is not completely overlapped to the other fluxes in the shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Observing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 9, the power stored in the magnetic field fluctuations is a bit higher than in the other two events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This will induce more particle scattering and therefore more isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' DISCUSSION ON THE VELOCITY FILTER CONDITION Following the interpretation proposed by le Roux & Webb (2012) (see also Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2019), we have derived the energetic particle flux by imposing the velocity filter condition, namely low energy particles cannot easily escape far upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Since the number of particles within a volume in phase space is defined as (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', Moraal 2013) dN = F(r, p, t)d3rd3p = F(r, p, t)d3rp2dpdΩ, (3) where F(r, p, t) is the particle distribution function and dΩ = sin αdαdφ is the element of solid angle in momentum space, we can define a differential density in momentum, namely the number of particles in a given volume d3r and with momentum within p and p + dp as 20 Perri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 normalized intensity 76 keV 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 130 keV far upstream close upstream downstream 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 200 keV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 normalized intensity 336 keV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 554 keV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0 MeV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 normalized intensity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='0 MeV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 2003-11-04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 MeV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='8 MeV Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Same as Figure 10 but for the shock crossing by the ACE spacecraft on 2003 November 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Up = p2 � Ω F(r, p, t)dΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (4) This implies that the number of particles becomes dN = Upd3rdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' We align the polar coordinate axis with the upstream magnetic field, so that α is the pitch angle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' considering that only particles with a sufficiently large parallel velocity can escape from the moving shock, we find that α can vary over a limited range of values, namely the particle velocity belongs to a limited spherical sector in phase space (see the cartoon in Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Then, we can calculate the differential density of upstream propagating particles as U′ p = 2πp2 � αmax 0 F(r, p, t) sin αdα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (5) Here, we assume that F(r, p, t) is isotropic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', that the particle distribution function depends on the momentum magnitude p only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' On the other hand, the velocity filter involves a condition on µ and therefore generates anisotropic distributions, as those which are commonly observed far upstream, see for examples Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='s 10-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In other words, near isotropy is assumed close upstream of the shock but the escaping particles will have an anisotropic distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' We also notice that the effect Flat energy spectra 21 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Cartoon of the spherical sector around the magnetic field direction with radius given by the particle velocity v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' of the velocity filter is easily quantified assuming an isotropic F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' conversely, if some dependence on µ would be given, the integration in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (5) would be less straightforward but a velocity filter would still be acting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' With the isotropy assumption, we can insert the omnidirectional distribution function f(r, p, t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' using the pitch-angle cosine µ = cos α, defining a minimum pitch-angle cosine µmin corresponding to αmax, and being dµ = − sin αdα, one can readily find that U′ p = 2πp2f(r, p, t)(1 − µmin) = 2πp2f(r, p, t) � 1 − V sh 1 sec θBn v � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (6) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (6) introduces a correction to the isotropic (in pitch-angle) flux of particles and the correcting factor comes from the condition for particle escape from the shock µv > V sh 1 sec θBn (Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Thus, the differential density is related to the flux J = U′ p ∝ p2f(p) (Moraal 2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' according to the DSA prediction, we have f(p) ∝ p−3r/(r−1), where r = n2/n1 is the compression ratio of the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' V uv αmax Velocity space OBn x Shock 22 Perri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In order to understand the effect of such a velocity filter on the particle energy spectrum observed upstream of shocks, we have computed the flux J over a broad energy range (from about 70 keV to 40 MeV) using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' First, we have fixed the plasma speed in the shock rest frame and have varied θBn from quasi-parallel to quasi-perpendicular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The left panel in Figure 14 shows a flattening at low energies for quasi-perpendicular shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This means that the flux of those particles is substantially reduced under those conditions and this promotes the observations of flat spectra upstream of the shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Notice that the observed reduction of the flux involves energy channels below about 400-600 keV, which are the channels where the upstream energetic particle fluxes are frequently seen to be overlapped (see also Zimbardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 2020, for the θBn dependence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 10 2 10 3 10 4 Energy (keV) 10 39 10 40 10 41 Flux (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' units) V1 sh=800 km/s Bn=15° Bn=55° Bn=75° Bn=78° 10 2 10 3 10 4 Energy (keV) 10 39 10 40 10 41 Flux (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' units) Bn=75° V1 sh=100 km/s V1 sh=300 km/s V1 sh=500 km/s V1 sh=700 km/s V1 sh=900 km/s Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Left panel: Particle energy spectrum computed fixing the upstream plasma speed in the shock reference frame at V sh 1 = 800 km/s and by varying the θBn angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The velocity filter acts for quasi- perpendicular shocks at low energies, causing a flattening in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Right panel: Particle energy spectrum computed fixing θBn = 75◦ and by varying V sh 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The velocity filter acts for high speed plasma flows at low energies, causing a flattening in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' We have further plotted the energy spectrum by fixing θBn = 75◦ and by varying the upstream plasma speed in the shock frame of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Thus, the right panel in Figure 14 displays a reduction of the low energy flux for high values of the plasma speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This is because the particle escape condition tends to be marginally fulfilled for high V sh 1 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Further, from the above condition it is possible to derive a limit value of θBn for particle upstream propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Namely, since the cosine of the pitch-angle is a bounded quantity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', −1 ≤ µ ≤ 1, the Flat energy spectra 23 0 500 1000 1500 Ek (keV) 76 78 80 82 84 86 88 90 c Bn V1 sh=100 km/s V1 sh=300 km/s V1 sh=500 km/s V1 sh=700 km/s V1 sh=900 km/s Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' θc Bn as a function of particle energy for several values of the upstream plasma speed (in the shock frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' particle escape condition when |µ| = 1 becomes v = V sh 1 sec θc Bn, so that a critical value for the angle between the magnetic field direction and the normal-to-the-shock direction can be easily found, θc Bn = arccos �V sh 1 v � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (7) Figure 15 displays the critical θc Bn as a function of particle energy for several values of the plasma speed in the shock frame: when θBn > θc Bn within a given energy range the particle escape condition cannot be satisfied and particles can be either confined close to the shock front or transmitted downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' As V sh 1 increases, the escape condition is marginally satisfied, especially at low ion energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' CONCLUSIONS In this paper we have reported peculiar observations of flat particle energy spectra upstream of three interplanetary shock waves, as detected by ACE and Wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In all of these events the spacecraft was well connected to the shock front, so that it has been possible to observe fluxes of particles previously accelerated at the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The analysis performed on the magnetic field measurements, by means of the computation of the PSD of the field components, has revealed that close upstream the PSD at the time scales in resonance with energetic particles tends to be bumped, and then 24 Perri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' bends over when going towards smaller time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This suggests the presence of newly injected magnetic field fluctuations, probably due to the presence of streaming energetic particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This, of course, favours particle scattering and diffusion near the shock front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Furthermore, Wind/3DP/SST observations of energetic ion fluxes in different pitch-angle bins have highlighted isotropic (in pitch- angle) distributions downstream of the shocks and isotropy is also recovered for lower energy particles close upstream, where magnetic fluctuations are enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Higher energy particles tend to be more anisotropic close upstream and far upstream, suggesting that they can escape more easily from the region in which lower energy particles are confined, thus promoting the formation of a flat energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This scenario also supports the numerical results by Ng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' We have explained these observations by elaborating the suggestion by Lario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' (2019) that a velocity filter, depending on both the particle speed and the pitch-angle, favours the upstream propagation of faster particles, leading to the flattening of the energy spectrum and the overlapping of the fluxes for moderate energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' We have derived the differential density of upstream propagating particles and we have found that for a given θBn, particles with higher energies/velocities and large µ tend to fulfill the velocity filter condition and can be easily detected far upstream, while lower energy particles with small µ cannot match that condition and tend to be confined at the shock front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' As a consequence, their fluxes are depleted far upstream and this leads to flat spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Such a mechanism would lead to particle anisotropy, and the degree of anisotropy should depend on both the velocity filter and the scattering undergone by energetic particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In addition, the energetic particle fluxes have been computed both as a function of the shock normal angle θBn and as a function of the shock speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' It is found that the effect of the velocity filter is the more relevant, the more perpendicular the shock is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Such a flattening compares well with the flux observations of the three shocks here analyzed, since the energy channels where a significant depletion of the flux is observed match the ones in which overlapped ion fluxes are detected in the observations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=', from 70 keV up to ∼ 600 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' On the other hand, these flat spectra are observed several hours prior to the shock arrival, with particles being accelerated by a shock having different properties than those of the shock crossing, and in particular different shock normal angles θBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In this regard, we can consider that for a CME Flat energy spectra 25 propagating not too far from the Sun-Earth line, and because of the typical shape of the Parker spiral magnetic field, spacecraft at L1 are connected with the westward part of the CME driven shock, which can be considered to be more “perpendicular” than the shock near to Sun-Earth line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This allows for an efficient active influence of the velocity filter effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' It is also interesting to note that the spectral flattening is maintained for a large range of distances between the approaching shock and Earth, something which suggests that the energetic particle transport properties might be nearly independent of the particle energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In summary, here we have reported and interpreted the observational feature of flat energy spectra upstream of some interplanetary shocks by using magnetic field, energetic particles, and plasma measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' This joint analysis has allowed us to devise the following scenario: close to the shock front magnetic field fluctuations are injected at the scales of energetic particles, this creates an environment in which lower energy particles can be efficiently scattered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Indeed, the pitch-angle distributions for those particles have been found to be almost isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Higher energy particles can easily go far upstream outward from the shock because of the velocity filter condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' For lower energy particles, the escaping condition is not satisfied in case of fast, quasi-perpendicular shocks (as it is almost the case for the three shocks analyzed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' Their flux is then depleted and this promotes the formation of a flat energy spectrum far upstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' In future works we foresee to support such a scenario with Solar Orbiter and Parker Solar Probe observations closer to the Sun and to investigate more deeply the role of the shock geometry on the velocity filter condition from in-situ measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' acknowledges support from NASA Living With a Star (LWS) programs NNH17ZDA001N- LWS and NNH19ZDA001N-LWS, the Goddard Space Flight Center Internal Scientist Funding Model (competitive work package) program, and the Heliophysics Innovation Fund (HIF) program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content=' The work was supported by the Geospace Environment Modeling (GEM) Focus Group ”Particle Heating and Thermalization in Collisionless Shocks in the MMS Era” led by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE5T4oBgHgl3EQfJA6J/content/2301.05454v1.pdf'} +page_content='B.' metadata={'source': 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100644 index 0000000000000000000000000000000000000000..d192c3b80ccaabf21523b73ce2bb029a1555db0d --- /dev/null +++ b/qdE5T4oBgHgl3EQfJQ70/content/tmp_files/2301.05457v1.pdf.txt @@ -0,0 +1,1830 @@ +STFC +Astronomy Advisory Panel +Roadmap 2022 +Stephen Serjeant (Open University, chair), James Bolton +(University of Nottingham), Poshak Gandhi (University of +Southampton), Ben Stappers (University of Manchester), Paolo +Mazzali (Liverpool John Moores University), AprajitaVerma +(University of Oxford), Noelia E. D. Noël (University of Surrey) + +STFC +Astronomy Advisory Panel +Roadmap 2022 +Astronomy Advisory Panel: Stephen Serjeant (Open University, +chair), James Bolton (University of Nottingham), Poshak Gandhi +(University of Southampton), Ben Stappers (University of +Manchester), Paolo Mazzali (Liverpool John Moores University), +Aprajita Verma (University of Oxford), Noelia E. D. Noël +(University of Surrey) +The cover image shows the first image of Sgr A*, the supermassive black hole at the centre of our +Galaxy , , , , , , . UK astronomers are part of the Event Horizon Telescope team that made this image. +1 2 3 4 5 6 7 +The observatories involved include the James Clerk Maxwell Telescope, with UK access funded by +PPRP and UK university contributions, and ALMA, where UK access is through ESO. UK ALMA +users in particular are also supported by the STFC-funded UK ALMA Regional Centre. Image credit: +Event Horizon Telescope collaboration. +7 The Event Horizon Telescope Collaboration. 5/12/2022. “First Sagittarius A* Event Horizon Telescope +Results. VI. Testing the Black Hole Metric .” The Astrophysical Journal Letters, 930, L17. +https://doi.org/10.3847/2041-8213/ac6756 +6 The Event Horizon Telescope Collaboration. 5/12/2022. “First Sagittarius A* Event Horizon Telescope +Results. V. Testing Astrophysical Models of the Galactic Center Black Hole.” The Astrophysical Journal Letters, +930, L16. https://doi.org/10.3847/2041-8213/ac6672 +5 The Event Horizon Telescope Collaboration. 5/12/2022. “First Sagittarius A* Event Horizon Telescope +Results. IV. Variability, Morphology, and Black Hole Mass.” The Astrophysical Journal Letters, 930, L15. +https://doi.org/10.3847/2041-8213/ac6736 +4 The Event Horizon Telescope Collaboration. 5/12/2022. “First Sagittarius A* Event Horizon Telescope +Results. III. Imaging of the Galactic Center Supermassive Black Hole .” The Astrophysical Journal Letters, 930, +L14. https://doi.org/10.3847/2041-8213/ac6429 +3 The Event Horizon Telescope Collaboration. 5/12/2022. “First Sagittarius A* Event Horizon Telescope +Results. II. EHT and Multiwavelength Observations, Data Processing, and Calibration .” The Astrophysical +Journal Letters, 930, L13. https://10.0.15.7/2041-8213/ac6675 +2 The Event Horizon Telescope Collaboration. 5/12/2022. “First Sagittarius A* Event Horizon Telescope +Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way.” The Astrophysical +Journal Letters, 930, L12. https://doi.org/10.3847/2041-8213/ac6674 +1See https://www.ukri.org/news/first-image-of-black-hole-in-the-centre-of-our-galaxy-unveiled/ +Page 1 of 37 + +STFC Astronomy Advisory Panel +Roadmap 2022 +Astronomy Advisory Panel: Stephen Serjeant (Open University, chair), James Bolton (University of +Nottingham), Poshak Gandhi (University of Southampton), Ben Stappers (University of Manchester), +Paolo Mazzali (Liverpool John Moores University), Aprajita Verma (University of Oxford), Noelia E. +D. Noël (University of Surrey) +Contents +1. Executive Summary +3 +2. International context and consultation process +7 +2.1 Background of previous AAP reviews +7 +2.2 US Decadal Review 2020-1: priorities and synergies with UK interests +7 +2.3 European ASTRONET review 2021-2022: priorities and synergies with UK interests +9 +2.3.1 Context and background +9 +2.3.2 Origin and Evolution of the Universe +9 +2.3.3 Formation and Evolution of Galaxies +10 +2.3.4 Formation and Evolution of Stars +11 +2.3.5 Formation and Evolution of Planetary Systems +11 +2.3.6 Understanding the solar system and conditions for life +12 +2.3.7 Computing - big data, high performance computing, data infrastructure +12 +2.3.8 Societal aspects – education, public engagement climate action, equality, diversity and +inclusion +13 +2.3.9 Extreme astrophysics +13 +2.4 Subsequent UK developments and the scope of the AAP roadmaps +14 +3. Summary of the 2021-2022 AAP Community Consultation +14 +3.1 Overview of the process +14 +3.2 Overview of white papers +15 +3.3 Top, overarching community priority in the consultation +17 +3.4 Highest rated science and technology themes/facilities in the consultation +18 +3.5 Emerging technologies or capabilities identified by the community for inclusion in the +roadmap +19 +3.6 Science challenges identified by the community as requiring major investments +19 +3.7 Equity, Diversity, Inclusion, and Careers +19 +4. UKRI Infrastructure Funding call: AAP and community responses +20 +5. Science and Technology Roadmap +21 +5.1 Overview +21 +5.2 Top priority science facilities +23 +5.2.1 European Southern Observatory +23 +Page 2 of 37 + +5.2.2 Square Kilometer Array +23 +5.2.3 European Space Agency and other space opportunities +24 +5.3 Very high priority science and technology +25 +5.3.1 High Performance Computing +25 +5.3.2 Vera Rubin Observatory +25 +5.3.3 Simons Observatory / CMB science +25 +5.4 High priority science and facilities +25 +5.4.1 WEAVE +26 +5.4.2 The Maunakea Spectroscopic Explorer and other 10-12m-class massively multiplexed +spectroscopy facilities +26 +5.4.3 LOFAR, the Low Frequency Array +26 +5.4.4 VLT BlueMUSE +26 +5.4.5 VLT CUBES +26 +5.4.6 UK National Radio Astronomical Observatory UKNRAO +27 +5.4.7 Submm Astronomy on Large Scales / AtLAST +27 +5.4.8 Magdalena Ridge Observatory Interferometer, MROI +27 +5.4.9 HARPS-3 +28 +5.4.10 Other facility capabilities +28 +5.5 Emerging community priorities +28 +5.5.1 Astronomy and Space Domain Awareness +28 +5.5.2 Line intensity mapping at millimetre/sub-millimetre wavelengths +29 +5.5.3 Development in Africa with Radio Astronomy, DARA +29 +5.5.4 Collaboration with Goonhilly Earth Station Ltd on Radio Astronomy and Space +Applications +29 +5.5.5 High-risk, high return science areas +29 +5.7 Projects with a moderate or low science return +29 +6. Scope for further community consultation +30 +7. Summary list of recommendations +30 +8. Appendix: List of acronyms +32 +1. Executive Summary +These science and technology roadmaps provide a brief overview of UK Astrophysics research, areas +of strength, opportunities for growth and areas for strategic investment. Observational and theoretical +astronomy are fast-paced fields in which progress depends on a broad range of long-term, large-scale +facility investments, from international observatories to high-performance computing. UK astronomy +performs exceptionally well internationally on a wide variety of metrics (for example: for refereed +astronomy papers published in 2000-2020, those with UK co-authors received 2.7M citations , +8 +compared to e.g. 7.1M co-authored from the USA from a community approximately 4.2 times +9 +9 Source: NASA Astrophysics Data System +8 Source: NASA Astrophysics Data System +Page 3 of 37 + +larger ), at least some of which may be attributable to strategic positioning within a wide range of +10 +long-term international projects. +The Astronomy Advisory Panel have therefore consulted with the UK community to compile +roadmaps that capture at least some of the breadth and depth of the current community aspirations. +However, it is equally important to emphasise what these roadmaps are not. Firstly, in a significant +change to the advisory panel remit since the last major roadmap refresh in 2012, the tensioning +between current spending priorities is now explicitly conducted through the advice of Balance of +Programmes reviews (BoP), the last of which +was in 2020. The roadmaps are not intended as a +11 +substitute for this process (or any potential astronomy-specific equivalent), but rather to highlight +current aspirations for hypothetical future spends. Secondly, the fast-paced nature of the field implies +that the aspirations are subject to change. For example, some of our community consultation +responses were explicit that it is impossible to identify unambiguously the technologies needed to +address STFC’s key science challenges in the coming ten years, and that therefore the only defensible +strategic choice that maximises the probable science return is to favour the breadth of the technology +programme. Finally, these roadmaps must not be used as the sole justification for the allocation of +new funding resources, without discussion with advisory panels about the consequences for other +parts of the programme and consultation on any subsequent developments in astronomy. +The top priority of the UK astronomy community remains exploitation grants, despite a very welcome +recent uplift of £2M year on year (leading to a £6M increase by 2024/25). This priority has been +consistent over the past ten years. Flat-cash STFC funding, combined with the increasing costs of +facility funding required to maintain a world-class astronomy programme, has resulted in a radical +erosion of PDRA funding. At current levels, a typical research-active academic might secure postdoc +funding three times in an entire career. Not only is this a bottleneck for science exploitation, it drives +an unplanned and potentially inequitable attrition of early-career talent from UK astrophysics. There +is evidence presented in the roadmaps that the community view of the balance of funding is evolving +in response to this chronic underfunding of exploitation. +Exploitation funding aside, UK astronomy has benefitted from long-term strategic facility and +technology funding that has driven a very wide range of high-profile science results. Unlike for +example particle physics, astronomy has a wide and diverse range of science goals from the +microscopic processes in astrochemistry, to exoplanetary science, to star formation and evolution, to +the evolution of galaxy populations and the large scale structure of the Universe, to the properties of +dark matter and dark energy, to the fundamental cosmological parameters and theories of gravity, and +more, all driven by UK access to world-leading facilities, most of which address a very broad range of +these science goals. The diversity in this portfolio of facilities is inevitable and essential, and is driven +in part by the wide range of physical processes that dominate in the various parts of the +electromagnetic spectrum. Currently, the UK astronomy community aspirations continue to follow a +similar pattern of strongly supporting these long-term investments, and have led to a very clear set of +recommendations for sustaining a vigorous timeline of UK astronomy capabilities. +Among the top community priorities is the Square Kilometer Array (SKA), which is supported by a +very broad section of the community and which promises transformative progress across a very wide +range of STFC priority areas. A high community priority is the UK role in the Low Frequency Array +(LOFAR), which as a pathfinder to the SKA is already producing world-leading survey results, and in +which the UK is very well positioned for future success. The UK involvement in the European +Southern Observatory (ESO) is critical to UK astronomy, and is another of our top community +priorities. ESO’s Very Large Telescopes (VLT) provides critical capability across almost the entirety +of the UK astronomy programme. ESO’s Extremely Large Telescope (ELT) is a top priority for the +long term UK astronomy capability and for nearer-term technology development, and will clearly +11 See https://www.ukri.org/wp-content/uploads/2022/03/STFC-210322-BalanceOfProgrammeExercise2020.pdf +10 Source: International Astronomical Union, https://www.iau.org/public/themes/member_statistics/ +Page 4 of 37 + +revolutionise astronomy throughout the UK programme. Through ESO, the UK also has access to the +~£1.3Bn Atacama Large Millimetre Array (ALMA) in which the UK submm/mm-wavelength +community have capitalised on their strong heritage in securing competitive allocations, supported by +the UK ALMA Regional Centre and with many programmes fed by the UK role in the James Clerk +Maxwell Telescope (JCMT) (funded via PPRP and university contributions). The UK also plays +leading roles in building facilities and/or associated instruments (e.g. ELT, VLT, SKA, ALMA, etc.), +often strategically placing the UK to trade unique technical capabilities, often mission-critical, for a +seat at the table setting the scientific agenda of a mission. +UKSA is responsible for much of the space mission programme, while the science strategy, early +technology, data challenges and exploitation are all within the remit of STFC; nevertheless, the +roadmaps would be incomplete if they failed to note that the ESA Science mandatory programme +covers projects supporting almost the entirety of the UK astronomy science and technology +communities, with missions comparable (or greater) in cost, impact and community support to our +top-rated ground-based projects. +There is a very strong track record in astronomy of major discoveries following the opening of new +astronomical parameter spaces. Time-domain astronomy is an excellent topical example, not just with +follow-ups of gravitational wave events (see PAAP roadmaps), fast radio bursts, gamma-ray bursts, +supernovae and other transients, but also in the explosion of interest and UK world-leadership in the +field of exoplanets. The Vera Rubin Observatory Legacy Survey of Space and Time (Rubin +LSST) will image the whole available sky twice a week for ten years at optical wavelengths, and is +one of the community’s clear Very High priorities. +A further Very High priority driven by the UK’s world-leading cosmic microwave background (CMB) +community is the UK role in the Simons Observatory, which has been funded outside the STFC core +programme through a UKRI infrastructure call. This is the top priority of the UK CMB community +and builds on a very strong heritage of international leadership in CMB cosmology. Also in the Very +High priority science category, and Very High in technology priority (our highest categorisation for +technology) is UK capability in High Performance Computing (HPC), which underpins a wide +range of theoretical astrophysics from planet formation to cosmology. +There is also support for many smaller facility projects. Here, ‘small’ should not be equated with ‘not +world-leading’; there is no place in these roadmaps for any facility that is not world leading. While +these projects may not enjoy the scientific breadth and almost-universal community involvement of +our top priorities, they nevertheless offer the prospect of transformative progress in particular areas. In +many cases it is not enough for there to be only access to the largest international facilities; national +access to world-leading smaller specialist facilities can provide feeder programmes to the larger +facilities and provide the UK with tactical and strategic advantages (e.g. JCMT feeding ALMA and +e-MERLIN feeding SKA). See Figure 1. +In summary, astronomy in all its forms is a UK strength. The UK strength is across the breadth of the +field, from planetary science to cosmology. There is a clear consensus on the key science questions to +be addressed, and these require access to a wide range of facilities and capabilities, many of which +address multiple key science questions. International partnerships (e.g. ESA, ESO, SKA and bilateral +programmes) are extremely important in order for the UK to play leading roles in international +science teams, including across Europe. A strong relationship with UKSA and ESA, and engagement +with their plans, is also essential. The UK astronomy programme is naturally split across several +STFC advisory panels (e.g. AAP, SSAP, PAAP), but there are many synergies, in terms of the need for +access to exploitation funds, the support for theoretical studies, training & skills, computing resources, +instrumentation and technology development, etc. The advisory panels are therefore not in +competition but rather are synergistic, and where there are overlapping interests there is concordance +over the science and technology aspirations. +Page 5 of 37 + +Figure 1: Schematic Gantt chart of a selection of significant facilities +, +, +, +, +, +, +, +, +, +featured in +12 13 14 15 16 17 18 19 20 21 +the AAP Roadmap. Timescales are indicative only and dependent on a wide variety of factors. Project +end dates do not indicate AAP decisions but in many cases reflect current review dates at the time of +writing. Only the facilities themselves are shown, and not individual instruments (e.g. WEAVE). +21 ELT timescales adapted from https://elt.eso.org/about/timeline/ +20 MSE: timescales adapted from White Paper submitted to AAP +19 GOTO, LT, NRT: adapted from indicative timescales and current funding commitments from C. Vincent, priv. +comm., November 2022, without precluding further future funding +18 For Simons Observatory and CMB-S4, see PAAP roadmap +17 JCMT timescale based on currently awarded STFC funding, without precluding further funding in future +https://gtr.ukri.org/projects?ref=ST%2FV000268%2F1 +16 AtLAST timescales based on https://arxiv.org/abs/2011.07974 with a purely illustrative ~6-year construction +(cf. approximately 4 years for JCMT, 8 for ALMA to achieve early science, approximately 9 for LMT) +15 Rubin timeline adapted from https://www.lsst.org/about/project-status +14 e-MERLIN timescales of existing funding from Simon Garrington, priv. comm., November 2022; there is no +horizon for the ERIC as contributions are expected to be renewed by the current partners and new partners are +being sought. +13 LOFAR 2.0 timescales from Philip Best, priv. comm., November 2022: Nov 7th 2022: Formal sign-off by +International LOFAR Telescope Board on procurement contracts; March 2023: delivery of equipment for three +stations; March 2024: system commissioning/testing with 3-station array; Sept 2024 to mid-2025: full system +roll-out. +12 SKA timeline adapted from https://www.skao.int/en/explore/construction-journey and +https://www.skao.int/en/science-users/159/scientific-timeline +Page 6 of 37 + +Key +Design +Construction +Operations +SKA +RADIO +LOFAR/LOFAR2.0 +e-MERLIN +Simons Observatory +CMB / (sub)mm +CMB-S4 +ALMA +AtLAST +CMT +Rubin +UV, Optical, near-IR +MSE +ELT +GOTO +T +NRT2. International context and consultation process +2.1 Background of previous AAP reviews +AAP conducted a call for white papers and consultation exercise with the UK astronomy community +via an online survey in November 2021. This was advertised via astrocommunity and other routes. We +greatly appreciate the help of our STFC colleagues in assembling the survey, providing us with the +results promptly in a digestible format and in providing other contextual information. The results of +the consultation are described in detail elsewhere , and summarised in Section 3 below. +22 +The AAP science roadmap last had a major update +in 2012, and the technology roadmap is even +23 +further outdated. In the meantime, there have been minor updates following AAP’s 2016 roadmap +consultation , and its 2018 consultation to feed into the second STFC Balance of Programmes +24 +review . The non-confidential parts of AAP’s 2018 consultation report were published +in +25 +26 +Astronomy and Geophysics. +AAP nevertheless opted to delay its most recent consultation and report for several reasons. Firstly, +the global health crisis meant that the capacities of both the panel and the wider community were +limited. Secondly, the outcomes of the European ASTRONET review could affect the policy decisions +for AAP. Finally, the US Decadal Review outcomes would also be likely to affect the AAP roadmap. +Both the ASTRONET and US Decadal reviews were also delayed partly due to the global health +crisis. AAP do not seek to duplicate these roadmaps, but rather present the UK-specific community +voice. +2.2 US Decadal Review 2020-1: priorities and synergies with UK interests +The US Decadal Review is a comprehensive survey of US community aspirations in both +ground-based and space-based astronomy for not just the coming decade but also outlining a roadmap +for future decades. In ground-based astronomy, the review prioritises participation in extremely large +telescopes (30 metre class), the next generation Stage 4 cosmic microwave background (CMB) +experiments, the next generation Very Large Array radio telescope ngVLA. Unlike the UK, the US is +not a participant in the Square Kilometer Array (SKA); instead the focus is on US-specific +complementary higher frequency Northern hemisphere capabilities in ngVLA. The US prioritisation +of extremely large telescopes is very similar to the UK community’s prioritisation of science +opportunities and technical developments in the European Southern Observatory’s Extremely Large +Telescope, ELT. The coming decades of ground-based optical and near-infrared astronomy will clearly +be dominated internationally by facilities of this class. The CMB aspirations are very well aligned +with those in the UK, and UK participation in the US-led Simons Observatory is now funded (see +below). The review also prioritised the Vera Rubin Observatory, which also features as a UK very +high priority (see below). US priorities in astroparticle physics include upgrading the neutrino detector +IceCube and gravitational wave technology development, though in UK terms these are overlapping +with PAAP interests. The review calls for an expansion in grants (referred to as the exploitation line in +STFC terminology), supporting data archives and curation, bolstering theory underpinnings, +26 See Serjeant, S., et al., 2019, Astronomy & Geophysics, Volume 60, Issue 2, Pages 2.13–2.17, +https://academic.oup.com/astrogeo/article/60/2/2.13/5380734 +25 See https://www.ukri.org/about-us/stfc/planning-strategy-reviews/balance-of-programmes/ +24 No longer available on the UKRI websites, but archived at +https://web.archive.org/web/20220305101010/https://stfc.ukri.org/files/aap-balance-report/ +23 No longer available on the UKRI websites, but archived at +https://www.yumpu.com/en/document/view/4206082/aap-prreport-submitted-nov22 +22 See https://www.ukri.org/publications/astronomy-advisory-panel-summary-of-2021-community-consultation/ +Page 7 of 37 + +advancing laboratory measurements (e.g. laboratory astrochemistry), and expanding basic technology +development. Most of these items feature in the high or very high UK community priorities discussed +below. The US prioritisation of data archives and curation echoes a European prioritisation of open +science under the “FAIR” principles: Findable, Accessible, Interoperable and Reusable. However, in +the UK our consultation highlighted not only a need for continued support for this area, but rather a +conspicuous lack of support for software, archiving, instrumentation and technical careers. The US is +prioritising an expansion in grants, while the UK has for many years been failing to make optimal use +of its astronomy facility investments due to a chronic underfunding of the exploitation line. +In terms of overarching AAP-area science themes, the US Decadal Review recognises several key +areas. “Pathways to Habitable Worlds” supports the work of the US exoplanet community, which has +a sizable UK counterpart, and in terms of ground-based astronomy requires support from ELTs, +resolution spectroscopy, high-performance adaptive optics, high-contrast imaging, as well as +laboratory and theoretical studies, all of which have been recognised as UK community priorities +below. The “New Messengers and New Physics” and “New Windows on the Dynamic Universe” +themes cover not just CMB cosmology but also the rapidly expanding discovery space of time-domain +astrophysics, including gravitational wave counterparts and other optical and radio transients, all of +which have sizeable UK community support. Ground-based technical capabilities required to support +these themes are very similar to those highlighted by the UK communities and include ELTs, next +generation CMB experiments and gravitational wave technology developments. The “Cosmic +Ecosystems” theme covers a wide range of galaxy evolution studies, requiring ground-based support +from the Vera Rubin Observatory as well as a wide range of space-based facilities including the +forthcoming ESA-led Euclid mission, while the “Unveiling the Drivers of Galaxy Growth” theme +pushes also to higher redshifts with the help of ELTs, new radio facilities, and space missions. Both +themes also recommend support for theoretical underpinnings. The UK has a vibrant extragalactic +community and prioritises a similar mix of ground-based technical capabilities, though the SKA +affords a slightly different and complementary set of opportunities. The consistent call for theoretical +underpinning is reflected also in the UK community prioritisation of continuously updating the +national high performance computing capability, discussed below. +The US Decadal Review also set out aspirations for space-based astronomy, the headlines of which +are a large infrared/optical/ultraviolet space telescope, X-ray and far-infrared missions. Together, +these missions cover the breadth of the review’s science themes. In the UK, space mission +involvements are the purview of UKSA and are formally outside the remit of STFC, except in some +limited situations discussed briefly below. Nevertheless, it is also the case in the UK that space +missions are central to a wide range of science themes, including the US priorities of JWST, Roman, +ESA Euclid, as well as other current and future ESA M-class and L-class missions. The limitations of +atmospheric stability, transmission and/or emissivity inevitably require space missions for many +science goals, including exoplanet discovery, high energy astrophysics, time-domain astrophysics and +physical processes obscured by dust. Although space projects are outside the AAP remit, and although +AAP are not being asked for as fine-grained detail on community priorities for space missions as we +are in ground-based astronomy, we felt it would be inappropriate if we did not capture the community +view by ranking the UK involvement in the ESA mandatory science programme as being collectively +equal to our highest ranked ground-based priorities, below. There is also a great deal of UK interest in +space bilateral opportunities, including with the US missions highlighted in the Decadal Review, but +this is largely beyond the remit of the AAP roadmaps. +The review also made a wide range of recommendations on societal impacts, career development, +equity, diversity and inclusion. Many of these are also appropriate for UK contexts and are discussed +in the context of our community consultation below. +Page 8 of 37 + +2.3 European ASTRONET review 2021-2022: priorities and synergies with UK +interests +2.3.1 Context and background +ASTRONET is a European consortium of funding agencies, research organisations and associated +bodies that was formed in 2005 with initial funding support from the European Commission. It is now +self-sustaining (i.e. without EC financial contributions). One of its principal objectives is the creation +of a community-driven strategic plan and infrastructure roadmap, covering all of European astronomy. +The remit covers not just the ground-based astronomy within the UK STFC AAP remit, but also solar +system science and space-based facilities across all of European astronomy. +ASTRONET last published its science vision and infrastructure roadmap +in 2015, after which its +27 +focus has been on the implementation through a series of reports on the European coordination of +research infrastructures; the integration of the European communities in mainstream astronomy; the +coordination of national funding agencies; and education, training, and public outreach. Most recently, +over the past two years ASTRONET has been working on a new Science Vision and Infrastructure +Roadmap. Again the aspiration is for this to include all aspects of European astronomy to aid national +and multi-national long-term astronomy planning. An extensive community consultation +on the +28 +drafts concluded in August 2022, and at the time of writing (November 2022) five of the eight science +themes exist in a final form, while the remaining three are still draft reports. Below we summarise the +recommendations in each theme and the UK relevance. Note the wide range of facilities and +instruments that are named as supporting several science challenges. Many UK colleagues +participated in the ASTRONET consultation; there are no obvious areas in which UK aspirations are +in conflict with those of this wider European strategy, but there are areas of particular UK interest. +2.3.2 Origin and Evolution of the Universe +This science theme outlines the current standard cosmological models and the community aspirations +for progress in the coming two decades. The report highlights a growing overlap or synergy between +fundamental physics and cosmology, through the physics of the early Universe and tracers of the +matter distribution around the epoch of recombination, as well as the large-scale distribution of +galaxies at lower redshifts, both of which are areas that have broad UK active communities with +world-leading track records. The UK had a representative in the working group that drafted this +chapter. Key science questions in those areas include: “Are there deviations from the standard model +of particle physics? Are there deviations from the standard cosmological model?” The emerging +tension in measurements of the Hubble constant hints either at unknown systematics or new physics, +while the accelerating expansion of the Universe inevitably signals new physics; the report poses +further key questions: “Are there deviations from general relativity, and on what scales? What is the +origin of the accelerated expansion?” A further inevitable signal of physics beyond the Standard +Model is the evidence for dark matter, all of which is currently astronomical, and a further key +question in this section is “What is the nature of dark matter?” The temperature and polarisation +anisotropy of the CMB probes physics between the LHC energy range of 104 GeV to Grand Unified +Theory scales of 1016 GeV, and a further key science question highlighted in the report is “Can we +identify specific observational signatures of inflation?” Finally, this section of the report covers the +PAAP-remit area of gravitational waves, posing the key questions “What can gravitational waves +observations reveal about dark energy, dark matter and modifications of gravity on cosmological +scales?” +In each of these key question areas, critical ground-based and space-based facilities are identified. In +terms of ground-based astronomy, the report highlights the importance of the SKA, Rubin, Simons +28 See https://www.ASTRONET-eu.org/forums/roadmap-community-consultation +27 See https://www.ASTRONET-eu.org/archives +Page 9 of 37 + +Observatory (and forthcoming CMB Stage 4), all of which are very high UK community priorities. +European CMB ground-based (e.g. QUBIC, QUIJOTE, LSPE-strip) and balloon-borne experiments +(e.g. LSPE-swipe, BISOU) highlighted as pathfinders for larger space and ground-based facilities. +Ground-based spectroscopic projects also feature prominently, such as the US-led DESI (with UK +member institutions), the Japanese Subaru-PFS, the Mauna Kea Spectroscopic Explorer (MSE, +recommended by AAP for UKRI infrastructure funding, see below), and the US-led MegaMapper. For +probing the epoch of reionization via 21cm cosmology, the report highlights LOFAR (with UK +involvement), MWA and the GMRT, and forthcoming facilities LOFAR2.0, MWA2, uGMRT, with +SKA already under construction and HERA operational. We note that UK involvement in HERA has +been considered for PPRP funding but so far has not been successful. Line intensity mapping also +features as a cosmological probe, tracing the large-scale structure of atomic, molecular and/or ionic +gas in the Universe, and requiring ground-based facility support from SKA, APEX/CONCERTO +(with ESO involvement), CCAT (with UK involvement), and BINGO (with UK involvement). The +report recommends investing in “small/medium-size facilities and new instruments on existing +platforms (e.g., WEAVE, 4MOST)” as pathfinders for more ambitious wide-field facilities (e.g. +MSE). Finally, the report chapter recommends investments in data processing, analysis and data +science to accompany new instrumentation, in concordance with similar, broader recommendations +from the US Decadal Review. +2.3.3 Formation and Evolution of Galaxies +At the time of writing, this chapter currently only exists in draft form. There is a very broad and active +community in both the UK and more widely across Europe in this science area. Accordingly, the +report chapter covers an extremely wide range of science questions and concomitant facilities. At very +early cosmic epochs, the chapter outlines the aspirations for understanding the physics during the +reionization epoch, including the sources of reionization, the topology of ionized gas, the properties of +the first stars and galaxies, and the origins of gas in galaxies that fuel star formation. At lower +redshifts, the report highlights aspirations for understanding critical processes driving galaxy +evolution and the growth of stellar mass, including positive and negative feedback processes from +starbursts and active nuclei, the dark matter halo processes (besides mass) that drive galaxy properties, +the state of gaseous halos surrounding galaxies, the processes that drive star formation out of dense +molecular gas, the processes that drive the star formation histories and morphological evolution of +galaxies, the role of environments and the accretion physics around supermassive black holes. The +report extends to science questions covering the Milky Way, including its accretion history (and what +information can be recovered about it), the assembly of its halo, the abundance patterns of its first +stars, and our very local environment. +The facility requirements and aspirations that accompany this vision are similarly broad. Many current +and future space missions are prioritised, though in UK contexts these are broadly in the remit of +UKSA rather than STFC. In terms of ground-based spectroscopy, the report chapter highlights +multiplexed spectroscopy (4MOST, ESO VLT/MOONS, WEAVE, Subaru PFS, CFHT/Mauna Kea +Spectroscopic Explorer, ESO spectroscopic facility), high-resolution spectroscopy (CUBES and +ELT/ANDES and MOSAIC), integral field spectroscopy (e.g. VLT/MUSE and BlueMUSE), and +21cm spectroscopy and line intensity mapping (SKA, HERA). Many of these have UK key roles, +including ESO instrumentation, VLT/MOONS, and WEAVE; the Mauna Kea Spectroscopic Explorer +was among the AAP priorities for UKRI infrastructure funding (see below). Ground-based imaging +capabilities are also prioritised with Rubin and ultimately ELT. The report supports the development +of interferometers such as LOFAR and including endorsing the aspirations in the ALMA 2030 +roadmap , as well as developments of the German/Spanish/French PdBI/NOEMA array and +29 +supporting new facilities such as the SKA, the US ngVLA (see above), and a future large single-dish +29 See https://www.almaobservatory.org/en/publications/the-alma-development-roadmap/ +Page 10 of 37 + +submm/mm-wave telescope AtLAST (which featured in an AAP recommendation for future UKRI +infrastructure funding, see below). High angular resolution studies are prioritised on both the ESO +VLTs and the future ELT (e.g. VLT/MAVIS, ELT/MICADO, ELT/HARMONI). High angular +resolution interferometry with ALMA (especially the long baseline extension), VLBI, VLTI, ngVLA, +SKA and its precursors such as MeerKAT are all prioritised, as is Rubin LSST. The Cherenkov +Telescope Array (CTA) is also an ASTRONET priority (in which there is UK technical interest in +instrumentation and UK science interests mainly in the prospects for multi-messenger / +multi-wavelength astronomy). Also highlighted are cosmological and extragalactic simulations, +including Illustris, the EAGLE simulations, the Hestia project, and many others. +2.3.4 Formation and Evolution of Stars +This is a wide-ranging science theme in which the UK also has large, vibrant research communities; +the UK chaired the working group that drafted this chapter and also had representation among its +membership. The science theme covers the birth, lifecycle and death of stars, the physics of stellar +remnants and the cosmological applications of transient events. Key questions in the area of stellar +birth include: “How do molecular clumps fragment? What effect does the local magnetic field have on +collapse? How do stars accrete material? What sets the upper mass limit of a star? Is the initial mass +function (IMF) universal? What is the relationship of the IMF to the core mass function? What sets +the initial multiplicity of stars, and how does this impact the IMF? How do young stars and their disks +set the properties of nascent planetary systems and their early evolution? What chemical pathways are +important in the production of complex organics?” The evolution of stars also poses a number of key +science questions, including: “What are the basic constituents of [stellar] matter? What happens to +matter in extreme conditions? What external factors influence the conditions required for life on +planets?” Finally, ASTRONET highlights a number of key science questions regarding the end-states +of stellar evolution, including: “What are the progenitors of type Ia supernovae? What is the Galactic +low-frequency GW background? What is the ultimate fate of planetary systems orbiting stellar +remnants?” Neutron stars pose a wide range of key science questions relating to their equations of +state, the origin of their magnetic fields and radio emission, their formation routes and interaction with +their environments. Stellar-mass black holes are important tests of General Relativity and there are +open science questions on their formation routes. +A wide range of ground-based instruments and facilities are prioritised to support these science goals, +including ELT, Rubin LSST, SKA, WEAVE, 4MOST, VLT/MOONS, VLT/CUBES, VLTI, as well as +smaller facilities specialising in time-domain science such as GOTO (prioritised by AAP in the +previous “priority projects” process and with UK involvement now funded by PPRP) and BlackGEM +(again with UK involvement). Multi-messenger astronomy features prominently in this theme. +Existing facilities with unique capabilities are also prioritised, including ALMA, VLT/VLTI, the +Spanish 10.4 metre telescope GTC, WHT, and the Italian 3.58-metre TNG telescope. Looking ahead, +the report prioritises new ground-based capabilities with 8-10 metre class facilities such as VLT +GRAVITY+ and VLT BlueMUSE, new highly multiplexed spectroscopic survey capabilities such as +wider European participation in the Mauna Kea Spectroscopic Explorer or similar (which featured in +the AAP prioritisation for UKRI infrastructure funding, see below), as well as highlighting the need +for a large single-dish sub-mm facility such as AtLAST (again featuring in AAP’s UKRI +infrastructure funding shortlist). +2.3.5 Formation and Evolution of Planetary Systems +Once again this is an area with a vibrant UK research community, and the UK had a representative on +the working group that drafted this chapter. This field is experiencing an extraordinary phase of +discovery since the first exoplanets were discovered only three decades ago. The overarching science +questions highlighted by ASTRONET that are driving the current and future research in this area are: +“What drives the enormous diversity in exoplanet systems? How special is Earth within all the +Page 11 of 37 + +possible worlds? What are the necessary conditions for life to emerge and thrive? What is the fate of +the Solar System?” and, “Are we alone…?” +A very wide range of ground-based instruments and facilities are prioritised to support this science, +covering planet formation, exoplanet discovery and demographics, exoplanet characterisation, and the +late-stage evolution of planetary systems, including: VLTI MIDI, MATISSE and GRAVITY+; VLT +instruments SPHERE, MUSE, ERIS, HiRISE (SPHERE/CRIRES+ coupling), RISTRETTO, and +possibly MAVIS; ALMA; Rubin LSST; ESO CRIRES+, NIRPS. Precision radial velocity +measurements are critically important to this field, with the CFHT SPIROU, ESO NIRPS, and the +Spanish/German CARMENES explicitly cited in the near-infrared, while in the optical the VLT +ESPRESSO, the US EXPRES facility, and HARPS3 on the Isaac Newton Telescope are all explicitly +cited. VLT and (later) ELT direct imaging are prioritised. LOFAR and SKA (both with substantial UK +interests) are cited for the study of star-planet interactions. The wide-area multi-object spectroscopic +surveys WEAVE, DESI, 4MOST and SDSS-V are particularly important for the study of white dwarf +planetary systems. Further in future the report prioritises the ELT instruments MICADO, HARMONI, +METIS, HIRES (now named ANDES), PCS. Furthermore, laboratory astrochemistry (also a UK area +of world-leading expertise) is explicitly prioritised in order to complement and interpret astrophysical +observations. Theoretical work with three-dimensional numerical simulations is similarly prioritised. +2.3.6 Understanding the solar system and conditions for life +This chapter of the ASTRONET roadmap also covers a wide range of UK research interests, and the +UK also participated in the working group that drafted this chapter. The scientific remit is solar +system science and heliospheric physics, and as such is much more closely aligned with the +specialism of SSAP than AAP. A large proportion of this review is also based on space missions that +fall within the UKSA remit. For both reasons we do not summarise this chapter here. Nevertheless, +many of the same priority terrestrial facilities and technologies are in common between these solar +system recommendations and those of AAP, including Rubin LSST and laboratory astrochemistry. +2.3.7 Computing - big data, high performance computing, data infrastructure +This overarching technology chapter of the ASTRONET roadmap covers high performance +computing, software engineering, data products, software tools, open science (including astronomical +use of the European Open Science Cloud, an interdisciplinary open science initiative of the European +Commission funded by approximately a third of a billion euros to date), and the carbon footprint of +astronomical computing. UK expertise was represented on the working group that drafted this chapter. +The report provides six main recommendations, which in summary are: +1. +Developing and investing in a professional software engineering / computational skills base in +Astronomy, including career development with clear progression pathways in academia and +improving the diversity of the workforce, and new measures to quantify the impact and +usefulness of computationally focussed outputs. +2. +Missions and facilities should plan an integrated approach for data products and software +tools including design, delivery, maintenance, development, and scientific data preservation. +3. +Creation of a “tiered” approach to data infrastructures, for all astrophysical data including +models, simulations and mock data. +4. +A fully collaborative, open and synergistic approach to the astronomy-computing ecosystem, +encompassing data, software, processing, analysing and modelling, and embedding reward +structures to realise this ambition. +5. +ASTRONET should produce or commission a biennial quantitative report to assess the carbon +footprint of computing in Astronomy. +6. +ASTRONET should develop specific actions to coordinate cross-cutting activities. +Page 12 of 37 + +These aspirations are shared by the UK community, but AAP notes these aspirations are not currently +being met. For example, they sit uneasily with the AAP consultation responses below that highlight +current UK deficiencies in the support of career pathways for technical roles in astronomy. See +Recommendation 3.4 below. +2.3.8 Societal aspects – education, public engagement climate action, equality, diversity and inclusion +At the time of writing (November 2022), this chapter of the ASTRONET roadmap is in draft form and +is not yet finalised. The draft report covers the social and cultural relevance of astronomy, education, +technology transfer from astronomy, public engagement, astronomy for sustainable development, +climate action, gender equality, and inequalities in astronomy. The educational and cultural aspects of +the report are not germane to our AAP roadmap so we do not summarise them here, but our AAP +consultation responses below share many of the same ideas as the diversity, equity and inclusion +themes of this report, including: “significant gender imbalance still exists within astronomy as a +profession [...] there is a significant imbalance in the inclusion of underrepresented and vulnerable +groups (persons with (dis)abilities and persons with racial, ethnic, religious, LGBTQI+ backgrounds) +[...] The European astronomical community needs to develop specific training [for] researchers and +advocate for equity and inclusion”. Furthermore, the draft report is currently recommending that +“European astronomy (community) should (at the very least) follow the European timeline towards +carbon-neutrality: 50% reduction of CO2 emissions by 2030 and 100% by 2050”; AAP are pleased to +note that the environmental impact was one of the criteria in assessing the 2020 UKRI infrastructure +funding bids, and AAP welcomes the 2020 UKRI Environmental Sustainability Strategy +that aims to +30 +achieve net zero carbon emissions from UKRI no later than 2040. +2.3.9 Extreme astrophysics +At the time of writing (November 2022) this chapter of the ASTRONET roadmap is currently in draft +form. This science theme again covers areas in which a wide UK community are active and in which +the UK has many leadership roles, and UK expertise was represented in the working group that +drafted this chapter. Extreme astrophysics here refers to the environments within and around compact +objects such as white dwarfs, neutron stars, and black holes. At the time of writing there is some +overlap with the science themes of the “Formation and evolution of stars” ASTRONET roadmap +chapter (see section 2.3.4 above). Key science questions include: What is the nature of matter at +nuclear densities? Where are the heavy elements made? How do compact objects produce energy and +accelerate particles at all scales? What is the origin of cosmic rays of all energies? How do compact +objects form and evolve? To what precision can general relativity describe gravity? What new +fundamental physics can be probed with extreme astrophysical objects? +Many of the key facilities supporting these science questions are space-based and therefore largely +outside the remit of the AAP roadmaps, but a wide range of ground-based facilities and +instrumentation are also prioritised. Current key ground-based facilities cited include: +LIGO/Virgo/KAGRA, neutrino detectors (e.g. the US-led IceCube and the European-led KM3NeT), +LOFAR, Pierre Auger, MAGIC, HESS, the South African SKA precursor MeerKAT, JVLA, EVN, +ground based 2-8 m telescopes including wide-field optical/IR facilities. Future ground-based +facilities that will drive future discoveries include: SKA, CTA, ELT, Rubin, Einstein Telescope, the +Italian ASTRI facility, the Global Cosmic Ray Observatory, the Giant Radio Array for Neutrino +Detection (GRAND), and the next generation Event Horizon Telescope ngEHT. The neutrino, +gravitational wave and cosmic ray capabilities fall within the PAAP remit, but the remainder have a +striking commonality with the facilities supporting other astronomy areas in both the ASTRONET +roadmap and the AAP consultation responses on community priorities below. +30 See https://www.ukri.org/wp-content/uploads/2020/10/UKRI-050920-SustainabilityStrategy.pdf +Page 13 of 37 + +2.4 Subsequent UK developments and the scope of the AAP roadmaps +Since the AAP consultation closed, UKRI postgraduates wrote an open letter to UKRI on the impact +of the cost of living crisis combined with stipends being calculated on the previous year’s inflation, +and requesting that stipends are uplifted to match the current inflation rate. AAP are very pleased to +see UKRI increase the 2022-3 stipends in response to this community input. +Also since the AAP consultation concluded, STFC launched a further consultation on the changes to +the consolidated grant system. This new consultation has just reported its findings, with applications +accepted for the new scheme in March 2023. +The present document condenses the call for white papers and the consultation into science and +technology roadmaps. However, AAP have some concerns about potential mis-uses of the +roadmaps. These roadmaps only describe the community’s present strengths and aspirations, and an +overview of the community’s near-term and mid-term priorities for science and technology +opportunities. The roadmaps are not suitable for drawing a funding threshold line if the core +programme funding is reduced, because that would need either an STFC-wide Balance of +Programmes review with an Astronomy Evaluation Panel, or an equivalent but Astronomy-specific +tensioning of spending commitments, which in either case would be driven by explicit evaluation +criteria and explicit funding envelope options, as well as having agreed expected near-term outcomes +on the basis of the recommendations. AAP do not believe it would be helpful to provide a +finer-grained long-term prioritisation than is presented in this roadmap given that the criteria, +prospective funding envelopes and outcomes are all nebulous, because (a) the nebulous constraints +leave too great a risk of mis-construing or mis-stating the community views (b) the evidence base for +community views and quantitative measures of facility outputs/impact and other as-yet-undetermined +assessment criteria either do not yet exist or have not yet been assembled, and (c) implementation of a +finer-grained priority list without considering the wider impacts on the core programme would expose +astronomy to the risks of unintended consequences in taking on new commitments or divesting from +existing ones. Therefore, the roadmaps must not be used to award funding to something that appears +to be a priority, without any discussion with AAP about the consequences for other parts of the +programme and any developments since the roadmaps were drafted. +Recommendation 2.1: The 2022 AAP Science and Technology Roadmaps should not be used in +isolation to evidence current community support without further reference to AAP, because +astronomy is a fast-paced and rapidly changing field. Nor should these roadmaps be used to +draw a funding threshold line in the event of changes to the core programme funding, because +the Balance of Programmes Review (or a hypothetical Astronomy-specific near-term tensioning +exercise) is the appropriate process for that. +3. Summary of the 2021-2022 AAP Community +Consultation +3.1 Overview of the process +In this section we summarise the response to the AAP community consultation in November 2021, in +which we called for community input on priorities, as well as for science and technology white papers +(see below). Many of the themes highlighted in the 2021 community consultation overlap with the +earlier AAP consultation in 2018 that fed into the Balance of Programmes review. AAP received only +81 responses compared to 293 in 2018. AAP suspects time pressures related to working from home +and the pandemic. There was particularly low representation among early career researchers. In view +Page 14 of 37 + +of this low response rate AAP also solicited further free-form feedback from the community in 2022. +The consultation itself is provided in a separate document +so we only summarise the findings here. +31 +3.2 Overview of white papers +An important part of developing our science and technology roadmaps was the call for community +white papers. These were intended to have the possibility of also serving a dual purpose of assisting +STFC with having a ‘bank’ of potential projects that could be put forward in the event of UKRI-wide +ad hoc funding opportunities (see below). AAP are supportive of STFC’s aspiration to make the +Infrastructure Funding more strategic with more frequent calls to the community, and less dependent +on last-minute calls. The STFC Visions process (which replaces the previous Priority Projects calls) is +explicitly intended for inter-disciplinary/cross-disciplinary large infrastructure projects across UKRI, +and the white papers discussed here did not explicitly require this cross-UKRI remit. At least some of +the white papers discussed in our roadmaps may fit the remit of a future STFC Infrastructure Fund. +From our community responses it is apparent that both the white paper and infrastructure funding +submissions are fundamentally only a snapshot of the ideas present in the community, so it is not +obvious that it is possible to create a long-term roadmap plan that maintains the flexibility and +responsiveness to changes in the scientific and technological landscapes. +AAP were impressed at the breadth and depth of ideas submitted by the community. AAP members +independently evaluated the relative priorities of the white paper submissions and where possible +determined consensus views. Besides the specific projects and facilities discussed in the roadmaps +below, AAP received a wide range of science themes in the white paper submissions with clear +evidence for UK leadership, including the following. +● +What is the chemical and mineralogical composition of astronomical dust? This +interdisciplinary white paper covers Galactic planetary and interstellar astronomy, +astrochemistry, planetary science and meteoritics, and infrared spectroscopy. The paper makes +a compelling case for the support of laboratory astrochemistry and astro-mineralogical +research to underpin the interpretation of major data sets from facilities that represent top +community priorities, e.g. those from JWST. The white paper covers areas of interest of both +AAP and SSAP, and is in concordance with the European ASTRONET priorities (section +2.3.5). +● +Euclid Science Exploitation in the UK is a white paper submitted by the Euclid:UK +coordination group. There are around 350 UK-based researchers working in the Euclid +Consortium. The science themes are extremely wide, covering cosmology, extragalactic +astronomy, Milky Way and the local universe, as well as SSAP areas of interest, and are very +well aligned with STFC Science Challenges +as well as with many of the ASTRONET +32 +themes (section 2.3). STFC funding of the science exploitation is within the remit of AGP. A +related white paper was received by AAP on the new era of strong gravitational lensing +science, covering not just the discovery space opened by Euclid but also that of Rubin LSST, +SKA, and the NASA Roman space telescope, with follow-ups facilitated by ground-based +4-metre class facilities (e.g. ESO NTT, 3.6m, WHT, VISTA/4MOST), 8-metre class +telescopes (e.g. ESO VLTs) and 30-metre class telescopes (ELTs first light 2027), as well as +space telescopes (e.g. JWST). The step change in available discovery space covers a wide +range of science goals, including cosmography, dark matter structure and substructure, quasar +accretion disc structure and highly magnified background source morphologies and transients. +The goals are in concordance with a broad range of ASTRONET aspirations (section 2.3). +32 See https://www.ukri.org/publications/stfc-science-challenges/stfc-science-challenges-in-frontier-physics/ +31 See https://www.ukri.org/publications/astronomy-advisory-panel-summary-of-2021-community-consultation/ +Page 15 of 37 + +● +The white paper submitted to AAP on Stellar Variability and an Extreme Precision Radial +Velocity Roadmap in the UK presented, in effect, a brief update to the 2015 STFC review of +exoplanet science . The UK has clearly established itself as a world leader in this area. The +33 +large author list argues for developing a stellar variability and extreme precision radial +velocity roadmap in liaison with other international bodies such as ESA, ESO, NASA, NSF +etc. The facilities driving progress in this area include HARPS-N at the Italian TNG, the Solar +Telescope, HARPS at the ESO La Silla 3.6m, CARMENES at the CAHA 3.5m, ESPRESSO +at the ESO VLT, NEID at the WIYN 3.5m (a US facility with limited UK access). For +characterising solar/stellar variability, the white paper highlights facilities in the SSAP remit +including SDO, BiSON, DKIST. Looking forward, the white paper cites the importance of the +UK-led HARPS3 at INT, led from the UK, currently under construction (first light expected +2023) and HIRES (now named ANDES) at ELT (under construction) which has passed Phase +A. +● +The exploration of free-floating planets. This science white paper aims to use VLT, ELT, +Rubin LSST, and space missions Roman and JWST. The science theme is an area of UK +strength that features only implicitly in the ASTRONET roadmap (section 2.3.4) and may +represent a key opportunity for UK international leadership. +● +REACH: Radio Experiment for the Analysis of Cosmic Hydrogen . This project aims to +34 +detect the signals of cosmic reionization through sky-averaged 21-cm mapping, addressing +the instrumental systematics present in other experiments, and stealing a march on the SKA. +REACH is a collaboration between the University of Cambridge and Stellenbosch University +in South Africa. Although the facility is not currently financially supported by STFC, AAP +support the aims of the project and are happy to include it among the high priority facilities +below that represent a selection of the current and aspirational portfolio of “small” UK +astronomy facilities. +● +The role of massive stars in galaxy evolution and nucleosynthesis, supporting facility use of +VLT, e-MERLIN, ALMA, ELT, SKA, 4MOST, WEAVE, etc., as well as space missions, and +with scientific aspirations in concordance with those of ASTRONET (section 2.3.4). +● +Eclipsing binary stars: high-precision probes of the physics of stars and planets. This white +paper covers an area of UK strength, and supports the use of WASP, NGTS etc in conjunction +with a variety of space missions. The science area is well aligned with ASTRONET +aspirations (sections 2.3.4 and 2.3.5). +● +Pulsar rotational glitches, a probe to the extreme physics of neutron stars; Precision pulsar +timing as a tool for probing fundamental physics; Radio emission from pulsars and the pulsar +magnetosphere; Revealing the Galactic neutron star population through pulsar searches. +These comprehensive and wide-ranging white papers cover fundamental physics from neutron +stars, supporting facility use of e.g. Jodrell Bank, Parkes, MeerKAT/MeerTime, +LOFAR/LOFAR2.0, SKA, as well as a variety of space missions. Compute is also a key +requirement, particularly also in an additional white paper on magneto-thermal evolution of +neutron stars. A further white paper on optical follow-up of binary neutron star systems made +a strong case for investigating the dynamics, geometry, and probes of the physical processes, +with the aid of ESO NTT, VLT, ELT, as well as GOTO, Rubin LSST, and the Spanish facility +GranTeCan. The white papers are in concordance with aspirations in ASTRONET (sections +2.3.4 and 2.3.9). +● +Multi-messenger astronomy featured in a wide range of white papers submitted to AAP. The +white paper on time-domain polarimetry as a unique discovery space for transients covered +the important emerging fields of multi-wavelength, multi-messenger time domain science. +Besides aspirations for new instrumentation, the facilities supported by this white paper +34 See e.g. de Lera Acedo, E., et al., 2022, Nature Astronomy, 6, 984, +https://doi.org/10.1038/s41550-022-01709-9 +33 See https://www.ukri.org/about-us/stfc/planning-strategy-reviews/exoplanets-uk-research-review/ +Page 16 of 37 + +include Rubin LSST, the Zwicky Transient Factory, Liverpool Telescope, NRT, VLT, WHT, +ALMA, SKA, and a variety of space missions. Multi-messenger astronomy is an emerging +priority of both ASTRONET (section 2.3.4) and APPEC , and naturally also features in the +35 +PAAP roadmap. A further white paper on the high energy transient Universe underlined the +importance of X-ray and gamma-ray space telescopes for multi-messenger astronomy +(formally outside AAP roadmap remit) and the importance of complementary ground-based +observations from e.g. SKA, ELTs, CTA, LOFAR2.0, GOTO; the science goals are broad and +cover emission mechanisms of gamma-ray bursts, relativistic jet physics, thermal transients, +and the early Universe. AAP also received a white paper on searching for fast radio bursts +(FRBs) and other radio transients. This white paper addresses a very topical research area, and +also covers the emerging important techniques of multi-wavelength, multi-messenger +astronomy (see also the ASTRONET roadmap above, section 2.3.4). The science case covers +FRBs as cosmological probes, potential FRB progenitors such as magnetars, slow Galactic +transients, and population phenomenology. Key facilities include MeerKAT, Jodrell Bank, +e-MERLIN, European VLBI Network (EVN), LOFAR and its 2.0 upgrade, SKA, and a wide +range of X-ray space telescopes. +● +Gravitational Wave Astronomy: Advanced LIGO+, Cosmic Explorer, LISA, Einstein +Telescope, Pulsar Timing Arrays. These science themes also feature in the PAAP and +ASTRONET roadmaps (section 2.3) and reflect a widespread interest in this new +astronomical window, with UK leadership in both science and technology. AAP also +welcomed a white paper on neutron star physics from gravitational waves, supporting also +multi-messenger / multi-wavelength observations with LOFAR/LOFAR2.0, Lovell Telescope, +and SKA, underlining the overlapping interests of PAAP and AAP. +Recommendation 3.1: STFC and UKSA should support the UK exoplanet community in +developing a stellar variability and extreme precision radial velocity roadmap in liaison with +other international bodies such as ESA, ESO, NASA, NSF etc. +3.3 Top, overarching community priority in the consultation +Exploitation, i.e. grants, remain the top community priority and are chronically underfunded, despite +a very welcome uplift of £2M year on year (leading to a £6M increase by 2024/25) that was +announced at the 2022 National Astronomy Meeting from the last Comprehensive Spending Review +outcome. This funding for human capacity is now the main limiting factor for our national science +capability in astronomy, but it is a UKRI-wide problem. As John Womersley put it six years ago, we +are still "paying for gym membership and being unable to afford the bus fare to get there". The +continued, chronic underfunding of exploitation has arguably led to the result that 53% of the +community are now dissatisfied or very dissatisfied with the balance between exploitation, operations +and development; in contrast, our 2018 exercise had 52% wanting the exploitation line to grow only if +there were new money. Astronomy’s UKRI exploitation funding gap has been partly counterbalanced +by EU funding, reflecting also our strong relationships with continental communities and beyond, so +our association to the Horizon programmes is critical, providing not just funding but also close +association with continental-scale research networks. +Several white papers (see above) mentioned the possibility of hypothecating postdoc/fellowship +funding, i.e. setting aside some PDRAs/fellows for particular science areas or to support particular +facilities. AAP believe the underlying driver for these submissions is that exploitation funding (i.e. +grants) are so chronically under-supported and that the limiting factor for the UK’s science research +capability in astronomy is human capacity; however, we suspect that allocating such a sparse resource +as postdoc/fellowship funding to particular science areas would prove too contentious and divisive to +35 See e.g. https://indico.ego-gw.it/event/199/ +Page 17 of 37 + +be implementable in practice within the core programme while maintaining community support. If +additional UKRI funding were found beyond that of the STFC core programme, then dedicated +subject-specific fellowship opportunities may prove less contentious. +Recommendation 3.2: STFC should prioritise increasing the exploitation line, which is the +community’s top priority. +Recommendation 3.3: UK association to the EU Horizon funding programme is critical for +astronomy. Replacing the funding alone would alleviate the exploitation pressure but would +disconnect the UK from continental-sized research networks. +Our white paper and consultation submissions highlighted the problem of a lack of career structure +and career opportunities in the cross-cutting areas of technology development, software, and +instrumentation. Universities can find it difficult to recruit and retain core technical teams. The white +papers suggested aspirations for early-career fellowships for instrumentation; expanding the number +of research software engineer fellowships in the STFC area; helping universities retain core technical +teams via grant assessment guidelines to facilitate employing staff on multiple grants. In the Equity, +Diversity and Inclusion section of our consultation, it was also highlighted that there is little +recognition of the software engineer / blended (software and astronomy) as a career path in an +environment, yet this is crucially important (and we may add that a large amount of the effort is done +by early-career researchers despite the whole community benefitting). The white papers aspired “to +see a visible shift of priority from the science exploitation of facilities to their delivery in order to +highlight instrumentation [and AAP would add software and project roles] as a viable career +specialism. Such a priority shift would also extend the reach of astronomy-based work to stakeholders +in other critically important domains.” We agree and see some of the underlying problems not just +being structural but also arising from the chronic underfunding of exploitation. There is no headroom. +This is not just limiting our science exploitation now, but is incurring long term, strategic costs. The +white paper submissions also correctly pointed out that the typical short-term (e.g. 3 year) project +grant durations are not commensurate with typical project lifetimes that are a factor of several longer, +creating an avoidable career precarity particularly for those in technical roles that are increasingly +important for STFC science exploitation. +Recommendation 3.4: STFC should review the career structure for instrumentation and +technical roles, both within and beyond astronomy and discuss with Universities as to how to +implement that within their structures. +3.4 Highest rated science and technology themes/facilities in the consultation +SKA (together with its pathfinders/precursors) and ESO (including ELT) remain highest rated for both +the science and technology roadmaps in our consultation. Besides the consultation response, the SKA +breadth of community support is also very well evidenced by UK participation in SKA working +groups +as well as regular town hall meetings +organised by the UK SKA Science Committee. AAP +36 +37 +are very pleased that the SKA has been awarded UKRI infrastructure funds for the development of the +UK SKA Regional Centre. ESO being among the highest community priorities is also in accord with +other obvious lines of evidence besides our consultation response, such as the recent evaluation +of +38 +UK membership of ESO, the breadth and depth of UK PI and co-I time awards +on current ESO +39 +facilities, the factor ~8 oversubscription +on Europe-node ALMA time (in which the UK is typically +40 +40 See https://almascience.eso.org/documents-and-tools/cycle9/cycle-9-proposal-submission-statistics +39 See https://www.eso.org/sci/observing/teles-alloc/all.html +38 See https://www.ukri.org/publications/socio-economic-impact-evaluation-of-the-uk-subscription-to-eso/ +37 See https://www.ukri.org/about-us/stfc/how-we-are-governed/advisory-boards/ukskasc/ +36 See https://astronomers.skatelescope.org/science-working-groups/ +Page 18 of 37 + +one of the largest national European winners of PI and co-I time), the breadth of UK involvement +in +41 +science working groups +and instruments +on the forthcoming Extremely Large Telescope (ELT), and +42 +43 +the community participation in UK ELT town hall meetings . +44 +Recommendation 3.5: STFC must maintain the UK role in the SKA and support the +development of the UK SKA Regional Centre. +Recommendation 3.6: The UK must remain a member of the European Southern Observatory +and play leading roles in its development of its world-class instrumentation, including the +second- and third-generation instrument suite of the ELT and the development of ALMA +instrumentation. +High performance and high throughput computing provision also remain a community priority. This +has received a welcome boost in 2020 in the form of £20m of capital funding from the UKRI World +Class Laboratories funding line, enabling the long-awaited deployment of the DiRAC-3 phase 1 +upgrade in 2021. The establishment of the IRIS +project, following capital grant funding from BEIS +45 +in 2018, has also provided a framework for linking the range of digital research infrastructure that +falls under the STFC remit. However, large oversubscription factors and the ever increasing data +requirements of state-of-the-art simulation codes and large-scale surveys mean that continued support +remains vital. +Recommendation 3.7: UK HPC capabilities such as DiRAC & IRIS underpin a wide range of +world-leading UK theoretical astrophysics and data science that must be continually supported +and upgraded to remain competitive. +3.5 Emerging technologies or capabilities identified by the community for +inclusion in the roadmap +The largest response was the development of improved detectors, in particular energy-sensitive +detectors (KIDS, MKIDS). Many replies highlight the importance of existing or upcoming projects. +ALMA, LOFAR, SKA, JWST, LSST, GRST are mentioned several times, while Euclid, CTA, new +instruments on the VLTs (in particular optical/NIR interferometry), gravitational wave research, +exoplanet science, high-energy instruments and improved radio telescopes were also mentioned. Big +Data and machine learning techniques were also mentioned as an opportunity. +3.6 Science challenges identified by the community as requiring major +investments +Many science challenges were identified by the community as requiring major investments, including +gravitational waves from space to Gamma-ray detectors, a new all-purpose space observatory, +supporting ESO, various space missions, studying planets, discovering life, strengthening staffing for +research, HPC and a new sub-millimetre telescope. +3.7 Equity, Diversity, Inclusion, and Careers +Science, including astronomy, does not always operate as a meritocracy, and there are many +well-recognised biases in accessibility, inclusion and career progression. Improving equality, equity , +46 +46 See e.g. https://social-change.co.uk/blog/2019-03-29-equality-and-equity +45 See https://www.iris.ac.uk/ +44 See https://www.elt-uk.org/astronomers/meetings/ +43 See https://elt.eso.org/instrument/ +42 See https://elt.eso.org/science/ +41 See https://www.elt-uk.org/astronomers/ +Page 19 of 37 + +diversity and inclusion (EDI) is therefore part of the core business of astrophysics, besides also having +a moral imperative. The consultation presented a wide range of community suggestions. See also +Recommendations 3.4 and 4.2. Some suggestions were within the remit of the Consolidated Grant +consultation process, which has now completed. Other suggestions that extend across UKRI and +beyond the remit of AAP included: +● +The career structure disadvantages some and negatively affects EDI objectives. For example: +differences in leave (including maternity) for professional staff, and no maternity leave on +grants; early career precarity, especially for those with low-income backgrounds; deficits in +support for disadvantaged/under-represented communities; the need to move overseas; the +need to improve leave systems for those with caring responsibilities, and lack of support for +families to attend conferences; ableism and the ability to travel; work-life balance; +unsafe/unhealthy working environments, i.e. not free of prejudice/abuse; insufficiently diverse +set of role models at all levels. +● +EDI progress evidence should be collected for both institutions and individuals, with +demonstrable outcomes to avoid tick boxes. EDI training should be mandatory and action +should be taken against staff who disregard EDI, while good EDI practices should be +recognised and rewarded. Why do we weight science track records highly, and not even ask +for EDI track records? +● +Research consortia should be expected to implement and adhere to their codes of conduct. +● +STFC should monitor and report on the diversity of its own advisory structures. +4. UKRI Infrastructure Funding call: AAP and +community responses +Subsequent to the consultation and white papers call, AAP solicited community input for the UKRI +Infrastructure funding “Preliminary Activity Wave 3” call in Summer 2022. Five bids were received, +of which AAP were only allowed to recommend three due to the proposal demand management: +1. +A 10-12-metre spectroscopic survey telescope. This design study aimed to prepare the way +for a very wide range of science including galactic archaeology, galaxy assembly and the +cosmic web, and transient science. This is a natural next step in Galactic and extragalactic +survey parameter space: the next generation imaging surveys by the Vera Rubin Observatory +and ESA Euclid will create targets that are too faint for 4-metre class observatories, and both +the number of targets and the near-all-sky coverage make follow-ups by current 8-10-metre +class observatories unfeasible. The project covered both the Mauna Kea Spectroscopic +Explorer and the proposed ESO Widefield Spectroscopic Telescope. This wide-field +spectroscopic capability has already been identified as a European community priority by +ASTRONET (see above), and additionally featured in the AAP call for white papers. +2. +UKELT: The next generation instrumentation suite for the Extremely Large Telescope. This +design study aimed to prepare the way for the second generation of ELT instruments: the +multi-object spectrograph MOSAIC, the high-resolution spectrograph ANDES, and the +planetary camera and spectrograph PCS. The UK is in a leading position for ELT +instrumentation, which in turn will be transformative across essentially all of UK +astrophysics. This study had previously had some of the phase A & PCS research and +development staffing costs funded by PPRP but not capital or hardware costs for the +instruments. Unlike previous VLT and first-generation ELT instruments, ESO will not provide +the hardware costs for second & third generation ELT instruments leaving the responsibility +of raising hardware funds to the instrument consortia. The project also featured in the AAP +call for white papers, further evidencing UK community interest. +3. +Scalable continuum cameras for sub-mm telescopes. This design study aimed to prepare for a +Page 20 of 37 + +first-generation instrument for the AtLAST large submillimetre-wave telescope, which in turn +has had a facility design study funded by the EU Horizon 2020 programme. In the process, +this project would also deliver a wide-field continuum camera for the JCMT, giving the UK a +tactical and strategic advantage for feeder programmes for ALMA across a wide range of +astronomy, including extragalactic surveys, Galactic star formation and protoplanetary discs. +The UK JCMT involvement has been jointly funded by PPRP and university contributions, +while AtLAST is a clear European priority across a wide range of science goals (see above) +and featured in the AAP call for white papers, further evidencing UK community interest. +4. +UK Extreme High Frequency Facility. This project aimed to create a technological +development hub for coherent detector development from radio to sub-millimetre +wavelengths. The project would support UK instrumentation interests for SKA, ALMA and +several other facilities, and would cover a wide range of science interests from star formation +to extragalactic surveys. +5. +Joint Rubin–Euclid data processing. The ground-based Rubin Legacy Survey of Space and +Time will image the whole available sky twice a week for ten years at optical wavelengths, +while the ESA Euclid mission will map around half the sky at higher resolution at optical +wavelengths, extend to near-infrared wavelengths and provide near-infrared spectroscopy. +The Rubin and Euclid consortia together have comprehensively studied +the wide-ranging +47 +science cases resulting from combining these data sets, and this project proposed to create +new value-added data products from the joint analysis. The astronomy science goals span +exoplanets, Milky Way, nearby galaxies, transient phenomena, galaxy formation and +evolution, and cosmology. +All five were excellent, and well-aligned with our roadmaps below. Numbers 1-3 (in no priority order) +were selected. (STFC then down-select all the proposals across all the advisory panels to a total of just +three to submit to UKRI.) We were advised by UKRI that our number 5 might not fit the remit of that +particular call, and number 4 was deferred to the next call based only on grounds of proposal +readiness at the time of the deadline. These submissions did however unveil two facets of the UK +community: firstly, the independent, internationally-excellent groups working on sub-mm/mm-wave +science and technology do not yet have a single consistent voice (see also section 5.5.2); secondly, the +difficulty in finding funding routes for joint facility data processing reveals a deficiency in support for +digital infrastructure, despite big data and machine learning being highlighted by the UK astronomy +community as important emerging technologies (see above). +Recommendation 4.1: STFC (via AAP) should commission a review of UK submm/mm-wave +science and technology, covering UK aspirations for current and future large single-dish +facilities that feed the major international interferometers, and the underpinning aspirations for +next-generation instrumentation, identifying areas of international excellence. +Recommendation 4.2: There should be increased support for digital infrastructure in UK +astronomy, including the creation of value-added data and software products, open science, and +the development and implementation of machine learning technologies. See also +Recommendation 3.4. +5. Science and Technology Roadmap +5.1 Overview +In constructing this roadmap, we refer both to large overarching projects (e.g. ESO, LSST, SKA) and +specific applications (e.g. joint LSST/Euclid data processing). As shown comprehensively above in +section 2, every facility supporting astronomy typically covers a very wide range of science themes, +47 See https://arxiv.org/abs/2201.03862 +Page 21 of 37 + +and the science themes themselves demand a diverse range of facilities and instrumental capabilities. +This is driven ultimately by the diversity of physical processes that occur in each science theme, and +the common ground-based technological approaches to investigating these processes throughout the +science themes (e.g. far-infrared to millimetre wavelengths for molecular astrophysics and dust; radio +astronomy for synchrotron and coherent processes and atomic gas; UV, optical and near-infrared +astronomy for ions, stellar populations and black hole accretion). Major discoveries in observational +astronomy are often driven by the opening of new observational parameter spaces and/or +developments in instrumentation. Therefore we have opted to combine the science and technology +roadmaps in a single roadmap for technology and facility needs and aspirations. +State-of-the-art instrumentation and facilities drive developments in areas throughout the STFC +“Frontier Physics” science challenges. There is a wealth and depth of key technologies where the UK +is a leading force, including: +● +Kinetic inductance detectors used for a range of aspects such as spectroscopy, imaging, +time-domain, CMB, galaxy evolution, etc. +● +Software, computing, receivers, and development for radio astronomy (Jodrell Bank +Observatory, e-MERLIN, Lovell Telescope, Square Kilometer Array). +● +Infrared detectors +● +High performance computing +AAP would add (astro)photonic technology to these community-driven suggestions. Furthermore, +many technologies have the capacity for cross-cutting or multi-disciplinary applications, such as: +areas that require mathematics and quantum devices development, time-domain astronomy, machine +learning, high performing computing, artificial intelligence, data science/big data, and +communications. Submm astronomy technology was identified by the community as having a wide +range of high priority science exploitation areas. There is a very wide range of opportunities for the +UK to play a leading role in large international astronomy projects, covering all wavelengths of +observational astronomy, including especially the Vera C. Rubin Observatory, SKA, Euclid, LISA, +and ELT. International leadership opportunities also exist in theoretical astrophysics, reflecting our +high priority technology item below. Although outside the STFC remit, there is also a very strong +community appetite for UK international leadership opportunities via space agency bilaterals (e.g. +NASA/China/Japan). +There is currently a gap in funding resources for exploratory instrumentally-focused work, compared +to that available for science exploitation and technology development with clear near-term goals. +There are also structural problems in supporting the career development of instrumentation specialists. +Some important technology roadmap conclusions were suggested directly by the community, and +which AAP endorses: +● +The strategic importance of trading unique UK technical capabilities, often mission-critical, +for a seat at the table setting the scientific agenda of a mission. +● +“Creating consistent opportunity for scientific leadership of future missions will require a +strategic plan to link university-led innovation to the longer-term resources needed to support +future major technical leadership bids.” +● +“We do not believe anyone can unambiguously identify the technologies needed to address +STFC’s key science challenges in the next decade. So to remain competitive there must be +strong UK investment in a broad program of advanced instrumentation and its supporting +technologies” which the authors suggested should include precision optomechanics, +nm-resolution metrology and both electromagnetic and non-electromagnetic sensor +development. +Page 22 of 37 + +Recommendation 5.1: There must be strong UK investment in a broad programme of advanced +instrumentation and its supporting technologies, including (but not limited to) precision +optomechanics, (astro)photonic technology, nm-resolution metrology, sensor development, +kinetic inductance detectors and infrared detectors in general, CCDs and CMOS detectors, +receiver development, low noise high-electron-mobility transistors, device fabrication +capabilities, software and computing. +AAP is pleased to note that there is now a dedicated STFC early TRL funding stream with follow-on +fund, which had its first round in 2022. This covers all the STFC core science programme but many +astronomy groups have applied . The call was associated with a capital call this year for lab +48 +equipment etc. (over £1M awarded) AAP are also pleased that STFC expect +the second joint +49 +UKSA-STFC early TRL Technology Fund for Space Science will shortly be announced offering up to +£1.6M for targeted projects in detectors, miniaturisation and robotics for science instruments, and that +it is expected this will be repeated in 2023 and 2024 with different themes. +The priorities listed below are for near- and mid-term timescales. Our consultations also asked the +community about the anticipated science themes on longer timescales. Community-identified science +themes to be addressed in the next two decades include “life elsewhere” and “exoplanet formation and +characterisation”, studies the nature and composition of Dark Matter and Dark Energy in order to +disentangle the processes involved in galaxy formation and evolution, high precision gravitational +tests of GR, gravitational waves, including different sources as well as waveform models ahead of +LISA. +5.2 Top priority science facilities +The top priority science items have wide-ranging and transformative science goals and world-leading +technology development, with strong and very wide-ranging / almost-universal community support. +5.2.1 European Southern Observatory +ESO (including ELT, ALMA) is a clear joint top priority. It has very clear evidence of widespread +community support both for near-term science (e.g. VLTs, ALMA) and mid-term (e.g. ELT first and +second generation instrumentation) with transformative science. See section 3.4, and see +Recommendation 3.6 above. The UK Programme for the Extremely Large Telescope positions a +wide portion of the UK community strategically in future decades for major science, including in the +following science and instrumentation areas: +● +Towards Earth: Direct Detection of Exoplanets with the Extremely Large Telescope Planetary +Camera and Spectrograph ELT-PCS; +● +HARMONI: the first light integral field spectrograph for the Extremely Large Telescope +● +ANDES : the High Resolution Spectrograph for the Extremely Large Telescope MOSAIC: the +multi-object spectrograph for the Extremely Large Telescope +● +METIS: The mid-IR imager and spectrograph for the Extremely Large Telescope +The latter four instruments together cover broad science cases ranging from exoplanets to the epoch of +reionization. +5.2.2 Square Kilometer Array +The SKA has very clear evidence of wide community support (including beyond ‘typical’ radio +astronomy groups) with field-changing science across a very wide range of astronomy. In 2022, SKA +Observatory was awarded £66.7 million from the Infrastructure Fund in total including future funding +years (following endorsement by AAP for funding via the “priority projects” process). See section +3.4, and see Recommendation 3.5 above. +49 Colin Vincent, priv. comm., 8 November 2022 +48 Colin Vincent, priv. comm., 8 November 2022 +Page 23 of 37 + +5.2.3 European Space Agency and other space opportunities +The ESA Science mandatory programme is formally outside the STFC remit, except e.g. in the +science exploitation of current or past missions in AGP, and occasionally in preparatory technology or +software infrastructure development for science in PPRP. Nevertheless it covers projects supporting +almost the entirety of the UK astronomy science and technology communities, with missions +comparable (or greater) in cost, impact and community support to our highest-rated ground-based +projects, e.g. JWST, Athena, Ariel, Plato, Euclid, LISA etc. As part of the “dual key” mechanism, +AAP advises STFC Science Board when UKSA requests the view of the STFC science community on +prospective astronomy missions. Our community consultations highlighted the community importance +of several major current and future space missions, despite them being in the UKSA remit. The +community support is also well-evidenced by the extensive UK signatories to the respective mission +study papers. Rather than consider these individually as with STFC-supported ground-based projects, +we have treated these collectively as a single item. At the time of writing there are a number of +external policy instabilities, and AAP would not want to send an inadvertent signal by omitting this +comparison. +There is a very wide astronomy community appetite for exploring bilateral space agreements and +other space opportunities. AAP received many excellent white paper ideas across the breadth of the +astronomy research portfolio, including: +● +The Next Generation of Space missions and their role to understand the most extreme objects +in our Universe: Black holes and Neutron Stars +● +Dark Ages and First Light: A case for space-based radio cosmology (a science case for +participation in, or UK leadership of, a range of potential future space-based 21cm cosmology +experiments) +● +The search for living worlds and the connection to our cosmic origins (a wide-ranging science +case supporting UK roles in exoplanet space missions in the NASA Decadal Review, +particularly a large optical/ultraviolet/infrared space telescope) +● +Gravitational Wave Astronomy with the Laser Interferometer Space Antenna (also within the +remit of PAAP) +● +The crucial missing piece in the multi-wavelength jigsaw of the Universe: the Far-Infrared; +Participation in a NASA Far Infrared Probe Mission. These white papers have wide-ranging +support, high science return, with strategic involvement in mission launching in over 10 years. +● +GaiaNIR: Near-Infrared Astrometry Revealing the Galactic Ecosystem +● +CASTOR: A Wide-Field, UV Space Telescope +● +UK opportunities for extrasolar research with the Twinkle space mission +● +Many white papers had extensive aspirations for use of X-ray space facilities. +These opportunities fall outside the STFC remit, but should UKSA seek the views of STFC Science +Board on the alignment of these projects with roadmap strategies, then there is already prima facie +evidence in their support. +AAP notes that there is currently some discussion of removing all proprietary time from NASA +federally funded missions +,. US PIs with NASA facility time allocations are often also awarded +50 +exploitation grant funding; in contrast, UK PIs have no such guarantee of post-doctoral support to +exploit data in these highly competitive allocations. The pressure on UK exploitation grants would +further disadvantage the UK community if this proposed policy is enacted (as well as having other +consequences). +50 “On 25 August, the White House’s Office of Science and Technology Policy ordered departments and +agencies to move toward making the results of all federally funded research freely and immediately available by +2026.” Clery, D.., 2022, Science, Volume 378, Issue 6619. +https://www.science.org/content/article/should-webb-telescope-s-data-be-open-all +Page 24 of 37 + +AAP also received a white paper submission on best practices for public engagement with UK space +missions. Recommendations included providing sustained engagement support pre- and post-launch, a +better awareness of equity, diversity and inclusion in UKSA resource allocation; better recognition of +outreach in career progression; retention of hardware created during mission development, for +museums; a central resource hub for STEM outreach; and funding for better coordination. These are +beyond the AAP remit but AAP encourages the authors to publish their recommendations and/or +submit them to the UKSA advisory structures. +Recommendation 5.2: The ESA mandatory science programme covers projects supporting +almost the entirety of the UK astronomy science and technology communities, with missions +comparable (or greater) in cost, impact and community support to our highest-rated +ground-based projects. The UK must continue ESA membership, and there is widespread +astronomy community appetite for exploring bilateral space agreements and other space +opportunities. +5.3 Very high priority science and technology +The very high priority science items have wide-ranging and important science goals and +world-leading technology development, with strong and wide-ranging community support. AAP +received a community submission in favour of Gravitational Wave Astronomy with Advanced LIGO+ +and Cosmic Explorer, and note that this very high priority area is already considered in the PAAP +roadmap. +5.3.1 High Performance Computing +High Performance Computing, in particular the DiRAC HPC Facility (including capital equipment, +operations, training & innovation) has clear UK leadership in many areas underpinning a wide range +of UK astronomy. See Recommendation 3.7 above. +5.3.2 Vera Rubin Observatory +The Vera Rubin Observatory Legacy Survey of Space and Time covers a very wide range of high +priority UK science themes (section 2) and is critical to UK leadership in many anticipated future +developments. The major role the UK is playing in the LSST will enable membership for a large +fraction of the UK astronomical community; we anticipate Rubin LSST soon achieving the +almost-universal breadth of support to rank alongside SKA and ESO. We note above in section 4 that +there is a deficit in support for digital infrastructure, such as in supporting the creation of value-added +data products from the scientific synergy with Rubin and Euclid; see Recommendation 4.2 above. +5.3.3 Simons Observatory / CMB science +Science from observations of the Cosmic Microwave Background are an obvious very high priority +with long-term historic UK leadership internationally. The large UK CMB community has a well +thought-out strategy with Simons Observatory (SO) as top mid-term priority. SO was not in the core +programme of the second BoP review due in part to lack of headroom for major new commitments, +but was endorsed by AAP for new funding under the “priority projects”; SO was awarded £12.6M of +new infrastructure funding from UKRI in 2022 over three years. The CMB community white paper +submission to AAP also describes ground-based aspirations beyond SO for CMB Stage 4 and the +UK-led CBASS experiment. Further details on CMB science priorities and further smaller facility +aspirations are covered in the PAAP roadmap. +5.4 High priority science and facilities +These are science areas and facilities that may not clearly satisfy all of the criteria for the “Very High +Priority” areas above, but include projects that could be developed to very high priority, including +Page 25 of 37 + +those with high science return over the coming 10 years. The science areas and projects listed in this +section cover a very wide range of costs, science areas and timescales, so we felt it would not be +useful to draw prioritisations among them under some hypothetical constraints. Rather, the Balance of +Programmes process (or an equivalent Astronomy-specific near-term tensioning exercise) should be +used in the light of real constraints. See Recommendation 2.1 above. +5.4.1 WEAVE +The WEAVE instrument is the next generation spectroscopy facility for the William Herschel +Telescope (WHT), built by an international STFC-led consortium. The science goals are +wide-ranging, covering Galactic archaeology, white dwarfs, neutral Hydrogen selected galaxies, +galaxy clusters, low-frequency radio sources from LOFAR, and quasars. WEAVE has also been +highlighted in several science themes of the ASTRONET roadmap (section 2.3 above). +5.4.2 The Maunakea Spectroscopic Explorer and other 10-12m-class massively multiplexed +spectroscopy facilities +MSE, or the Maunakea Spectroscopic Explorer, is a 11.25-metre optical spectroscopy facility +proposed to replace the Canada-France-Hawaii Telescope (CFHT) and sit at the location of the CFHT +in an enclosure only slightly larger than the existing dome. The science case is wide-ranging, and +spectroscopic facilities of this class are essential for the science exploitation of the coming generation +of very wide-area optical and near-infrared imaging surveys from Rubin LSST and ESA Euclid. More +details can be found in section 4. AAP supported MSE as a priority for potential UKRI infrastructure +funding, and spectroscopy facilities of this class have also been highlighted in several science themes +of the ASTRONET roadmap (section 2.3 above). +5.4.3 LOFAR, the Low Frequency Array +LOFAR is a new-generation radio telescope operating at low radio frequencies of ~15 to 240 MHz. +LOFAR is a precursor facility to the SKA, which is one of the community’s top priorities (see above). +The science areas covered by LOFAR are vast, from Solar System science to cosmological scales. 23 +UK universities are participating in the UK LOFAR consortium. LOFAR is currently operational and +highly productive, with UK leadership roles and strong UK science exploitation. LOFAR’s staged +upgrades, known as LOFAR2.0, are already in progress and the initial stages have approved STFC +funding; this will maintain LOFAR’s position as the most powerful very-low-frequency and +long-baseline radio interferometer internationally until at least 2030. LOFAR features in several +science themes of the ASTRONET roadmap (section 2.3 above). +5.4.4 VLT BlueMUSE +BlueMUSE is a panoramic 3-D spectroscopy instrument in the UV/blue for the ESO VLT, addressing +a very wide range of science from the Solar System to Galactic astronomy to redshifts of z~6. +BlueMUSE has been approved by ESO for a Phase A start in January 2023. The consortium is led +from Lyon, with the UK bidding for a key role in assembling, aligning and testing the spectrographs. +BlueMUSE features in several science themes of the ASTRONET roadmap (section 2.3 above). +5.4.5 VLT CUBES +CUBES is a near-ultraviolet spectrograph being built for the ESO VLTs, offering a much higher +throughput and sensitivity to the current state-of-the-art VLT UVES instrument (under 5 percent +throughput including telescope, atmosphere and instrument). This wavelength range is rich in +diagnostic information covering a wide range of science goals covering Solar System science, +Galactic and extragalactic astronomy. The project passed its Phase A study review with ESO in June +2021 and started construction in early 2022, with operations planned to start in 2028. CUBES is led +by Italy with the UK leading the Science and Detector work packages. CUBES also features in several +science themes of the ASTRONET roadmap (section 2.3 above). +Page 26 of 37 + +5.4.6 UK National Radio Astronomical Observatory UKNRAO +UKNRAO is a bottom-up proposal to create a national radio observatory that would be a S&T/R&D +coordinating centre for radio astronomy, a timely initiative designed to fully exploit the UK’s £200M+ +investment in the SKA. This white paper came about partly in response to previous AAP feedback on +the diversity of radio proposals in the “priority projects” process, and AAP are very pleased to see that +the radio community has engaged in careful, consultative strategic planning to create this white paper. +UKNRAO would be established as a Company Limited by Guarantee with a governance structure +similar to the highly successful Alan Turing Institute, bringing together all the major players in UK +radio astronomy. The science goals addressed by radio astronomy are very broad, as seen e.g. in +section 2, and the UK has a large and vibrant radio community. There are excellent synergies with the +UK ALMA Regional Centre co-located at one of the UKNRAO sites. The aspirations include a UK +SKA Regional Centre, recently funded by UKRI infrastructure funding and perfectly aligned with one +of the community top priorities (see above). As well as enhanced national and international standing, +UKNRAO would also host key UK instrument and technology development, including a flagship +“Receiver Factory” and the UK Extreme High Frequency Facility that was submitted to AAP for the +most recent UKRI infrastructure call (section 4). +5.4.7 Submm Astronomy on Large Scales / AtLAST +Two white papers made a strong case to AAP for wide-field mapping at sub-millimetre and millimetre +wavelengths, to provide the UK with tactical and strategic advantages in creating feeder programmes +for the heavily competitive time allocations on the ~£1.3billion ALMA facility. The broad science +goals cover star formation, the interstellar medium, galaxies and cosmology. More than 90% of the +energy output from newly formed stars and galaxies is hidden by dust and only accessible to these +facilities. ALMA excels at deep pointed observations, but even the largest ALMA programmes cover +only a few square arcminutes, so wide-field mapping is essential to feed ALMA with targets. The +largest submm telescope in the world is the James Clerk Maxwell Telescope (JCMT, with UK roles +funded by PPRP and university contributions) and is likely to remain the largest for the next decade, +until the AtLAST facility is available; the UK also has collaborative access to the Mexican Large +Millimetre Telescope. The UK has a very strong track record in submm/mm-wave astronomy, a broad +and vibrant community, and many leadership roles in JCMT Large Programs. The UK also has +international leadership in related detector technologies, such as KIDS. Wide-field (sub)-millimetre +mapping also features in several science areas of the ASTRONET roadmap (section 2.3 above). AAP +recommended the AtLAST instrument feasibility study and JCMT instrumentation as a priority for +UKRI Infrastructure funds (section 4). +5.4.8 Magdalena Ridge Observatory Interferometer, MROI +The Magdalena Ridge Observatory Interferometer MROI has a broad, transformative science +application, from Galactic star and planet formation, to precision stellar astrophysics, to AGN +formation. This white paper aspires for ten 1.4m diameter telescopes at MROI; with seven or more +telescopes the scientific capabilities of the MROI will far exceed the capabilities of both the +state-of-the-art facilities VLTI and CHARA, e.g. double the angular resolution of VLTI and triple the +imaging field of view of CHARA. The limiting sensitivity would be 2-5 magnitudes fainter than the +current state-of-the-art. The first three telescopes of MROI are currently funded by the US Air Force. +The expansion to 10 telescopes was one of the priorities for optical/IR interferometry in the 2021 US +Decadal Review. The required technology development would position the UK scientifically (see +Recommendation 5.1) and establish UK expertise for the Planet Formation Imager (timescales: after +ELT first light) and ELT first-generation instruments. +Page 27 of 37 + +5.4.9 HARPS-3 +HARPS-3 is a new generation high resolution, high stability echelle fibre spectrograph, supporting the +UK’s world-leading roles +in exoplanet science. HARPS-3 is to be installed on the Isaac Newton +51 +Telescope, and aims to conduct precise Doppler spectroscopy of exoplanet systems for the discovery +of Earth analogues. The UK-led instrument is part-funded by PPRP and will achieve first light in +2023, and full operations are planned to start in 2024. Half of the observing time will be available to +the Isaac Newton Group community. The instrument also features as an exoplanet priority in the +ASTRONET roadmap (section 2.3.5). +5.4.10 Other facility capabilities +The above high priority facilities include the projects for which dedicated white papers were +submitted in our consultation, but there are several others where comparable community support, +breadth of science goals and high science impact may also be evidenced. Examples include: UK +involvements in DESI, CCAT, 4MOST; VLT/MOONS (six-country collaboration, managed by STFC +at UK ATC); GOTO (prioritised by AAP in the previous “priority projects” process and with UK +involvement now funded by PPRP); CTA (UK technical interests in instrumentation and science +interests in multi-messenger astronomy); laboratory astrochemistry (UK international science +leadership underpinning a wide range of science themes); crowd-sourced astrophysics (UK has taken +leadership through e.g. the Zooniverse platform). Funding only the top priority facilities is not +sufficient for the UK to achieve a scientific return on the investment; the “small” facilities are an +essential part of the UK astronomy programme. A facility or instrument being “small” in cost terms is +not inconsistent with it being world-leading. +Recommendation 5.3: Maintaining UK access to the top priority facilities is not sufficient to +ensure returns on these major investments, so STFC should maintain and develop a broad +portfolio of high/very high priority science facilities, including (but not limited to) WEAVE, +LOFAR, JCMT, GOTO, LT/NRT, WASP, NGTS, DIRAC, IRIS, e-MERLIN, Lovell Telescope, +among current facilities, and UK roles in e.g. DESI, CCAT, 4MOST, BlackGEM, as well as +fostering roles in currently non-STFC-supported projects such as REACH; looking further +ahead, HARPS-3, Rubin LSST, Simons Observatory, AtLAST, MSE, CTA, VLT/MOONS. As +part of this broad portfolio supporting the highest priority science, AAP further recommends +support for laboratory astrochemistry. +5.5 Emerging community priorities +AAP received a number of white papers covering important science and/or technology developments +with a significant degree of community support, including the following. +5.5.1 Astronomy and Space Domain Awareness +This white paper highlights the importance of space situational awareness, i.e. the study of satellites +and debris in Earth orbit. In the UK alone, satellite services underpin +£360 billion of economic +52 +activity per year. ESA estimates that around one million objects larger than 1 cm are in orbit, all of +which are capable of damaging a satellite. Through the STFC 21st Century Challenge Network+, +STFC have funded the Global Network on Sustainability in Space, GNOSIS , which has successfully +53 +facilitated interdisciplinary approaches to the problems of space domain awareness. More broadly, +there is arguably good evidence that the 21st Century Challenges have been very effective for +fostering societal impact of STFC science, but this is beyond the scope of the AAP roadmaps. +53 See https://gnosisnetwork.org/ +52 See e.g. https://publications.parliament.uk/pa/cm5803/cmselect/cmsctech/100/report.html +51 See e.g. https://www.nobelprize.org/prizes/physics/2019/summary/ +Page 28 of 37 + +5.5.2 Line intensity mapping at millimetre/sub-millimetre wavelengths +The white paper on Line Intensity Mapping at millimetre/sub-millimetre wavelengths offers a +roadmap for a new and complementary probe of early universe cosmology. The science goals cover +inflation and primordial non-Gaussianity, dark energy through baryonic acoustic oscillations, the +growth of large scale structure, and fundamental physics (neutrinos and light relics). The technique +also features in the ASTRONET roadmap (see section 2.3 above). While it does not currently have the +breadth of science goals and UK community support of the high priority facilities and facility +aspirations above, the science is manifestly internationally excellent. A bespoke instrument for the +JCMT (where UK access is currently funded by PPRP and university contributions) would provide +UK leadership in this area. To support the development of an evidence base for wider community +support, AAP recommends that this is considered as part of the wider portfolio UK science and +technology in (sub)-millimetre astronomy; see Recommendation 4.1 above. +5.5.3 Development in Africa with Radio Astronomy, DARA +The Development in Africa with Radio Astronomy programme, DARA +is a radio +54 +astronomy training project funded by international development funding, through the Global +Challenges Research Fund and the Newton Fund, as well as securing matching funding from South +Africa’s NRF. DARA has also been supported by the EU Jumping JIVE project. Since 2015 it has +been delivering basic training in radio astronomy and associated technologies to young people across +the eight DAC-list countries partnering with South Africa in hosting SKA-MID. A sister programme, +DARA Big Data, has provided MSc and PhD-level advanced training in Africa and in UK +universities. AAP commends the creativity and resourcefulness of the DARA team in using +astronomy training to address societal challenges in DAC-list countries, but also notes the anticipated +cuts to UK ODA funding that will severely curtail these activities. The white paper asks STFC to +continue its strategic partnership +with NRF in training and placements of researchers and +55 +technicians, to which AAP is happy to concur; the five-year agreement started in 2021. +5.5.4 Collaboration with Goonhilly Earth Station Ltd on Radio Astronomy and Space Applications +This white paper was submitted on behalf of the CUGAS consortium, the Consortium of Universities +for Goonhilly Astronomy and Space. The science goals covered in this collaboration include radio +astronomy, interferometry, satellite communications, and space situational awareness. The project +seeks to integrate the Goonhilly dishes into e-MERLIN, providing much needed long baselines and +higher angular resolutions, while state-of-the-art low noise instrumentation benefits commercial +satellite communication roles. Discussions are underway on providing a sovereign UK space +situational awareness capability. Further funds are needed, either from CUGAS members or from +STFC, for the cost of the data link and hydrogen maser time standard. AAP commend the CUGAS +consortium for their creativity and resourcefulness in pursuing this industry engagement. +5.5.5 High-risk, high return science areas +AAP were very pleased to receive a white paper on searching for techno-signatures via anomalies in +astronomical data, and are happy to record the community support for this area. A proportion of +resources must always be spent on high-risk, high-return science. +5.7 Projects with a moderate or low science return +No projects submitted to AAP fall into this category, whether through the consultation or white papers +or the call for UKRI infrastructure projects. +55 See https://www.ukri.org/news/forging-a-new-partnership-with-south-africa/ +54 See www.dara-project.org +Page 29 of 37 + +6. Scope for further community consultation +AAP have considered whether there is scope and/or community capacity for further major +consultations at this stage, following the model e.g. of the PPAP review of the entirety of UK particle +physics. With the ASTRONET review still in the process of reporting the last of its findings on the +science roadmap for European astronomy (section 2.3), and with the US Decadal Review recently +completed (section 2.2), there does not appear to be an obviously compelling case for a UK-specific +review of all astronomy, because it would duplicate at least some of this recent and ongoing effort. +Furthermore, the reduced survey return rate in the AAP consultation may indicate a decreased UK +astronomy community capacity or appetite for further consultation activities, though the global health +crisis may also have been a factor. Nevertheless, there are some focussed areas where further +consultations may be useful, such as an overview of community aspirations and priorities in +sub-mm/mm-wave science and technology (Recommendation 4.1), and an STFC review of the career +structure of instrumentation and technical roles both within and beyond astronomy +(Recommendation 3.4). The UK CMB community have already organised their own bottom-up +community roadmap, updated in the CMB community white paper submitted to AAP and also +addressed in the PAAP roadmaps. The UK exoplanet community last had an STFC community +consultation +in 2015, and our white paper submissions clearly evidence a desire of the UK exoplanet +56 +community to develop a stellar variability and extreme precision radial velocity roadmap in liaison +with other international bodies such as ESA, ESO, NASA, NSF etc. (Recommendation 3.1); such a +roadmap would cover areas of interest of both AAP and SSAP, as well as UKSA. Radio astronomy +was last reviewed +by STFC in 2017 following the 2016 STFC Balance of Programmes Review, and +57 +among several conclusions recommended a review of e-MERLIN from 2022; the radio community +has since created a vision for a UKNRAO (section 5.4.6) so the landscape has changed since the 2017 +review. Furthermore, the JIVE ERIC, linked to the pursuance of e-MERLIN, is currently three years +into its five-year window; e-MERLIN is nominally operational at least until SKA is operational and +the role for other radio contributions can be reviewed. +7. Summary list of recommendations +Recommendation 2.1: The 2022 AAP Science and Technology Roadmaps should not be used in +isolation to evidence current community support without further reference to AAP, because +astronomy is a fast-paced and rapidly changing field. Nor should these roadmaps be used to +draw a funding threshold line in the event of changes to the core programme funding, because +the Balance of Programmes Review (or a hypothetical Astronomy-specific near-term tensioning +exercise) is the appropriate process for that. +Recommendation 3.1: STFC and UKSA should support the UK exoplanet community in +developing a stellar variability and extreme precision radial velocity roadmap in liaison with +other international bodies such as ESA, ESO, NASA, NSF etc. +Recommendation 3.2: STFC should prioritise increasing the exploitation line, which is the +community’s top priority. +57 See https://www.ukri.org/about-us/stfc/planning-strategy-reviews/2017-uk-radio-astronomy-strategic-review/ +56 See https://www.ukri.org/about-us/stfc/planning-strategy-reviews/exoplanets-uk-research-review/ +Page 30 of 37 + +Recommendation 3.3: UK association to the EU Horizon funding programme is critical for +astronomy. Replacing the funding alone would alleviate the exploitation pressure but would +disconnect the UK from continental-sized research networks. +Recommendation 3.4: STFC should review the career structure for instrumentation and +technical roles, both within and beyond astronomy and discuss with Universities as to how to +implement that within their structures. +Recommendation 3.5: STFC must maintain the UK role in the SKA and support the +development of the UK SKA Regional Centre. +Recommendation 3.6: The UK must remain a member of the European Southern Observatory +and play leading roles in its development of its world-class instrumentation, including the +second- and third-generation instrument suite of the ELT and the development of ALMA +instrumentation. +Recommendation 3.7: UK HPC capabilities such as DiRAC & IRIS underpin a wide range of +world-leading UK theoretical astrophysics and data science that must be continually supported +and upgraded to remain competitive. +Recommendation 4.1: STFC (via AAP) should commission a review of UK submm/mm-wave +science and technology, covering UK aspirations for current and future large single-dish +facilities that feed the major international interferometers, and the underpinning aspirations for +next-generation instrumentation, identifying areas of international excellence. +Recommendation 4.2: There should be increased support for digital infrastructure in UK +astronomy, including the creation of value-added data and software products, open science, and +the development and implementation of machine learning technologies. See also +Recommendation 3.4. +Recommendation 5.1: There must be strong UK investment in a broad programme of advanced +instrumentation and its supporting technologies, including (but not limited to) precision +optomechanics, (astro)photonic technology, nm-resolution metrology, sensor development, +kinetic inductance detectors and infrared detectors in general, CCDs and CMOS detectors, +receiver development, low noise high-electron-mobility transistors, device fabrication +capabilities, software and computing. +Recommendation 5.2: The ESA mandatory science programme covers projects supporting +almost the entirety of the UK astronomy science and technology communities, with missions +comparable (or greater) in cost, impact and community support to our highest-rated +ground-based projects. The UK must continue ESA membership, and there is widespread +astronomy community appetite for exploring bilateral space agreements and other space +opportunities. +Recommendation 5.3: Maintaining UK access to the top priority facilities is not sufficient to +ensure returns on these major investments, so STFC should maintain and develop a broad +portfolio of high/very high priority science facilities, including (but not limited to) WEAVE, +LOFAR, JCMT, GOTO, LT/NRT, WASP, NGTS, DIRAC, IRIS, e-MERLIN, Lovell Telescope, +among current facilities, and UK roles in e.g. DESI, CCAT, 4MOST, BlackGEM, as well as +fostering roles in currently non-STFC-supported projects such as REACH; looking further +ahead, HARPS-3, Rubin LSST, Simons Observatory, AtLAST, MSE, CTA, VLT/MOONS. As +Page 31 of 37 + +part of this broad portfolio supporting the highest priority science, AAP further recommends +support for laboratory astrochemistry. +8. Appendix: List of acronyms +4MOST: 4-metre Multi-Object Spectroscopic Telescope +AAP: Astronomy Advisory Panel +AGP: Astronomy Grants Panel +ALMA: Atacama Large Millimetre Array +ANDES: ArmazoNes high Dispersion Echelle Spectrograph (formally known as HIRES) +AO: Adaptive Optics +APEX: Atacama Pathfinder Experiment +APPEC: Astroparticle Physics European Consortium +ASTRONET: a network of European funding agencies and research organisations in astronomy (not +an acronym, but capitalised) +ASTRI: Astrofisica con Specchi a Tecnologia replicante Italiana +ATC: Astronomy Technology Centre +AtLAST: Atacama Large Aperture Submillimeter Telescope +BINGO: Baryon Acoustic Oscillations from Integrated Neutral Gas Observations +BiSON: Birmingham Solar Oscillations Network +BISOU: Balloon Interferometer for Spectral Observations of the Universe +BlackGEM: a wide-field array of optical telescopes to be located at ESO’s La Silla Observatory in +Chile’s Atacama desert (not an acronym ) +58 +BlueMUSE: Blue Multi Unit Spectroscopic Explorer +BoP: Balance of Programmes +CAHA: Centro Astronómico Hispano-Alemán +CARMENES: Calar Alto high-Resolution search for M dwarfs with Exoearths with Near-infrared and +optical Échelle Spectrographs +CBASS: C-Band All Sky Survey +CCAT: Cerro Chajnantor Atacama Telescope +CCD: Charge Coupled Device +CFHT: Canada-France-Hawaii Telescope +CHARA: Center for High Angular Resolution Astronomy +CMB: Cosmic Microwave Background +58 See https://www.eso.org/public/about-eso/acronyms/ +Page 32 of 37 + +CMB-S4: Stage Four Cosmic Microwave Background experiment +CMOS: Complementary metal–oxide–semiconductor +Co-I: Co-investigator +CONCERTO: Carbon CII line in post-reionisation and reionisation epoch +CRIRES, CRIRES+: Cryogenic high-resolution cross-dispersed infrared echelle spectrograph +CSA: Canadian Space Agency +CTA: Cherenkov Telescope Array +CUBES: Cassegrain U-Band Efficient Spectrograph +CUGAS: Consortium of Universities for Goonhilly Astronomy and Space +DESI: Dark Energy Spectroscopic Instrument +DiRAC: Distributed Research using Advanced Computing +DKIST: Daniel K. Inouye Solar Telescope +EAGLE: Evolution and Assembly of GaLaxies and their Environments +EC: European Commission +EDI: Equity, Diversity and Inclusion +EHT: Event Horizon Telescope +ELT: Extremely Large Telescope +e-MERLIN: enhanced Multi Element Remotely Linked Interferometer Network +EOSC: European Open Science Cloud +ERIC: European Research Infrastructure Consortium +ERIS: Enhanced Resolution Imager and Spectrograph +ESA: European Space Agency +ESO: European Southern Observatory +ESPRESSO: Echelle SPectrograph for Rocky Exoplanets and Stable Spectroscopic Observations +EVN: European VLBI Network +FAIR: Findable, Accessible, Interoperable, Reusable +FRB: Fast Radio Burst +GMRT: Giant Metrewave Radio Telescope +GNOSIS: Global Network on Sustainability in Space +GOTO: Gravitational-wave Optical Transient Observer +GRAND: Giant Radio Array for Neutrino Detection +GranTeCan: Gran Telescopio Canarias +Page 33 of 37 + +GRAVITY, GRAVITY+: a 4-beam interferometric combiner at VLTI operating in K-band (not an +acronym ) +59 +GRST: Fermi Gamma-Ray Space Telescope +GTM: Gran Telescopio Milimétrico Alfonso Serrano +GW: Gravitational Wave +HARMONI: High Angular Resolution Monolithic Optical and Near-infrared Integral field +spectrograph +HARPS, HARPS-3: High Accuracy Radial velocity Planet Searcher +HARPS-N: High Accuracy Radial velocity Planet Searcher, North +HERA: Hydrogen Epoch of Reionization Array +HESS: High Energy Stereoscopic System +HIRES: high-resolution spectrograph (ELT instrument now known as ANDES) +HiRISE: High-Resolution Imaging and Spectroscopy of Exoplanets +HPC: High Performance Computing +IMF: Initial Mass Function +ING: Isaac Newton Group +INT: Isaac Newton Telescope +IRIS: eInfrastructure for Research and Innovation in STFC +JCMT: James Clerk Maxwell Telescope +JIVE: Joint Institute for VLBI ERIC +JWST: NASA/ESA/CSA next-generation infrared space telescope. (Following the provisional best +practice outlined by the Royal Astronomical Society , we do not expand the acronym.) +60 +JVLA: Jansky Very Large Array +KAGRA: Kamioka Gravitational Wave Detector +KIDS: Kinetic Inductance Detectors +LGBTQI+: Lesbian, Gay, Bisexual, Transgender, Queer, Intersex +LHC: Large Hadron Collider +LIGO: Laser Interferometer Gravitational-Wave Observatory +LISA: Laser Interferometer Space Antenna +LMT: Large Millimetre Telescope (in Spanish: GTM, Gran Telescopio Milimétrico Alfonso Serrano) +LOFAR: Low Frequency Array +LSPE-strip: Large-Scale Polarization Explorer/Survey Tenerife Polarimeter +60 See https://ras.ac.uk/news-and-press/news/ras-and-jwst +59 See https://www.eso.org/public/about-eso/acronyms/ +Page 34 of 37 + +LSPE-swipe: Large-Scale Polarization Explorer/Short Wavelength Instrument for the Polarization +Explorer +LSST: Legacy Survey of Space and Time +LT: Liverpool Telescope +MAGIC: Major Atmospheric Gamma Imaging Cherenkov telescopes +MATISSE: Multi AperTure mid-Infrared SpectroScopic Experiment +MAVIS: MCAO-Assisted Visible Imager and Spectrograph +MCAO: Multi-Conjugate Adaptive Optics +MeerKAT: originally the Karoo Array Telescope (not an acronym) +MeerLICHT: the Dutch translation for ‘more light’, is an astronomical project which aims to provide a +simultaneous, real-time optical view of the radio (transient) sky as observed by MeerKAT. (Not an +acronym) +MeerTime: a five-year pulsar timing program on the MeerKAT array (not an acronym) +METIS: Mid-Infrared ELT Imager and Spectrograph +MICADO: Multi-AO Imaging Camera for Deep Observations +MIDI: Mid-infrared interferometric instrument +MKIDS: Microwave Kinetic Inductance Detectors +MOSAIC: Multi-Object Spectrograph (for ELT) +MOONS: Multi-Object Optical and Near-infrared Spectrograph +MROI: Magdalena Ridge Observatory Interferometer +MSE: Mauna Kea Spectroscopic Explorer +MUSE: Multi Unit Spectroscopic Explorer +MWA: Murchison Widefield Array +NASA: National Aeronautics and Space Administration +NEID: NN-explore Exoplanet Investigations with Doppler spectroscopy +ngEHT: Next generation Event Horizon Telescope +NGTS: Next-Generation Transit Survey +ngVLA: Next Generation Very Large Array +NIR: Near Infra-Red +NIRPS: Near Infra Red Planet Searcher +NN-Explore: NASA-NSF Explore, the joint exoplanet programme of NASA and the NSF +NRT: New Robotic Telescope +NSF: National Science Foundation +NTT: New Technology Telescope +Page 35 of 37 + +PAAP: Particle Astrophysics Advisory Panel +PCS: Planetary Camera and Spectrograph +PDRA: Post-Doctoral Research Associate +PI: Principal Investigator +PFS: Prime Focus Spectrograph +PPRP: Projects Peer Review Panel +QUBIC: Q-U Bolometric Interferometer for Cosmology +QUIJOTE: Q-U-I JOint TEnerife CMB experiment +REACH: Radio Experiment for the Analysis of Cosmic Hydrogen +RISTRETTO: high-Resolution Integral-field Spectrograph for the Tomography of Resolved +Exoplanets Through Timely Observations61 +SDO: Solar Dynamics Observatory +SDSS: Sloan Digital Sky Survey +SDSS-V: the fifth Sloan Digital Sky Survey +SKA: Square Kilometer Array +SO: Simons Observatory +SPHERE: Spectro-Polarimetic High contrast imager for Exoplanets REsearch +SPIROU: SPectropolarimètre InfraROUge +SSAP: Solar System Advisory Panel +STFC: Science and Technology Facilities Council +TRL: Technology Readiness Level +TNG: Telescopio Nazionale Galileo +uGMRT: upgraded Giant Metrewave Radio Telescope +UK: United Kingdom +UKNRAO: UK National Radio Astronomical Observatory +UKRI: UK Research and Innovation +UKSA: UK Space Agency +US: United States +UV: Ultraviolet +UVES: Ultraviolet and Visual Echelle Spectrograph +WASP: Wide-Angle Search for Planets +WEAVE: WHT Enhanced Area Velocity Explorer +61 See e.g. https://zenodo.org/record/3356296 +Page 36 of 37 + +WHT: William Herschel Telescope +WIYN: Telescope operated by the University of Wisconsin–Madison (W), Indiana University (I), Yale +University (Y), and the National Optical Astronomy Observatories (N). +VISTA: Visible and Infrared Survey Telescope for Astronomy +VLA: Very Large Array +VLBI: Very Long Baseline Interferometry +VLT: Very Large Telescope +VLTI: Very Large Telescope Interferometer +Page 37 of 37 + diff --git a/qdE5T4oBgHgl3EQfJQ70/content/tmp_files/load_file.txt b/qdE5T4oBgHgl3EQfJQ70/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7079425d3e4a0586debdbed3b4881ed41cfd96d5 --- /dev/null +++ b/qdE5T4oBgHgl3EQfJQ70/content/tmp_files/load_file.txt @@ -0,0 +1,1329 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf,len=1328 +page_content='STFC Astronomy Advisory Panel Roadmap 2022 Stephen Serjeant (Open University, chair), James Bolton (University of Nottingham), Poshak Gandhi (University of Southampton), Ben Stappers (University of Manchester), Paolo Mazzali (Liverpool John Moores University), AprajitaVerma (University of Oxford), Noelia E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Noël (University of Surrey) STFC Astronomy Advisory Panel Roadmap 2022 Astronomy Advisory Panel: Stephen Serjeant (Open University, chair), James Bolton (University of Nottingham), Poshak Gandhi (University of Southampton), Ben Stappers (University of Manchester), Paolo Mazzali (Liverpool John Moores University), Aprajita Verma (University of Oxford), Noelia E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Noël (University of Surrey) The cover image shows the first image of Sgr A*, the supermassive black hole at the centre of our Galaxy , , , , , , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' UK astronomers are part of the Event Horizon Telescope team that made this image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 1 2 3 4 5 6 7 The observatories involved include the James Clerk Maxwell Telescope, with UK access funded by PPRP and UK university contributions, and ALMA, where UK access is through ESO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' UK ALMA users in particular are also supported by the STFC-funded UK ALMA Regional Centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Image credit: Event Horizon Telescope collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 7 The Event Horizon Telescope Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5/12/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' “First Sagittarius A* Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Testing the Black Hole Metric .” The Astrophysical Journal Letters, 930, L17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3847/2041-8213/ac6756 6 The Event Horizon Telescope Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5/12/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' “First Sagittarius A* Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Testing Astrophysical Models of the Galactic Center Black Hole.” The Astrophysical Journal Letters, 930, L16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3847/2041-8213/ac6672 5 The Event Horizon Telescope Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5/12/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' “First Sagittarius A* Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Variability, Morphology, and Black Hole Mass.” The Astrophysical Journal Letters, 930, L15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3847/2041-8213/ac6736 4 The Event Horizon Telescope Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5/12/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' “First Sagittarius A* Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Imaging of the Galactic Center Supermassive Black Hole .” The Astrophysical Journal Letters, 930, L14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3847/2041-8213/ac6429 3 The Event Horizon Telescope Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5/12/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' “First Sagittarius A* Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' EHT and Multiwavelength Observations, Data Processing, and Calibration .” The Astrophysical Journal Letters, 930, L13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' https://10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='7/2041-8213/ac6675 2 The Event Horizon Telescope Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5/12/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' “First Sagittarius A* Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The Shadow of the Supermassive Black Hole in the Center of the Milky Way.” The Astrophysical Journal Letters, 930, L12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3847/2041-8213/ac6674 1See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ukri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/news/first-image-of-black-hole-in-the-centre-of-our-galaxy-unveiled/ Page 1 of 37 STFC Astronomy Advisory Panel Roadmap 2022 Astronomy Advisory Panel: Stephen Serjeant (Open University, chair), James Bolton (University of Nottingham), Poshak Gandhi (University of Southampton), Ben Stappers (University of Manchester), Paolo Mazzali (Liverpool John Moores University), Aprajita Verma (University of Oxford), Noelia E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Noël (University of Surrey) Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Executive Summary 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' International context and consultation process 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 Background of previous AAP reviews 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 US Decadal Review 2020-1: priorities and synergies with UK interests 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 European ASTRONET review 2021-2022: priorities and synergies with UK interests 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 Context and background 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 Origin and Evolution of the Universe 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 Formation and Evolution of Galaxies 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 Formation and Evolution of Stars 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5 Formation and Evolution of Planetary Systems 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='6 Understanding the solar system and conditions for life 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='7 Computing - big data, high performance computing, data infrastructure 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='8 Societal aspects – education, public engagement climate action, equality, diversity and inclusion 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='9 Extreme astrophysics 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 Subsequent UK developments and the scope of the AAP roadmaps 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Summary of the 2021-2022 AAP Community Consultation 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 Overview of the process 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 Overview of white papers 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 Top, overarching community priority in the consultation 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 Highest rated science and technology themes/facilities in the consultation 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5 Emerging technologies or capabilities identified by the community for inclusion in the roadmap 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='6 Science challenges identified by the community as requiring major investments 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='7 Equity, Diversity, Inclusion, and Careers 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' UKRI Infrastructure Funding call: AAP and community responses 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Science and Technology Roadmap 21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 Overview 21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 Top priority science facilities 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 European Southern Observatory 23 Page 2 of 37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 Square Kilometer Array 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 European Space Agency and other space opportunities 24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 Very high priority science and technology 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 High Performance Computing 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 Vera Rubin Observatory 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 Simons Observatory / CMB science 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 High priority science and facilities 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 WEAVE 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 The Maunakea Spectroscopic Explorer and other 10-12m-class massively multiplexed spectroscopy facilities 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 LOFAR, the Low Frequency Array 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 VLT BlueMUSE 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5 VLT CUBES 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='6 UK National Radio Astronomical Observatory UKNRAO 27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='7 Submm Astronomy on Large Scales / AtLAST 27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='8 Magdalena Ridge Observatory Interferometer, MROI 27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='9 HARPS-3 28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='10 Other facility capabilities 28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5 Emerging community priorities 28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 Astronomy and Space Domain Awareness 28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 Line intensity mapping at millimetre/sub-millimetre wavelengths 29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 Development in Africa with Radio Astronomy, DARA 29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 Collaboration with Goonhilly Earth Station Ltd on Radio Astronomy and Space Applications 29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5 High-risk, high return science areas 29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='7 Projects with a moderate or low science return 29 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Scope for further community consultation 30 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Summary list of recommendations 30 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Appendix: List of acronyms 32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Executive Summary These science and technology roadmaps provide a brief overview of UK Astrophysics research, areas of strength, opportunities for growth and areas for strategic investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Observational and theoretical astronomy are fast-paced fields in which progress depends on a broad range of long-term, large-scale facility investments, from international observatories to high-performance computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' UK astronomy performs exceptionally well internationally on a wide variety of metrics (for example: for refereed astronomy papers published in 2000-2020, those with UK co-authors received 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='7M citations , 8 compared to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1M co-authored from the USA from a community approximately 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 times 9 9 Source: NASA Astrophysics Data System 8 Source: NASA Astrophysics Data System Page 3 of 37 larger ), at least some of which may be attributable to strategic positioning within a wide range of 10 long-term international projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The Astronomy Advisory Panel have therefore consulted with the UK community to compile roadmaps that capture at least some of the breadth and depth of the current community aspirations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' However, it is equally important to emphasise what these roadmaps are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Firstly, in a significant change to the advisory panel remit since the last major roadmap refresh in 2012, the tensioning between current spending priorities is now explicitly conducted through the advice of Balance of Programmes reviews (BoP), the last of which was in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The roadmaps are not intended as a 11 substitute for this process (or any potential astronomy-specific equivalent), but rather to highlight current aspirations for hypothetical future spends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Secondly, the fast-paced nature of the field implies that the aspirations are subject to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' For example, some of our community consultation responses were explicit that it is impossible to identify unambiguously the technologies needed to address STFC’s key science challenges in the coming ten years, and that therefore the only defensible strategic choice that maximises the probable science return is to favour the breadth of the technology programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Finally, these roadmaps must not be used as the sole justification for the allocation of new funding resources, without discussion with advisory panels about the consequences for other parts of the programme and consultation on any subsequent developments in astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The top priority of the UK astronomy community remains exploitation grants, despite a very welcome recent uplift of £2M year on year (leading to a £6M increase by 2024/25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This priority has been consistent over the past ten years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Flat-cash STFC funding, combined with the increasing costs of facility funding required to maintain a world-class astronomy programme, has resulted in a radical erosion of PDRA funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' At current levels, a typical research-active academic might secure postdoc funding three times in an entire career.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Not only is this a bottleneck for science exploitation, it drives an unplanned and potentially inequitable attrition of early-career talent from UK astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' There is evidence presented in the roadmaps that the community view of the balance of funding is evolving in response to this chronic underfunding of exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Exploitation funding aside, UK astronomy has benefitted from long-term strategic facility and technology funding that has driven a very wide range of high-profile science results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Unlike for example particle physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' astronomy has a wide and diverse range of science goals from the microscopic processes in astrochemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' to exoplanetary science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' to star formation and evolution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' to the evolution of galaxy populations and the large scale structure of the Universe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' to the properties of dark matter and dark energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' to the fundamental cosmological parameters and theories of gravity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' and more,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' all driven by UK access to world-leading facilities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' most of which address a very broad range of these science goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The diversity in this portfolio of facilities is inevitable and essential, and is driven in part by the wide range of physical processes that dominate in the various parts of the electromagnetic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Currently, the UK astronomy community aspirations continue to follow a similar pattern of strongly supporting these long-term investments, and have led to a very clear set of recommendations for sustaining a vigorous timeline of UK astronomy capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Among the top community priorities is the Square Kilometer Array (SKA), which is supported by a very broad section of the community and which promises transformative progress across a very wide range of STFC priority areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' A high community priority is the UK role in the Low Frequency Array (LOFAR), which as a pathfinder to the SKA is already producing world-leading survey results, and in which the UK is very well positioned for future success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The UK involvement in the European Southern Observatory (ESO) is critical to UK astronomy, and is another of our top community priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ESO’s Very Large Telescopes (VLT) provides critical capability across almost the entirety of the UK astronomy programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ESO’s Extremely Large Telescope (ELT) is a top priority for the long term UK astronomy capability and for nearer-term technology development, and will clearly 11 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ukri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/wp-content/uploads/2022/03/STFC-210322-BalanceOfProgrammeExercise2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='pdf 10 Source: International Astronomical Union, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='iau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/public/themes/member_statistics/ Page 4 of 37 revolutionise astronomy throughout the UK programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Through ESO, the UK also has access to the ~£1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3Bn Atacama Large Millimetre Array (ALMA) in which the UK submm/mm-wavelength community have capitalised on their strong heritage in securing competitive allocations, supported by the UK ALMA Regional Centre and with many programmes fed by the UK role in the James Clerk Maxwell Telescope (JCMT) (funded via PPRP and university contributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The UK also plays leading roles in building facilities and/or associated instruments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ELT, VLT, SKA, ALMA, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ), often strategically placing the UK to trade unique technical capabilities, often mission-critical, for a seat at the table setting the scientific agenda of a mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' UKSA is responsible for much of the space mission programme, while the science strategy, early technology, data challenges and exploitation are all within the remit of STFC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' nevertheless, the roadmaps would be incomplete if they failed to note that the ESA Science mandatory programme covers projects supporting almost the entirety of the UK astronomy science and technology communities, with missions comparable (or greater) in cost, impact and community support to our top-rated ground-based projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' There is a very strong track record in astronomy of major discoveries following the opening of new astronomical parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Time-domain astronomy is an excellent topical example, not just with follow-ups of gravitational wave events (see PAAP roadmaps), fast radio bursts, gamma-ray bursts, supernovae and other transients, but also in the explosion of interest and UK world-leadership in the field of exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The Vera Rubin Observatory Legacy Survey of Space and Time (Rubin LSST) will image the whole available sky twice a week for ten years at optical wavelengths, and is one of the community’s clear Very High priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' A further Very High priority driven by the UK’s world-leading cosmic microwave background (CMB) community is the UK role in the Simons Observatory, which has been funded outside the STFC core programme through a UKRI infrastructure call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This is the top priority of the UK CMB community and builds on a very strong heritage of international leadership in CMB cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Also in the Very High priority science category, and Very High in technology priority (our highest categorisation for technology) is UK capability in High Performance Computing (HPC), which underpins a wide range of theoretical astrophysics from planet formation to cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' There is also support for many smaller facility projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Here, ‘small’ should not be equated with ‘not world-leading’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' there is no place in these roadmaps for any facility that is not world leading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' While these projects may not enjoy the scientific breadth and almost-universal community involvement of our top priorities, they nevertheless offer the prospect of transformative progress in particular areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' In many cases it is not enough for there to be only access to the largest international facilities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' national access to world-leading smaller specialist facilities can provide feeder programmes to the larger facilities and provide the UK with tactical and strategic advantages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' JCMT feeding ALMA and e-MERLIN feeding SKA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' See Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' In summary, astronomy in all its forms is a UK strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The UK strength is across the breadth of the field, from planetary science to cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' There is a clear consensus on the key science questions to be addressed, and these require access to a wide range of facilities and capabilities, many of which address multiple key science questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' International partnerships (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ESA, ESO, SKA and bilateral programmes) are extremely important in order for the UK to play leading roles in international science teams, including across Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' A strong relationship with UKSA and ESA, and engagement with their plans, is also essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The UK astronomy programme is naturally split across several STFC advisory panels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP, SSAP, PAAP), but there are many synergies, in terms of the need for access to exploitation funds, the support for theoretical studies, training & skills, computing resources, instrumentation and technology development, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The advisory panels are therefore not in competition but rather are synergistic, and where there are overlapping interests there is concordance over the science and technology aspirations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Page 5 of 37 Figure 1: Schematic Gantt chart of a selection of significant facilities , , , , , , , , , featured in 12 13 14 15 16 17 18 19 20 21 the AAP Roadmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Timescales are indicative only and dependent on a wide variety of factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Project end dates do not indicate AAP decisions but in many cases reflect current review dates at the time of writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Only the facilities themselves are shown, and not individual instruments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' WEAVE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 21 ELT timescales adapted from https://elt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/about/timeline/ 20 MSE: timescales adapted from White Paper submitted to AAP 19 GOTO, LT, NRT: adapted from indicative timescales and current funding commitments from C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Vincent, priv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=', November 2022, without precluding further future funding 18 For Simons Observatory and CMB-S4, see PAAP roadmap 17 JCMT timescale based on currently awarded STFC funding, without precluding further funding in future https://gtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ukri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/projects?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ref=ST%2FV000268%2F1 16 AtLAST timescales based on https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/abs/2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='07974 with a purely illustrative ~6-year construction (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' approximately 4 years for JCMT, 8 for ALMA to achieve early science, approximately 9 for LMT) 15 Rubin timeline adapted from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='lsst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/about/project-status 14 e-MERLIN timescales of existing funding from Simon Garrington, priv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=', November 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' there is no horizon for the ERIC as contributions are expected to be renewed by the current partners and new partners are being sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 13 LOFAR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='0 timescales from Philip Best, priv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=', November 2022: Nov 7th 2022: Formal sign-off by International LOFAR Telescope Board on procurement contracts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' March 2023: delivery of equipment for three stations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' March 2024: system commissioning/testing with 3-station array;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Sept 2024 to mid-2025: full system roll-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 12 SKA timeline adapted from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='skao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='int/en/explore/construction-journey and https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='skao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='int/en/science-users/159/scientific-timeline Page 6 of 37 Key Design Construction Operations SKA RADIO LOFAR/LOFAR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='0 e-MERLIN Simons Observatory CMB / (sub)mm CMB-S4 ALMA AtLAST CMT Rubin UV, Optical, near-IR MSE ELT GOTO T NRT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' International context and consultation process 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 Background of previous AAP reviews AAP conducted a call for white papers and consultation exercise with the UK astronomy community via an online survey in November 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This was advertised via astrocommunity and other routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' We greatly appreciate the help of our STFC colleagues in assembling the survey, providing us with the results promptly in a digestible format and in providing other contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The results of the consultation are described in detail elsewhere , and summarised in Section 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 22 The AAP science roadmap last had a major update in 2012, and the technology roadmap is even 23 further outdated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' In the meantime, there have been minor updates following AAP’s 2016 roadmap consultation , and its 2018 consultation to feed into the second STFC Balance of Programmes 24 review .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The non-confidential parts of AAP’s 2018 consultation report were published in 25 26 Astronomy and Geophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP nevertheless opted to delay its most recent consultation and report for several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Firstly, the global health crisis meant that the capacities of both the panel and the wider community were limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Secondly, the outcomes of the European ASTRONET review could affect the policy decisions for AAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Finally, the US Decadal Review outcomes would also be likely to affect the AAP roadmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Both the ASTRONET and US Decadal reviews were also delayed partly due to the global health crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP do not seek to duplicate these roadmaps, but rather present the UK-specific community voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 US Decadal Review 2020-1: priorities and synergies with UK interests The US Decadal Review is a comprehensive survey of US community aspirations in both ground-based and space-based astronomy for not just the coming decade but also outlining a roadmap for future decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' In ground-based astronomy, the review prioritises participation in extremely large telescopes (30 metre class), the next generation Stage 4 cosmic microwave background (CMB) experiments, the next generation Very Large Array radio telescope ngVLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Unlike the UK, the US is not a participant in the Square Kilometer Array (SKA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' instead the focus is on US-specific complementary higher frequency Northern hemisphere capabilities in ngVLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The US prioritisation of extremely large telescopes is very similar to the UK community’s prioritisation of science opportunities and technical developments in the European Southern Observatory’s Extremely Large Telescope, ELT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The coming decades of ground-based optical and near-infrared astronomy will clearly be dominated internationally by facilities of this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The CMB aspirations are very well aligned with those in the UK, and UK participation in the US-led Simons Observatory is now funded (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The review also prioritised the Vera Rubin Observatory, which also features as a UK very high priority (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' US priorities in astroparticle physics include upgrading the neutrino detector IceCube and gravitational wave technology development, though in UK terms these are overlapping with PAAP interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The review calls for an expansion in grants (referred to as the exploitation line in STFC terminology), supporting data archives and curation, bolstering theory underpinnings, 26 See Serjeant, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=', 2019, Astronomy & Geophysics, Volume 60, Issue 2, Pages 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='13–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='17, https://academic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='oup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='com/astrogeo/article/60/2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='13/5380734 25 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ukri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/about-us/stfc/planning-strategy-reviews/balance-of-programmes/ 24 No longer available on the UKRI websites, but archived at https://web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/web/20220305101010/https://stfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ukri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/files/aap-balance-report/ 23 No longer available on the UKRI websites, but archived at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='yumpu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='com/en/document/view/4206082/aap-prreport-submitted-nov22 22 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ukri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/publications/astronomy-advisory-panel-summary-of-2021-community-consultation/ Page 7 of 37 advancing laboratory measurements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' laboratory astrochemistry), and expanding basic technology development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Most of these items feature in the high or very high UK community priorities discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The US prioritisation of data archives and curation echoes a European prioritisation of open science under the “FAIR” principles: Findable, Accessible, Interoperable and Reusable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' However, in the UK our consultation highlighted not only a need for continued support for this area, but rather a conspicuous lack of support for software, archiving, instrumentation and technical careers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The US is prioritising an expansion in grants, while the UK has for many years been failing to make optimal use of its astronomy facility investments due to a chronic underfunding of the exploitation line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' In terms of overarching AAP-area science themes, the US Decadal Review recognises several key areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' “Pathways to Habitable Worlds” supports the work of the US exoplanet community, which has a sizable UK counterpart, and in terms of ground-based astronomy requires support from ELTs, resolution spectroscopy, high-performance adaptive optics, high-contrast imaging, as well as laboratory and theoretical studies, all of which have been recognised as UK community priorities below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The “New Messengers and New Physics” and “New Windows on the Dynamic Universe” themes cover not just CMB cosmology but also the rapidly expanding discovery space of time-domain astrophysics, including gravitational wave counterparts and other optical and radio transients, all of which have sizeable UK community support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Ground-based technical capabilities required to support these themes are very similar to those highlighted by the UK communities and include ELTs, next generation CMB experiments and gravitational wave technology developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The “Cosmic Ecosystems” theme covers a wide range of galaxy evolution studies, requiring ground-based support from the Vera Rubin Observatory as well as a wide range of space-based facilities including the forthcoming ESA-led Euclid mission, while the “Unveiling the Drivers of Galaxy Growth” theme pushes also to higher redshifts with the help of ELTs, new radio facilities, and space missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Both themes also recommend support for theoretical underpinnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The UK has a vibrant extragalactic community and prioritises a similar mix of ground-based technical capabilities, though the SKA affords a slightly different and complementary set of opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The consistent call for theoretical underpinning is reflected also in the UK community prioritisation of continuously updating the national high performance computing capability, discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The US Decadal Review also set out aspirations for space-based astronomy, the headlines of which are a large infrared/optical/ultraviolet space telescope, X-ray and far-infrared missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Together, these missions cover the breadth of the review’s science themes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' In the UK, space mission involvements are the purview of UKSA and are formally outside the remit of STFC, except in some limited situations discussed briefly below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Nevertheless, it is also the case in the UK that space missions are central to a wide range of science themes, including the US priorities of JWST, Roman, ESA Euclid, as well as other current and future ESA M-class and L-class missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The limitations of atmospheric stability, transmission and/or emissivity inevitably require space missions for many science goals, including exoplanet discovery, high energy astrophysics, time-domain astrophysics and physical processes obscured by dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Although space projects are outside the AAP remit, and although AAP are not being asked for as fine-grained detail on community priorities for space missions as we are in ground-based astronomy, we felt it would be inappropriate if we did not capture the community view by ranking the UK involvement in the ESA mandatory science programme as being collectively equal to our highest ranked ground-based priorities, below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' There is also a great deal of UK interest in space bilateral opportunities, including with the US missions highlighted in the Decadal Review, but this is largely beyond the remit of the AAP roadmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The review also made a wide range of recommendations on societal impacts, career development, equity, diversity and inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Many of these are also appropriate for UK contexts and are discussed in the context of our community consultation below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Page 8 of 37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 European ASTRONET review 2021-2022: priorities and synergies with UK interests 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 Context and background ASTRONET is a European consortium of funding agencies, research organisations and associated bodies that was formed in 2005 with initial funding support from the European Commission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' It is now self-sustaining (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' without EC financial contributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' One of its principal objectives is the creation of a community-driven strategic plan and infrastructure roadmap, covering all of European astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The remit covers not just the ground-based astronomy within the UK STFC AAP remit, but also solar system science and space-based facilities across all of European astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ASTRONET last published its science vision and infrastructure roadmap in 2015, after which its 27 focus has been on the implementation through a series of reports on the European coordination of research infrastructures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the integration of the European communities in mainstream astronomy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the coordination of national funding agencies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' and education, training, and public outreach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Most recently, over the past two years ASTRONET has been working on a new Science Vision and Infrastructure Roadmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Again the aspiration is for this to include all aspects of European astronomy to aid national and multi-national long-term astronomy planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' An extensive community consultation on the 28 drafts concluded in August 2022, and at the time of writing (November 2022) five of the eight science themes exist in a final form, while the remaining three are still draft reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Below we summarise the recommendations in each theme and the UK relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Note the wide range of facilities and instruments that are named as supporting several science challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Many UK colleagues participated in the ASTRONET consultation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' there are no obvious areas in which UK aspirations are in conflict with those of this wider European strategy, but there are areas of particular UK interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 Origin and Evolution of the Universe This science theme outlines the current standard cosmological models and the community aspirations for progress in the coming two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The report highlights a growing overlap or synergy between fundamental physics and cosmology, through the physics of the early Universe and tracers of the matter distribution around the epoch of recombination, as well as the large-scale distribution of galaxies at lower redshifts, both of which are areas that have broad UK active communities with world-leading track records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The UK had a representative in the working group that drafted this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Key science questions in those areas include: “Are there deviations from the standard model of particle physics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Are there deviations from the standard cosmological model?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The emerging tension in measurements of the Hubble constant hints either at unknown systematics or new physics, while the accelerating expansion of the Universe inevitably signals new physics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the report poses further key questions: “Are there deviations from general relativity, and on what scales?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' What is the origin of the accelerated expansion?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' A further inevitable signal of physics beyond the Standard Model is the evidence for dark matter, all of which is currently astronomical, and a further key question in this section is “What is the nature of dark matter?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The temperature and polarisation anisotropy of the CMB probes physics between the LHC energy range of 104 GeV to Grand Unified Theory scales of 1016 GeV, and a further key science question highlighted in the report is “Can we identify specific observational signatures of inflation?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Finally, this section of the report covers the PAAP-remit area of gravitational waves, posing the key questions “What can gravitational waves observations reveal about dark energy, dark matter and modifications of gravity on cosmological scales?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' In each of these key question areas, critical ground-based and space-based facilities are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' In terms of ground-based astronomy, the report highlights the importance of the SKA, Rubin, Simons 28 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ASTRONET-eu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/forums/roadmap-community-consultation 27 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ASTRONET-eu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/archives Page 9 of 37 Observatory (and forthcoming CMB Stage 4), all of which are very high UK community priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' European CMB ground-based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' QUBIC, QUIJOTE, LSPE-strip) and balloon-borne experiments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' LSPE-swipe, BISOU) highlighted as pathfinders for larger space and ground-based facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Ground-based spectroscopic projects also feature prominently, such as the US-led DESI (with UK member institutions), the Japanese Subaru-PFS, the Mauna Kea Spectroscopic Explorer (MSE, recommended by AAP for UKRI infrastructure funding, see below), and the US-led MegaMapper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' For probing the epoch of reionization via 21cm cosmology, the report highlights LOFAR (with UK involvement), MWA and the GMRT, and forthcoming facilities LOFAR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='0, MWA2, uGMRT, with SKA already under construction and HERA operational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' We note that UK involvement in HERA has been considered for PPRP funding but so far has not been successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Line intensity mapping also features as a cosmological probe, tracing the large-scale structure of atomic, molecular and/or ionic gas in the Universe, and requiring ground-based facility support from SKA, APEX/CONCERTO (with ESO involvement), CCAT (with UK involvement), and BINGO (with UK involvement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The report recommends investing in “small/medium-size facilities and new instruments on existing platforms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=', WEAVE, 4MOST)” as pathfinders for more ambitious wide-field facilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' MSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Finally, the report chapter recommends investments in data processing, analysis and data science to accompany new instrumentation, in concordance with similar, broader recommendations from the US Decadal Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 Formation and Evolution of Galaxies At the time of writing, this chapter currently only exists in draft form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' There is a very broad and active community in both the UK and more widely across Europe in this science area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Accordingly, the report chapter covers an extremely wide range of science questions and concomitant facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' At very early cosmic epochs, the chapter outlines the aspirations for understanding the physics during the reionization epoch, including the sources of reionization, the topology of ionized gas, the properties of the first stars and galaxies, and the origins of gas in galaxies that fuel star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' At lower redshifts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the report highlights aspirations for understanding critical processes driving galaxy evolution and the growth of stellar mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' including positive and negative feedback processes from starbursts and active nuclei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the dark matter halo processes (besides mass) that drive galaxy properties,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the state of gaseous halos surrounding galaxies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the processes that drive star formation out of dense molecular gas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the processes that drive the star formation histories and morphological evolution of galaxies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the role of environments and the accretion physics around supermassive black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The report extends to science questions covering the Milky Way, including its accretion history (and what information can be recovered about it), the assembly of its halo, the abundance patterns of its first stars, and our very local environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The facility requirements and aspirations that accompany this vision are similarly broad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Many current and future space missions are prioritised, though in UK contexts these are broadly in the remit of UKSA rather than STFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' In terms of ground-based spectroscopy, the report chapter highlights multiplexed spectroscopy (4MOST, ESO VLT/MOONS, WEAVE, Subaru PFS, CFHT/Mauna Kea Spectroscopic Explorer, ESO spectroscopic facility), high-resolution spectroscopy (CUBES and ELT/ANDES and MOSAIC), integral field spectroscopy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' VLT/MUSE and BlueMUSE), and 21cm spectroscopy and line intensity mapping (SKA, HERA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Many of these have UK key roles, including ESO instrumentation, VLT/MOONS, and WEAVE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the Mauna Kea Spectroscopic Explorer was among the AAP priorities for UKRI infrastructure funding (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Ground-based imaging capabilities are also prioritised with Rubin and ultimately ELT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The report supports the development of interferometers such as LOFAR and including endorsing the aspirations in the ALMA 2030 roadmap , as well as developments of the German/Spanish/French PdBI/NOEMA array and 29 supporting new facilities such as the SKA, the US ngVLA (see above), and a future large single-dish 29 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='almaobservatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/en/publications/the-alma-development-roadmap/ Page 10 of 37 submm/mm-wave telescope AtLAST (which featured in an AAP recommendation for future UKRI infrastructure funding, see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' High angular resolution studies are prioritised on both the ESO VLTs and the future ELT (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' VLT/MAVIS, ELT/MICADO, ELT/HARMONI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' High angular resolution interferometry with ALMA (especially the long baseline extension), VLBI, VLTI, ngVLA, SKA and its precursors such as MeerKAT are all prioritised, as is Rubin LSST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The Cherenkov Telescope Array (CTA) is also an ASTRONET priority (in which there is UK technical interest in instrumentation and UK science interests mainly in the prospects for multi-messenger / multi-wavelength astronomy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Also highlighted are cosmological and extragalactic simulations, including Illustris, the EAGLE simulations, the Hestia project, and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 Formation and Evolution of Stars This is a wide-ranging science theme in which the UK also has large, vibrant research communities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the UK chaired the working group that drafted this chapter and also had representation among its membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The science theme covers the birth, lifecycle and death of stars, the physics of stellar remnants and the cosmological applications of transient events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Key questions in the area of stellar birth include: “How do molecular clumps fragment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' What effect does the local magnetic field have on collapse?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' How do stars accrete material?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' What sets the upper mass limit of a star?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Is the initial mass function (IMF) universal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' What is the relationship of the IMF to the core mass function?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' What sets the initial multiplicity of stars, and how does this impact the IMF?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' How do young stars and their disks set the properties of nascent planetary systems and their early evolution?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' What chemical pathways are important in the production of complex organics?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The evolution of stars also poses a number of key science questions, including: “What are the basic constituents of [stellar] matter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' What happens to matter in extreme conditions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' What external factors influence the conditions required for life on planets?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Finally, ASTRONET highlights a number of key science questions regarding the end-states of stellar evolution, including: “What are the progenitors of type Ia supernovae?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' What is the Galactic low-frequency GW background?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' What is the ultimate fate of planetary systems orbiting stellar remnants?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Neutron stars pose a wide range of key science questions relating to their equations of state, the origin of their magnetic fields and radio emission, their formation routes and interaction with their environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Stellar-mass black holes are important tests of General Relativity and there are open science questions on their formation routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' A wide range of ground-based instruments and facilities are prioritised to support these science goals, including ELT, Rubin LSST, SKA, WEAVE, 4MOST, VLT/MOONS, VLT/CUBES, VLTI, as well as smaller facilities specialising in time-domain science such as GOTO (prioritised by AAP in the previous “priority projects” process and with UK involvement now funded by PPRP) and BlackGEM (again with UK involvement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Multi-messenger astronomy features prominently in this theme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Existing facilities with unique capabilities are also prioritised, including ALMA, VLT/VLTI, the Spanish 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 metre telescope GTC, WHT, and the Italian 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='58-metre TNG telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Looking ahead,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the report prioritises new ground-based capabilities with 8-10 metre class facilities such as VLT GRAVITY+ and VLT BlueMUSE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' new highly multiplexed spectroscopic survey capabilities such as wider European participation in the Mauna Kea Spectroscopic Explorer or similar (which featured in the AAP prioritisation for UKRI infrastructure funding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' see below),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' as well as highlighting the need for a large single-dish sub-mm facility such as AtLAST (again featuring in AAP’s UKRI infrastructure funding shortlist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5 Formation and Evolution of Planetary Systems Once again this is an area with a vibrant UK research community, and the UK had a representative on the working group that drafted this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This field is experiencing an extraordinary phase of discovery since the first exoplanets were discovered only three decades ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The overarching science questions highlighted by ASTRONET that are driving the current and future research in this area are: “What drives the enormous diversity in exoplanet systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' How special is Earth within all the Page 11 of 37 possible worlds?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' What are the necessary conditions for life to emerge and thrive?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' What is the fate of the Solar System?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' and, “Are we alone…?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' A very wide range of ground-based instruments and facilities are prioritised to support this science, covering planet formation, exoplanet discovery and demographics, exoplanet characterisation, and the late-stage evolution of planetary systems, including: VLTI MIDI, MATISSE and GRAVITY+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' VLT instruments SPHERE, MUSE, ERIS, HiRISE (SPHERE/CRIRES+ coupling), RISTRETTO, and possibly MAVIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ALMA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Rubin LSST;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ESO CRIRES+, NIRPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Precision radial velocity measurements are critically important to this field, with the CFHT SPIROU, ESO NIRPS, and the Spanish/German CARMENES explicitly cited in the near-infrared, while in the optical the VLT ESPRESSO, the US EXPRES facility, and HARPS3 on the Isaac Newton Telescope are all explicitly cited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' VLT and (later) ELT direct imaging are prioritised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' LOFAR and SKA (both with substantial UK interests) are cited for the study of star-planet interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The wide-area multi-object spectroscopic surveys WEAVE, DESI, 4MOST and SDSS-V are particularly important for the study of white dwarf planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Further in future the report prioritises the ELT instruments MICADO, HARMONI, METIS, HIRES (now named ANDES), PCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Furthermore, laboratory astrochemistry (also a UK area of world-leading expertise) is explicitly prioritised in order to complement and interpret astrophysical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Theoretical work with three-dimensional numerical simulations is similarly prioritised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='6 Understanding the solar system and conditions for life This chapter of the ASTRONET roadmap also covers a wide range of UK research interests, and the UK also participated in the working group that drafted this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The scientific remit is solar system science and heliospheric physics, and as such is much more closely aligned with the specialism of SSAP than AAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' A large proportion of this review is also based on space missions that fall within the UKSA remit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' For both reasons we do not summarise this chapter here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Nevertheless, many of the same priority terrestrial facilities and technologies are in common between these solar system recommendations and those of AAP, including Rubin LSST and laboratory astrochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='7 Computing - big data, high performance computing, data infrastructure This overarching technology chapter of the ASTRONET roadmap covers high performance computing, software engineering, data products, software tools, open science (including astronomical use of the European Open Science Cloud, an interdisciplinary open science initiative of the European Commission funded by approximately a third of a billion euros to date), and the carbon footprint of astronomical computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' UK expertise was represented on the working group that drafted this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The report provides six main recommendations, which in summary are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Developing and investing in a professional software engineering / computational skills base in Astronomy, including career development with clear progression pathways in academia and improving the diversity of the workforce, and new measures to quantify the impact and usefulness of computationally focussed outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Missions and facilities should plan an integrated approach for data products and software tools including design, delivery, maintenance, development, and scientific data preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Creation of a “tiered” approach to data infrastructures, for all astrophysical data including models, simulations and mock data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' A fully collaborative, open and synergistic approach to the astronomy-computing ecosystem, encompassing data, software, processing, analysing and modelling, and embedding reward structures to realise this ambition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ASTRONET should produce or commission a biennial quantitative report to assess the carbon footprint of computing in Astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ASTRONET should develop specific actions to coordinate cross-cutting activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Page 12 of 37 These aspirations are shared by the UK community, but AAP notes these aspirations are not currently being met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' For example, they sit uneasily with the AAP consultation responses below that highlight current UK deficiencies in the support of career pathways for technical roles in astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' See Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='8 Societal aspects – education, public engagement climate action, equality, diversity and inclusion At the time of writing (November 2022), this chapter of the ASTRONET roadmap is in draft form and is not yet finalised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The draft report covers the social and cultural relevance of astronomy, education, technology transfer from astronomy, public engagement, astronomy for sustainable development, climate action, gender equality, and inequalities in astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The educational and cultural aspects of the report are not germane to our AAP roadmap so we do not summarise them here, but our AAP consultation responses below share many of the same ideas as the diversity, equity and inclusion themes of this report, including: “significant gender imbalance still exists within astronomy as a profession [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='] there is a significant imbalance in the inclusion of underrepresented and vulnerable groups (persons with (dis)abilities and persons with racial, ethnic, religious, LGBTQI+ backgrounds) [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='] The European astronomical community needs to develop specific training [for] researchers and advocate for equity and inclusion”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Furthermore, the draft report is currently recommending that “European astronomy (community) should (at the very least) follow the European timeline towards carbon-neutrality: 50% reduction of CO2 emissions by 2030 and 100% by 2050”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP are pleased to note that the environmental impact was one of the criteria in assessing the 2020 UKRI infrastructure funding bids, and AAP welcomes the 2020 UKRI Environmental Sustainability Strategy that aims to 30 achieve net zero carbon emissions from UKRI no later than 2040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='9 Extreme astrophysics At the time of writing (November 2022) this chapter of the ASTRONET roadmap is currently in draft form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This science theme again covers areas in which a wide UK community are active and in which the UK has many leadership roles, and UK expertise was represented in the working group that drafted this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Extreme astrophysics here refers to the environments within and around compact objects such as white dwarfs, neutron stars, and black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' At the time of writing there is some overlap with the science themes of the “Formation and evolution of stars” ASTRONET roadmap chapter (see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Key science questions include: What is the nature of matter at nuclear densities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Where are the heavy elements made?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' How do compact objects produce energy and accelerate particles at all scales?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' What is the origin of cosmic rays of all energies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' How do compact objects form and evolve?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' To what precision can general relativity describe gravity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' What new fundamental physics can be probed with extreme astrophysical objects?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Many of the key facilities supporting these science questions are space-based and therefore largely outside the remit of the AAP roadmaps, but a wide range of ground-based facilities and instrumentation are also prioritised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Current key ground-based facilities cited include: LIGO/Virgo/KAGRA, neutrino detectors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the US-led IceCube and the European-led KM3NeT), LOFAR, Pierre Auger, MAGIC, HESS, the South African SKA precursor MeerKAT, JVLA, EVN, ground based 2-8 m telescopes including wide-field optical/IR facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Future ground-based facilities that will drive future discoveries include: SKA, CTA, ELT, Rubin, Einstein Telescope, the Italian ASTRI facility, the Global Cosmic Ray Observatory, the Giant Radio Array for Neutrino Detection (GRAND), and the next generation Event Horizon Telescope ngEHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The neutrino, gravitational wave and cosmic ray capabilities fall within the PAAP remit, but the remainder have a striking commonality with the facilities supporting other astronomy areas in both the ASTRONET roadmap and the AAP consultation responses on community priorities below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 30 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ukri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/wp-content/uploads/2020/10/UKRI-050920-SustainabilityStrategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='pdf Page 13 of 37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 Subsequent UK developments and the scope of the AAP roadmaps Since the AAP consultation closed, UKRI postgraduates wrote an open letter to UKRI on the impact of the cost of living crisis combined with stipends being calculated on the previous year’s inflation, and requesting that stipends are uplifted to match the current inflation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP are very pleased to see UKRI increase the 2022-3 stipends in response to this community input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Also since the AAP consultation concluded, STFC launched a further consultation on the changes to the consolidated grant system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This new consultation has just reported its findings, with applications accepted for the new scheme in March 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The present document condenses the call for white papers and the consultation into science and technology roadmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' However, AAP have some concerns about potential mis-uses of the roadmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' These roadmaps only describe the community’s present strengths and aspirations, and an overview of the community’s near-term and mid-term priorities for science and technology opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The roadmaps are not suitable for drawing a funding threshold line if the core programme funding is reduced, because that would need either an STFC-wide Balance of Programmes review with an Astronomy Evaluation Panel, or an equivalent but Astronomy-specific tensioning of spending commitments, which in either case would be driven by explicit evaluation criteria and explicit funding envelope options, as well as having agreed expected near-term outcomes on the basis of the recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP do not believe it would be helpful to provide a finer-grained long-term prioritisation than is presented in this roadmap given that the criteria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' prospective funding envelopes and outcomes are all nebulous,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' because (a) the nebulous constraints leave too great a risk of mis-construing or mis-stating the community views (b) the evidence base for community views and quantitative measures of facility outputs/impact and other as-yet-undetermined assessment criteria either do not yet exist or have not yet been assembled,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' and (c) implementation of a finer-grained priority list without considering the wider impacts on the core programme would expose astronomy to the risks of unintended consequences in taking on new commitments or divesting from existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Therefore, the roadmaps must not be used to award funding to something that appears to be a priority, without any discussion with AAP about the consequences for other parts of the programme and any developments since the roadmaps were drafted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1: The 2022 AAP Science and Technology Roadmaps should not be used in isolation to evidence current community support without further reference to AAP, because astronomy is a fast-paced and rapidly changing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Nor should these roadmaps be used to draw a funding threshold line in the event of changes to the core programme funding, because the Balance of Programmes Review (or a hypothetical Astronomy-specific near-term tensioning exercise) is the appropriate process for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Summary of the 2021-2022 AAP Community Consultation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 Overview of the process In this section we summarise the response to the AAP community consultation in November 2021, in which we called for community input on priorities, as well as for science and technology white papers (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Many of the themes highlighted in the 2021 community consultation overlap with the earlier AAP consultation in 2018 that fed into the Balance of Programmes review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP received only 81 responses compared to 293 in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP suspects time pressures related to working from home and the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' There was particularly low representation among early career researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' In view Page 14 of 37 of this low response rate AAP also solicited further free-form feedback from the community in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The consultation itself is provided in a separate document so we only summarise the findings here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 Overview of white papers An important part of developing our science and technology roadmaps was the call for community white papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' These were intended to have the possibility of also serving a dual purpose of assisting STFC with having a ‘bank’ of potential projects that could be put forward in the event of UKRI-wide ad hoc funding opportunities (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP are supportive of STFC’s aspiration to make the Infrastructure Funding more strategic with more frequent calls to the community, and less dependent on last-minute calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The STFC Visions process (which replaces the previous Priority Projects calls) is explicitly intended for inter-disciplinary/cross-disciplinary large infrastructure projects across UKRI, and the white papers discussed here did not explicitly require this cross-UKRI remit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' At least some of the white papers discussed in our roadmaps may fit the remit of a future STFC Infrastructure Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' From our community responses it is apparent that both the white paper and infrastructure funding submissions are fundamentally only a snapshot of the ideas present in the community, so it is not obvious that it is possible to create a long-term roadmap plan that maintains the flexibility and responsiveness to changes in the scientific and technological landscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP were impressed at the breadth and depth of ideas submitted by the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP members independently evaluated the relative priorities of the white paper submissions and where possible determined consensus views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Besides the specific projects and facilities discussed in the roadmaps below, AAP received a wide range of science themes in the white paper submissions with clear evidence for UK leadership, including the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' What is the chemical and mineralogical composition of astronomical dust?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This interdisciplinary white paper covers Galactic planetary and interstellar astronomy, astrochemistry, planetary science and meteoritics, and infrared spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The paper makes a compelling case for the support of laboratory astrochemistry and astro-mineralogical research to underpin the interpretation of major data sets from facilities that represent top community priorities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' those from JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The white paper covers areas of interest of both AAP and SSAP, and is in concordance with the European ASTRONET priorities (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Euclid Science Exploitation in the UK is a white paper submitted by the Euclid:UK coordination group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' There are around 350 UK-based researchers working in the Euclid Consortium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The science themes are extremely wide, covering cosmology, extragalactic astronomy, Milky Way and the local universe, as well as SSAP areas of interest, and are very well aligned with STFC Science Challenges as well as with many of the ASTRONET 32 themes (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' STFC funding of the science exploitation is within the remit of AGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' A related white paper was received by AAP on the new era of strong gravitational lensing science, covering not just the discovery space opened by Euclid but also that of Rubin LSST, SKA, and the NASA Roman space telescope, with follow-ups facilitated by ground-based 4-metre class facilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ESO NTT, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='6m, WHT, VISTA/4MOST), 8-metre class telescopes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ESO VLTs) and 30-metre class telescopes (ELTs first light 2027), as well as space telescopes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' JWST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The step change in available discovery space covers a wide range of science goals, including cosmography, dark matter structure and substructure, quasar accretion disc structure and highly magnified background source morphologies and transients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The goals are in concordance with a broad range of ASTRONET aspirations (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 32 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ukri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/publications/stfc-science-challenges/stfc-science-challenges-in-frontier-physics/ 31 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ukri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/publications/astronomy-advisory-panel-summary-of-2021-community-consultation/ Page 15 of 37 The white paper submitted to AAP on Stellar Variability and an Extreme Precision Radial Velocity Roadmap in the UK presented, in effect, a brief update to the 2015 STFC review of exoplanet science .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The UK has clearly established itself as a world leader in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The 33 large author list argues for developing a stellar variability and extreme precision radial velocity roadmap in liaison with other international bodies such as ESA, ESO, NASA, NSF etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The facilities driving progress in this area include HARPS-N at the Italian TNG, the Solar Telescope, HARPS at the ESO La Silla 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='6m, CARMENES at the CAHA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5m, ESPRESSO at the ESO VLT, NEID at the WIYN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5m (a US facility with limited UK access).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' For characterising solar/stellar variability, the white paper highlights facilities in the SSAP remit including SDO, BiSON, DKIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Looking forward, the white paper cites the importance of the UK-led HARPS3 at INT, led from the UK, currently under construction (first light expected 2023) and HIRES (now named ANDES) at ELT (under construction) which has passed Phase A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The exploration of free-floating planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This science white paper aims to use VLT, ELT, Rubin LSST, and space missions Roman and JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The science theme is an area of UK strength that features only implicitly in the ASTRONET roadmap (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4) and may represent a key opportunity for UK international leadership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' REACH: Radio Experiment for the Analysis of Cosmic Hydrogen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This project aims to 34 detect the signals of cosmic reionization through sky-averaged 21-cm mapping, addressing the instrumental systematics present in other experiments, and stealing a march on the SKA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' REACH is a collaboration between the University of Cambridge and Stellenbosch University in South Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Although the facility is not currently financially supported by STFC, AAP support the aims of the project and are happy to include it among the high priority facilities below that represent a selection of the current and aspirational portfolio of “small” UK astronomy facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The role of massive stars in galaxy evolution and nucleosynthesis, supporting facility use of VLT, e-MERLIN, ALMA, ELT, SKA, 4MOST, WEAVE, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=', as well as space missions, and with scientific aspirations in concordance with those of ASTRONET (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Eclipsing binary stars: high-precision probes of the physics of stars and planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This white paper covers an area of UK strength, and supports the use of WASP, NGTS etc in conjunction with a variety of space missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The science area is well aligned with ASTRONET aspirations (sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Pulsar rotational glitches, a probe to the extreme physics of neutron stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Precision pulsar timing as a tool for probing fundamental physics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Radio emission from pulsars and the pulsar magnetosphere;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Revealing the Galactic neutron star population through pulsar searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' These comprehensive and wide-ranging white papers cover fundamental physics from neutron stars, supporting facility use of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Jodrell Bank, Parkes, MeerKAT/MeerTime, LOFAR/LOFAR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='0, SKA, as well as a variety of space missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Compute is also a key requirement, particularly also in an additional white paper on magneto-thermal evolution of neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' A further white paper on optical follow-up of binary neutron star systems made a strong case for investigating the dynamics, geometry, and probes of the physical processes, with the aid of ESO NTT, VLT, ELT, as well as GOTO, Rubin LSST, and the Spanish facility GranTeCan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The white papers are in concordance with aspirations in ASTRONET (sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Multi-messenger astronomy featured in a wide range of white papers submitted to AAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The white paper on time-domain polarimetry as a unique discovery space for transients covered the important emerging fields of multi-wavelength, multi-messenger time domain science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Besides aspirations for new instrumentation, the facilities supported by this white paper 34 See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' de Lera Acedo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=', 2022, Nature Astronomy, 6, 984, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1038/s41550-022-01709-9 33 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ukri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/about-us/stfc/planning-strategy-reviews/exoplanets-uk-research-review/ Page 16 of 37 include Rubin LSST, the Zwicky Transient Factory, Liverpool Telescope, NRT, VLT, WHT, ALMA, SKA, and a variety of space missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Multi-messenger astronomy is an emerging priority of both ASTRONET (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4) and APPEC , and naturally also features in the 35 PAAP roadmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' A further white paper on the high energy transient Universe underlined the importance of X-ray and gamma-ray space telescopes for multi-messenger astronomy (formally outside AAP roadmap remit) and the importance of complementary ground-based observations from e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' SKA, ELTs, CTA, LOFAR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='0, GOTO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the science goals are broad and cover emission mechanisms of gamma-ray bursts, relativistic jet physics, thermal transients, and the early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP also received a white paper on searching for fast radio bursts (FRBs) and other radio transients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This white paper addresses a very topical research area, and also covers the emerging important techniques of multi-wavelength, multi-messenger astronomy (see also the ASTRONET roadmap above, section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The science case covers FRBs as cosmological probes, potential FRB progenitors such as magnetars, slow Galactic transients, and population phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Key facilities include MeerKAT, Jodrell Bank, e-MERLIN, European VLBI Network (EVN), LOFAR and its 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='0 upgrade, SKA, and a wide range of X-ray space telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Gravitational Wave Astronomy: Advanced LIGO+, Cosmic Explorer, LISA, Einstein Telescope, Pulsar Timing Arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' These science themes also feature in the PAAP and ASTRONET roadmaps (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3) and reflect a widespread interest in this new astronomical window, with UK leadership in both science and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP also welcomed a white paper on neutron star physics from gravitational waves, supporting also multi-messenger / multi-wavelength observations with LOFAR/LOFAR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='0, Lovell Telescope, and SKA, underlining the overlapping interests of PAAP and AAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1: STFC and UKSA should support the UK exoplanet community in developing a stellar variability and extreme precision radial velocity roadmap in liaison with other international bodies such as ESA, ESO, NASA, NSF etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 Top, overarching community priority in the consultation Exploitation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' grants, remain the top community priority and are chronically underfunded, despite a very welcome uplift of £2M year on year (leading to a £6M increase by 2024/25) that was announced at the 2022 National Astronomy Meeting from the last Comprehensive Spending Review outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This funding for human capacity is now the main limiting factor for our national science capability in astronomy, but it is a UKRI-wide problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' As John Womersley put it six years ago, we are still "paying for gym membership and being unable to afford the bus fare to get there".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The continued, chronic underfunding of exploitation has arguably led to the result that 53% of the community are now dissatisfied or very dissatisfied with the balance between exploitation, operations and development;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' in contrast, our 2018 exercise had 52% wanting the exploitation line to grow only if there were new money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Astronomy’s UKRI exploitation funding gap has been partly counterbalanced by EU funding, reflecting also our strong relationships with continental communities and beyond, so our association to the Horizon programmes is critical, providing not just funding but also close association with continental-scale research networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Several white papers (see above) mentioned the possibility of hypothecating postdoc/fellowship funding, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' setting aside some PDRAs/fellows for particular science areas or to support particular facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP believe the underlying driver for these submissions is that exploitation funding (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' grants) are so chronically under-supported and that the limiting factor for the UK’s science research capability in astronomy is human capacity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' however, we suspect that allocating such a sparse resource as postdoc/fellowship funding to particular science areas would prove too contentious and divisive to 35 See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' https://indico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ego-gw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='it/event/199/ Page 17 of 37 be implementable in practice within the core programme while maintaining community support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' If additional UKRI funding were found beyond that of the STFC core programme, then dedicated subject-specific fellowship opportunities may prove less contentious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2: STFC should prioritise increasing the exploitation line, which is the community’s top priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3: UK association to the EU Horizon funding programme is critical for astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Replacing the funding alone would alleviate the exploitation pressure but would disconnect the UK from continental-sized research networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Our white paper and consultation submissions highlighted the problem of a lack of career structure and career opportunities in the cross-cutting areas of technology development, software, and instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Universities can find it difficult to recruit and retain core technical teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The white papers suggested aspirations for early-career fellowships for instrumentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' expanding the number of research software engineer fellowships in the STFC area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' helping universities retain core technical teams via grant assessment guidelines to facilitate employing staff on multiple grants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' In the Equity, Diversity and Inclusion section of our consultation, it was also highlighted that there is little recognition of the software engineer / blended (software and astronomy) as a career path in an environment, yet this is crucially important (and we may add that a large amount of the effort is done by early-career researchers despite the whole community benefitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The white papers aspired “to see a visible shift of priority from the science exploitation of facilities to their delivery in order to highlight instrumentation [and AAP would add software and project roles] as a viable career specialism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Such a priority shift would also extend the reach of astronomy-based work to stakeholders in other critically important domains.” We agree and see some of the underlying problems not just being structural but also arising from the chronic underfunding of exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' There is no headroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This is not just limiting our science exploitation now, but is incurring long term, strategic costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The white paper submissions also correctly pointed out that the typical short-term (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 3 year) project grant durations are not commensurate with typical project lifetimes that are a factor of several longer, creating an avoidable career precarity particularly for those in technical roles that are increasingly important for STFC science exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4: STFC should review the career structure for instrumentation and technical roles, both within and beyond astronomy and discuss with Universities as to how to implement that within their structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 Highest rated science and technology themes/facilities in the consultation SKA (together with its pathfinders/precursors) and ESO (including ELT) remain highest rated for both the science and technology roadmaps in our consultation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Besides the consultation response, the SKA breadth of community support is also very well evidenced by UK participation in SKA working groups as well as regular town hall meetings organised by the UK SKA Science Committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP 36 37 are very pleased that the SKA has been awarded UKRI infrastructure funds for the development of the UK SKA Regional Centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ESO being among the highest community priorities is also in accord with other obvious lines of evidence besides our consultation response, such as the recent evaluation of 38 UK membership of ESO, the breadth and depth of UK PI and co-I time awards on current ESO 39 facilities, the factor ~8 oversubscription on Europe-node ALMA time (in which the UK is typically 40 40 See https://almascience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/documents-and-tools/cycle9/cycle-9-proposal-submission-statistics 39 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/sci/observing/teles-alloc/all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='html 38 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ukri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/publications/socio-economic-impact-evaluation-of-the-uk-subscription-to-eso/ 37 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ukri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/about-us/stfc/how-we-are-governed/advisory-boards/ukskasc/ 36 See https://astronomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='skatelescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/science-working-groups/ Page 18 of 37 one of the largest national European winners of PI and co-I time), the breadth of UK involvement in 41 science working groups and instruments on the forthcoming Extremely Large Telescope (ELT), and 42 43 the community participation in UK ELT town hall meetings .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 44 Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5: STFC must maintain the UK role in the SKA and support the development of the UK SKA Regional Centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='6: The UK must remain a member of the European Southern Observatory and play leading roles in its development of its world-class instrumentation, including the second- and third-generation instrument suite of the ELT and the development of ALMA instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' High performance and high throughput computing provision also remain a community priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This has received a welcome boost in 2020 in the form of £20m of capital funding from the UKRI World Class Laboratories funding line, enabling the long-awaited deployment of the DiRAC-3 phase 1 upgrade in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The establishment of the IRIS project, following capital grant funding from BEIS 45 in 2018, has also provided a framework for linking the range of digital research infrastructure that falls under the STFC remit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' However, large oversubscription factors and the ever increasing data requirements of state-of-the-art simulation codes and large-scale surveys mean that continued support remains vital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='7: UK HPC capabilities such as DiRAC & IRIS underpin a wide range of world-leading UK theoretical astrophysics and data science that must be continually supported and upgraded to remain competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5 Emerging technologies or capabilities identified by the community for inclusion in the roadmap The largest response was the development of improved detectors, in particular energy-sensitive detectors (KIDS, MKIDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Many replies highlight the importance of existing or upcoming projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ALMA, LOFAR, SKA, JWST, LSST, GRST are mentioned several times, while Euclid, CTA, new instruments on the VLTs (in particular optical/NIR interferometry), gravitational wave research, exoplanet science, high-energy instruments and improved radio telescopes were also mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Big Data and machine learning techniques were also mentioned as an opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='6 Science challenges identified by the community as requiring major investments Many science challenges were identified by the community as requiring major investments, including gravitational waves from space to Gamma-ray detectors, a new all-purpose space observatory, supporting ESO, various space missions, studying planets, discovering life, strengthening staffing for research, HPC and a new sub-millimetre telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='7 Equity, Diversity, Inclusion, and Careers Science, including astronomy, does not always operate as a meritocracy, and there are many well-recognised biases in accessibility, inclusion and career progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Improving equality, equity , 46 46 See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' https://social-change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='uk/blog/2019-03-29-equality-and-equity 45 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='iris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='uk/ 44 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='elt-uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/astronomers/meetings/ 43 See https://elt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/instrument/ 42 See https://elt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/science/ 41 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='elt-uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/astronomers/ Page 19 of 37 diversity and inclusion (EDI) is therefore part of the core business of astrophysics, besides also having a moral imperative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The consultation presented a wide range of community suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' See also Recommendations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Some suggestions were within the remit of the Consolidated Grant consultation process, which has now completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Other suggestions that extend across UKRI and beyond the remit of AAP included: The career structure disadvantages some and negatively affects EDI objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' For example: differences in leave (including maternity) for professional staff, and no maternity leave on grants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' early career precarity, especially for those with low-income backgrounds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' deficits in support for disadvantaged/under-represented communities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the need to move overseas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the need to improve leave systems for those with caring responsibilities, and lack of support for families to attend conferences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ableism and the ability to travel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' work-life balance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' unsafe/unhealthy working environments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' not free of prejudice/abuse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' insufficiently diverse set of role models at all levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' EDI progress evidence should be collected for both institutions and individuals, with demonstrable outcomes to avoid tick boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' EDI training should be mandatory and action should be taken against staff who disregard EDI, while good EDI practices should be recognised and rewarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Why do we weight science track records highly, and not even ask for EDI track records?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Research consortia should be expected to implement and adhere to their codes of conduct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' STFC should monitor and report on the diversity of its own advisory structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' UKRI Infrastructure Funding call: AAP and community responses Subsequent to the consultation and white papers call, AAP solicited community input for the UKRI Infrastructure funding “Preliminary Activity Wave 3” call in Summer 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Five bids were received, of which AAP were only allowed to recommend three due to the proposal demand management: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' A 10-12-metre spectroscopic survey telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This design study aimed to prepare the way for a very wide range of science including galactic archaeology, galaxy assembly and the cosmic web, and transient science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This is a natural next step in Galactic and extragalactic survey parameter space: the next generation imaging surveys by the Vera Rubin Observatory and ESA Euclid will create targets that are too faint for 4-metre class observatories, and both the number of targets and the near-all-sky coverage make follow-ups by current 8-10-metre class observatories unfeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The project covered both the Mauna Kea Spectroscopic Explorer and the proposed ESO Widefield Spectroscopic Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This wide-field spectroscopic capability has already been identified as a European community priority by ASTRONET (see above), and additionally featured in the AAP call for white papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' UKELT: The next generation instrumentation suite for the Extremely Large Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This design study aimed to prepare the way for the second generation of ELT instruments: the multi-object spectrograph MOSAIC, the high-resolution spectrograph ANDES, and the planetary camera and spectrograph PCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The UK is in a leading position for ELT instrumentation, which in turn will be transformative across essentially all of UK astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This study had previously had some of the phase A & PCS research and development staffing costs funded by PPRP but not capital or hardware costs for the instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Unlike previous VLT and first-generation ELT instruments, ESO will not provide the hardware costs for second & third generation ELT instruments leaving the responsibility of raising hardware funds to the instrument consortia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The project also featured in the AAP call for white papers, further evidencing UK community interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Scalable continuum cameras for sub-mm telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This design study aimed to prepare for a Page 20 of 37 first-generation instrument for the AtLAST large submillimetre-wave telescope, which in turn has had a facility design study funded by the EU Horizon 2020 programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' In the process, this project would also deliver a wide-field continuum camera for the JCMT, giving the UK a tactical and strategic advantage for feeder programmes for ALMA across a wide range of astronomy, including extragalactic surveys, Galactic star formation and protoplanetary discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The UK JCMT involvement has been jointly funded by PPRP and university contributions, while AtLAST is a clear European priority across a wide range of science goals (see above) and featured in the AAP call for white papers, further evidencing UK community interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' UK Extreme High Frequency Facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This project aimed to create a technological development hub for coherent detector development from radio to sub-millimetre wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The project would support UK instrumentation interests for SKA, ALMA and several other facilities, and would cover a wide range of science interests from star formation to extragalactic surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Joint Rubin–Euclid data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The ground-based Rubin Legacy Survey of Space and Time will image the whole available sky twice a week for ten years at optical wavelengths, while the ESA Euclid mission will map around half the sky at higher resolution at optical wavelengths, extend to near-infrared wavelengths and provide near-infrared spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The Rubin and Euclid consortia together have comprehensively studied the wide-ranging 47 science cases resulting from combining these data sets, and this project proposed to create new value-added data products from the joint analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The astronomy science goals span exoplanets, Milky Way, nearby galaxies, transient phenomena, galaxy formation and evolution, and cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' All five were excellent, and well-aligned with our roadmaps below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Numbers 1-3 (in no priority order) were selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' (STFC then down-select all the proposals across all the advisory panels to a total of just three to submit to UKRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=') We were advised by UKRI that our number 5 might not fit the remit of that particular call, and number 4 was deferred to the next call based only on grounds of proposal readiness at the time of the deadline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' These submissions did however unveil two facets of the UK community: firstly, the independent, internationally-excellent groups working on sub-mm/mm-wave science and technology do not yet have a single consistent voice (see also section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' secondly, the difficulty in finding funding routes for joint facility data processing reveals a deficiency in support for digital infrastructure, despite big data and machine learning being highlighted by the UK astronomy community as important emerging technologies (see above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1: STFC (via AAP) should commission a review of UK submm/mm-wave science and technology, covering UK aspirations for current and future large single-dish facilities that feed the major international interferometers, and the underpinning aspirations for next-generation instrumentation, identifying areas of international excellence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2: There should be increased support for digital infrastructure in UK astronomy, including the creation of value-added data and software products, open science, and the development and implementation of machine learning technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' See also Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Science and Technology Roadmap 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 Overview In constructing this roadmap, we refer both to large overarching projects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ESO, LSST, SKA) and specific applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' joint LSST/Euclid data processing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' As shown comprehensively above in section 2, every facility supporting astronomy typically covers a very wide range of science themes, 47 See https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/abs/2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='03862 Page 21 of 37 and the science themes themselves demand a diverse range of facilities and instrumental capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This is driven ultimately by the diversity of physical processes that occur in each science theme, and the common ground-based technological approaches to investigating these processes throughout the science themes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' far-infrared to millimetre wavelengths for molecular astrophysics and dust;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' radio astronomy for synchrotron and coherent processes and atomic gas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' UV, optical and near-infrared astronomy for ions, stellar populations and black hole accretion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Major discoveries in observational astronomy are often driven by the opening of new observational parameter spaces and/or developments in instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Therefore we have opted to combine the science and technology roadmaps in a single roadmap for technology and facility needs and aspirations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' State-of-the-art instrumentation and facilities drive developments in areas throughout the STFC “Frontier Physics” science challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' There is a wealth and depth of key technologies where the UK is a leading force, including: Kinetic inductance detectors used for a range of aspects such as spectroscopy, imaging, time-domain, CMB, galaxy evolution, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Software, computing, receivers, and development for radio astronomy (Jodrell Bank Observatory, e-MERLIN, Lovell Telescope, Square Kilometer Array).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Infrared detectors High performance computing AAP would add (astro)photonic technology to these community-driven suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Furthermore, many technologies have the capacity for cross-cutting or multi-disciplinary applications, such as: areas that require mathematics and quantum devices development, time-domain astronomy, machine learning, high performing computing, artificial intelligence, data science/big data, and communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Submm astronomy technology was identified by the community as having a wide range of high priority science exploitation areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' There is a very wide range of opportunities for the UK to play a leading role in large international astronomy projects, covering all wavelengths of observational astronomy, including especially the Vera C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Rubin Observatory, SKA, Euclid, LISA, and ELT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' International leadership opportunities also exist in theoretical astrophysics, reflecting our high priority technology item below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Although outside the STFC remit, there is also a very strong community appetite for UK international leadership opportunities via space agency bilaterals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' NASA/China/Japan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' There is currently a gap in funding resources for exploratory instrumentally-focused work, compared to that available for science exploitation and technology development with clear near-term goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' There are also structural problems in supporting the career development of instrumentation specialists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Some important technology roadmap conclusions were suggested directly by the community, and which AAP endorses: The strategic importance of trading unique UK technical capabilities, often mission-critical, for a seat at the table setting the scientific agenda of a mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' “Creating consistent opportunity for scientific leadership of future missions will require a strategic plan to link university-led innovation to the longer-term resources needed to support future major technical leadership bids.” “We do not believe anyone can unambiguously identify the technologies needed to address STFC’s key science challenges in the next decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' So to remain competitive there must be strong UK investment in a broad program of advanced instrumentation and its supporting technologies” which the authors suggested should include precision optomechanics, nm-resolution metrology and both electromagnetic and non-electromagnetic sensor development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Page 22 of 37 Recommendation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1: There must be strong UK investment in a broad programme of advanced instrumentation and its supporting technologies, including (but not limited to) precision optomechanics, (astro)photonic technology, nm-resolution metrology, sensor development, kinetic inductance detectors and infrared detectors in general, CCDs and CMOS detectors, receiver development, low noise high-electron-mobility transistors, device fabrication capabilities, software and computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP is pleased to note that there is now a dedicated STFC early TRL funding stream with follow-on fund, which had its first round in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This covers all the STFC core science programme but many astronomy groups have applied .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The call was associated with a capital call this year for lab 48 equipment etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' (over £1M awarded) AAP are also pleased that STFC expect the second joint 49 UKSA-STFC early TRL Technology Fund for Space Science will shortly be announced offering up to £1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='6M for targeted projects in detectors, miniaturisation and robotics for science instruments, and that it is expected this will be repeated in 2023 and 2024 with different themes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The priorities listed below are for near- and mid-term timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Our consultations also asked the community about the anticipated science themes on longer timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Community-identified science themes to be addressed in the next two decades include “life elsewhere” and “exoplanet formation and characterisation”, studies the nature and composition of Dark Matter and Dark Energy in order to disentangle the processes involved in galaxy formation and evolution, high precision gravitational tests of GR, gravitational waves, including different sources as well as waveform models ahead of LISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 Top priority science facilities The top priority science items have wide-ranging and transformative science goals and world-leading technology development, with strong and very wide-ranging / almost-universal community support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 European Southern Observatory ESO (including ELT, ALMA) is a clear joint top priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' It has very clear evidence of widespread community support both for near-term science (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' VLTs, ALMA) and mid-term (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ELT first and second generation instrumentation) with transformative science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' See section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4, and see Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='6 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The UK Programme for the Extremely Large Telescope positions a wide portion of the UK community strategically in future decades for major science, including in the following science and instrumentation areas: Towards Earth: Direct Detection of Exoplanets with the Extremely Large Telescope Planetary Camera and Spectrograph ELT-PCS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' HARMONI: the first light integral field spectrograph for the Extremely Large Telescope ANDES : the High Resolution Spectrograph for the Extremely Large Telescope MOSAIC: the multi-object spectrograph for the Extremely Large Telescope METIS: The mid-IR imager and spectrograph for the Extremely Large Telescope The latter four instruments together cover broad science cases ranging from exoplanets to the epoch of reionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 Square Kilometer Array The SKA has very clear evidence of wide community support (including beyond ‘typical’ radio astronomy groups) with field-changing science across a very wide range of astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' In 2022, SKA Observatory was awarded £66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='7 million from the Infrastructure Fund in total including future funding years (following endorsement by AAP for funding via the “priority projects” process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' See section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4, and see Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 49 Colin Vincent, priv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=', 8 November 2022 48 Colin Vincent, priv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=', 8 November 2022 Page 23 of 37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 European Space Agency and other space opportunities The ESA Science mandatory programme is formally outside the STFC remit, except e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' in the science exploitation of current or past missions in AGP, and occasionally in preparatory technology or software infrastructure development for science in PPRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Nevertheless it covers projects supporting almost the entirety of the UK astronomy science and technology communities, with missions comparable (or greater) in cost, impact and community support to our highest-rated ground-based projects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' JWST, Athena, Ariel, Plato, Euclid, LISA etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' As part of the “dual key” mechanism, AAP advises STFC Science Board when UKSA requests the view of the STFC science community on prospective astronomy missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Our community consultations highlighted the community importance of several major current and future space missions, despite them being in the UKSA remit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The community support is also well-evidenced by the extensive UK signatories to the respective mission study papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Rather than consider these individually as with STFC-supported ground-based projects, we have treated these collectively as a single item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' At the time of writing there are a number of external policy instabilities, and AAP would not want to send an inadvertent signal by omitting this comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' There is a very wide astronomy community appetite for exploring bilateral space agreements and other space opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP received many excellent white paper ideas across the breadth of the astronomy research portfolio,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' including: The Next Generation of Space missions and their role to understand the most extreme objects in our Universe: Black holes and Neutron Stars Dark Ages and First Light: A case for space-based radio cosmology (a science case for participation in,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' or UK leadership of,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' a range of potential future space-based 21cm cosmology experiments) The search for living worlds and the connection to our cosmic origins (a wide-ranging science case supporting UK roles in exoplanet space missions in the NASA Decadal Review,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' particularly a large optical/ultraviolet/infrared space telescope) Gravitational Wave Astronomy with the Laser Interferometer Space Antenna (also within the remit of PAAP) The crucial missing piece in the multi-wavelength jigsaw of the Universe: the Far-Infrared;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Participation in a NASA Far Infrared Probe Mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' These white papers have wide-ranging support, high science return, with strategic involvement in mission launching in over 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' GaiaNIR: Near-Infrared Astrometry Revealing the Galactic Ecosystem CASTOR: A Wide-Field, UV Space Telescope UK opportunities for extrasolar research with the Twinkle space mission Many white papers had extensive aspirations for use of X-ray space facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' These opportunities fall outside the STFC remit, but should UKSA seek the views of STFC Science Board on the alignment of these projects with roadmap strategies, then there is already prima facie evidence in their support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP notes that there is currently some discussion of removing all proprietary time from NASA federally funded missions ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' US PIs with NASA facility time allocations are often also awarded 50 exploitation grant funding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' in contrast, UK PIs have no such guarantee of post-doctoral support to exploit data in these highly competitive allocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The pressure on UK exploitation grants would further disadvantage the UK community if this proposed policy is enacted (as well as having other consequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 50 “On 25 August, the White House’s Office of Science and Technology Policy ordered departments and agencies to move toward making the results of all federally funded research freely and immediately available by 2026.” Clery, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='., 2022, Science, Volume 378, Issue 6619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/content/article/should-webb-telescope-s-data-be-open-all Page 24 of 37 AAP also received a white paper submission on best practices for public engagement with UK space missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendations included providing sustained engagement support pre- and post-launch, a better awareness of equity, diversity and inclusion in UKSA resource allocation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' better recognition of outreach in career progression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' retention of hardware created during mission development, for museums;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' a central resource hub for STEM outreach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' and funding for better coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' These are beyond the AAP remit but AAP encourages the authors to publish their recommendations and/or submit them to the UKSA advisory structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2: The ESA mandatory science programme covers projects supporting almost the entirety of the UK astronomy science and technology communities, with missions comparable (or greater) in cost, impact and community support to our highest-rated ground-based projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The UK must continue ESA membership, and there is widespread astronomy community appetite for exploring bilateral space agreements and other space opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 Very high priority science and technology The very high priority science items have wide-ranging and important science goals and world-leading technology development, with strong and wide-ranging community support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP received a community submission in favour of Gravitational Wave Astronomy with Advanced LIGO+ and Cosmic Explorer, and note that this very high priority area is already considered in the PAAP roadmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 High Performance Computing High Performance Computing, in particular the DiRAC HPC Facility (including capital equipment, operations, training & innovation) has clear UK leadership in many areas underpinning a wide range of UK astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' See Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='7 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 Vera Rubin Observatory The Vera Rubin Observatory Legacy Survey of Space and Time covers a very wide range of high priority UK science themes (section 2) and is critical to UK leadership in many anticipated future developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The major role the UK is playing in the LSST will enable membership for a large fraction of the UK astronomical community;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' we anticipate Rubin LSST soon achieving the almost-universal breadth of support to rank alongside SKA and ESO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' We note above in section 4 that there is a deficit in support for digital infrastructure, such as in supporting the creation of value-added data products from the scientific synergy with Rubin and Euclid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' see Recommendation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 Simons Observatory / CMB science Science from observations of the Cosmic Microwave Background are an obvious very high priority with long-term historic UK leadership internationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The large UK CMB community has a well thought-out strategy with Simons Observatory (SO) as top mid-term priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' SO was not in the core programme of the second BoP review due in part to lack of headroom for major new commitments, but was endorsed by AAP for new funding under the “priority projects”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' SO was awarded £12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='6M of new infrastructure funding from UKRI in 2022 over three years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The CMB community white paper submission to AAP also describes ground-based aspirations beyond SO for CMB Stage 4 and the UK-led CBASS experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Further details on CMB science priorities and further smaller facility aspirations are covered in the PAAP roadmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 High priority science and facilities These are science areas and facilities that may not clearly satisfy all of the criteria for the “Very High Priority” areas above, but include projects that could be developed to very high priority, including Page 25 of 37 those with high science return over the coming 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The science areas and projects listed in this section cover a very wide range of costs, science areas and timescales, so we felt it would not be useful to draw prioritisations among them under some hypothetical constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Rather, the Balance of Programmes process (or an equivalent Astronomy-specific near-term tensioning exercise) should be used in the light of real constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' See Recommendation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 WEAVE The WEAVE instrument is the next generation spectroscopy facility for the William Herschel Telescope (WHT), built by an international STFC-led consortium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The science goals are wide-ranging, covering Galactic archaeology, white dwarfs, neutral Hydrogen selected galaxies, galaxy clusters, low-frequency radio sources from LOFAR, and quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' WEAVE has also been highlighted in several science themes of the ASTRONET roadmap (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 The Maunakea Spectroscopic Explorer and other 10-12m-class massively multiplexed spectroscopy facilities MSE, or the Maunakea Spectroscopic Explorer, is a 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='25-metre optical spectroscopy facility proposed to replace the Canada-France-Hawaii Telescope (CFHT) and sit at the location of the CFHT in an enclosure only slightly larger than the existing dome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The science case is wide-ranging, and spectroscopic facilities of this class are essential for the science exploitation of the coming generation of very wide-area optical and near-infrared imaging surveys from Rubin LSST and ESA Euclid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' More details can be found in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP supported MSE as a priority for potential UKRI infrastructure funding, and spectroscopy facilities of this class have also been highlighted in several science themes of the ASTRONET roadmap (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 LOFAR, the Low Frequency Array LOFAR is a new-generation radio telescope operating at low radio frequencies of ~15 to 240 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' LOFAR is a precursor facility to the SKA, which is one of the community’s top priorities (see above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The science areas covered by LOFAR are vast, from Solar System science to cosmological scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 23 UK universities are participating in the UK LOFAR consortium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' LOFAR is currently operational and highly productive, with UK leadership roles and strong UK science exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' LOFAR’s staged upgrades, known as LOFAR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='0, are already in progress and the initial stages have approved STFC funding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' this will maintain LOFAR’s position as the most powerful very-low-frequency and long-baseline radio interferometer internationally until at least 2030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' LOFAR features in several science themes of the ASTRONET roadmap (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 VLT BlueMUSE BlueMUSE is a panoramic 3-D spectroscopy instrument in the UV/blue for the ESO VLT, addressing a very wide range of science from the Solar System to Galactic astronomy to redshifts of z~6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' BlueMUSE has been approved by ESO for a Phase A start in January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The consortium is led from Lyon, with the UK bidding for a key role in assembling, aligning and testing the spectrographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' BlueMUSE features in several science themes of the ASTRONET roadmap (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5 VLT CUBES CUBES is a near-ultraviolet spectrograph being built for the ESO VLTs, offering a much higher throughput and sensitivity to the current state-of-the-art VLT UVES instrument (under 5 percent throughput including telescope, atmosphere and instrument).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This wavelength range is rich in diagnostic information covering a wide range of science goals covering Solar System science, Galactic and extragalactic astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The project passed its Phase A study review with ESO in June 2021 and started construction in early 2022, with operations planned to start in 2028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' CUBES is led by Italy with the UK leading the Science and Detector work packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' CUBES also features in several science themes of the ASTRONET roadmap (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Page 26 of 37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='6 UK National Radio Astronomical Observatory UKNRAO UKNRAO is a bottom-up proposal to create a national radio observatory that would be a S&T/R&D coordinating centre for radio astronomy, a timely initiative designed to fully exploit the UK’s £200M+ investment in the SKA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This white paper came about partly in response to previous AAP feedback on the diversity of radio proposals in the “priority projects” process, and AAP are very pleased to see that the radio community has engaged in careful, consultative strategic planning to create this white paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' UKNRAO would be established as a Company Limited by Guarantee with a governance structure similar to the highly successful Alan Turing Institute, bringing together all the major players in UK radio astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The science goals addressed by radio astronomy are very broad, as seen e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' in section 2, and the UK has a large and vibrant radio community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' There are excellent synergies with the UK ALMA Regional Centre co-located at one of the UKNRAO sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The aspirations include a UK SKA Regional Centre, recently funded by UKRI infrastructure funding and perfectly aligned with one of the community top priorities (see above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' As well as enhanced national and international standing, UKNRAO would also host key UK instrument and technology development, including a flagship “Receiver Factory” and the UK Extreme High Frequency Facility that was submitted to AAP for the most recent UKRI infrastructure call (section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='7 Submm Astronomy on Large Scales / AtLAST Two white papers made a strong case to AAP for wide-field mapping at sub-millimetre and millimetre wavelengths, to provide the UK with tactical and strategic advantages in creating feeder programmes for the heavily competitive time allocations on the ~£1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3billion ALMA facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The broad science goals cover star formation, the interstellar medium, galaxies and cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' More than 90% of the energy output from newly formed stars and galaxies is hidden by dust and only accessible to these facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ALMA excels at deep pointed observations, but even the largest ALMA programmes cover only a few square arcminutes, so wide-field mapping is essential to feed ALMA with targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The largest submm telescope in the world is the James Clerk Maxwell Telescope (JCMT, with UK roles funded by PPRP and university contributions) and is likely to remain the largest for the next decade, until the AtLAST facility is available;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the UK also has collaborative access to the Mexican Large Millimetre Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The UK has a very strong track record in submm/mm-wave astronomy, a broad and vibrant community, and many leadership roles in JCMT Large Programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The UK also has international leadership in related detector technologies, such as KIDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Wide-field (sub)-millimetre mapping also features in several science areas of the ASTRONET roadmap (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP recommended the AtLAST instrument feasibility study and JCMT instrumentation as a priority for UKRI Infrastructure funds (section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='8 Magdalena Ridge Observatory Interferometer, MROI The Magdalena Ridge Observatory Interferometer MROI has a broad, transformative science application, from Galactic star and planet formation, to precision stellar astrophysics, to AGN formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' This white paper aspires for ten 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4m diameter telescopes at MROI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' with seven or more telescopes the scientific capabilities of the MROI will far exceed the capabilities of both the state-of-the-art facilities VLTI and CHARA, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' double the angular resolution of VLTI and triple the imaging field of view of CHARA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The limiting sensitivity would be 2-5 magnitudes fainter than the current state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The first three telescopes of MROI are currently funded by the US Air Force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The expansion to 10 telescopes was one of the priorities for optical/IR interferometry in the 2021 US Decadal Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The required technology development would position the UK scientifically (see Recommendation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1) and establish UK expertise for the Planet Formation Imager (timescales: after ELT first light) and ELT first-generation instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Page 27 of 37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='9 HARPS-3 HARPS-3 is a new generation high resolution, high stability echelle fibre spectrograph, supporting the UK’s world-leading roles in exoplanet science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' HARPS-3 is to be installed on the Isaac Newton 51 Telescope, and aims to conduct precise Doppler spectroscopy of exoplanet systems for the discovery of Earth analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The UK-led instrument is part-funded by PPRP and will achieve first light in 2023, and full operations are planned to start in 2024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Half of the observing time will be available to the Isaac Newton Group community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The instrument also features as an exoplanet priority in the ASTRONET roadmap (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='10 Other facility capabilities The above high priority facilities include the projects for which dedicated white papers were submitted in our consultation, but there are several others where comparable community support, breadth of science goals and high science impact may also be evidenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Examples include: UK involvements in DESI, CCAT, 4MOST;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' VLT/MOONS (six-country collaboration, managed by STFC at UK ATC);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' GOTO (prioritised by AAP in the previous “priority projects” process and with UK involvement now funded by PPRP);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' CTA (UK technical interests in instrumentation and science interests in multi-messenger astronomy);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' laboratory astrochemistry (UK international science leadership underpinning a wide range of science themes);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' crowd-sourced astrophysics (UK has taken leadership through e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the Zooniverse platform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Funding only the top priority facilities is not sufficient for the UK to achieve a scientific return on the investment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the “small” facilities are an essential part of the UK astronomy programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' A facility or instrument being “small” in cost terms is not inconsistent with it being world-leading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3: Maintaining UK access to the top priority facilities is not sufficient to ensure returns on these major investments, so STFC should maintain and develop a broad portfolio of high/very high priority science facilities, including (but not limited to) WEAVE, LOFAR, JCMT, GOTO, LT/NRT, WASP, NGTS, DIRAC, IRIS, e-MERLIN, Lovell Telescope, among current facilities, and UK roles in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' DESI, CCAT, 4MOST, BlackGEM, as well as fostering roles in currently non-STFC-supported projects such as REACH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' looking further ahead, HARPS-3, Rubin LSST, Simons Observatory, AtLAST, MSE, CTA, VLT/MOONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' As part of this broad portfolio supporting the highest priority science, AAP further recommends support for laboratory astrochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5 Emerging community priorities AAP received a number of white papers covering important science and/or technology developments with a significant degree of community support, including the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 Astronomy and Space Domain Awareness This white paper highlights the importance of space situational awareness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the study of satellites and debris in Earth orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' In the UK alone, satellite services underpin £360 billion of economic 52 activity per year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' ESA estimates that around one million objects larger than 1 cm are in orbit, all of which are capable of damaging a satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Through the STFC 21st Century Challenge Network+, STFC have funded the Global Network on Sustainability in Space, GNOSIS , which has successfully 53 facilitated interdisciplinary approaches to the problems of space domain awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' More broadly, there is arguably good evidence that the 21st Century Challenges have been very effective for fostering societal impact of STFC science, but this is beyond the scope of the AAP roadmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 53 See https://gnosisnetwork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/ 52 See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' https://publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='parliament.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='uk/pa/cm5803/cmselect/cmsctech/100/report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='html 51 See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='nobelprize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/prizes/physics/2019/summary/ Page 28 of 37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2 Line intensity mapping at millimetre/sub-millimetre wavelengths The white paper on Line Intensity Mapping at millimetre/sub-millimetre wavelengths offers a roadmap for a new and complementary probe of early universe cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The science goals cover inflation and primordial non-Gaussianity, dark energy through baryonic acoustic oscillations, the growth of large scale structure, and fundamental physics (neutrinos and light relics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The technique also features in the ASTRONET roadmap (see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' While it does not currently have the breadth of science goals and UK community support of the high priority facilities and facility aspirations above, the science is manifestly internationally excellent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' A bespoke instrument for the JCMT (where UK access is currently funded by PPRP and university contributions) would provide UK leadership in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' To support the development of an evidence base for wider community support, AAP recommends that this is considered as part of the wider portfolio UK science and technology in (sub)-millimetre astronomy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' see Recommendation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3 Development in Africa with Radio Astronomy, DARA The Development in Africa with Radio Astronomy programme, DARA is a radio 54 astronomy training project funded by international development funding, through the Global Challenges Research Fund and the Newton Fund, as well as securing matching funding from South Africa’s NRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' DARA has also been supported by the EU Jumping JIVE project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Since 2015 it has been delivering basic training in radio astronomy and associated technologies to young people across the eight DAC-list countries partnering with South Africa in hosting SKA-MID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' A sister programme, DARA Big Data, has provided MSc and PhD-level advanced training in Africa and in UK universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP commends the creativity and resourcefulness of the DARA team in using astronomy training to address societal challenges in DAC-list countries, but also notes the anticipated cuts to UK ODA funding that will severely curtail these activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The white paper asks STFC to continue its strategic partnership with NRF in training and placements of researchers and 55 technicians, to which AAP is happy to concur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the five-year agreement started in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4 Collaboration with Goonhilly Earth Station Ltd on Radio Astronomy and Space Applications This white paper was submitted on behalf of the CUGAS consortium, the Consortium of Universities for Goonhilly Astronomy and Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The science goals covered in this collaboration include radio astronomy, interferometry, satellite communications, and space situational awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The project seeks to integrate the Goonhilly dishes into e-MERLIN, providing much needed long baselines and higher angular resolutions, while state-of-the-art low noise instrumentation benefits commercial satellite communication roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Discussions are underway on providing a sovereign UK space situational awareness capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Further funds are needed, either from CUGAS members or from STFC, for the cost of the data link and hydrogen maser time standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' AAP commend the CUGAS consortium for their creativity and resourcefulness in pursuing this industry engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5 High-risk, high return science areas AAP were very pleased to receive a white paper on searching for techno-signatures via anomalies in astronomical data, and are happy to record the community support for this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' A proportion of resources must always be spent on high-risk, high-return science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='7 Projects with a moderate or low science return No projects submitted to AAP fall into this category, whether through the consultation or white papers or the call for UKRI infrastructure projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 55 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ukri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/news/forging-a-new-partnership-with-south-africa/ 54 See www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='dara-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org Page 29 of 37 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Scope for further community consultation AAP have considered whether there is scope and/or community capacity for further major consultations at this stage, following the model e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' of the PPAP review of the entirety of UK particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' With the ASTRONET review still in the process of reporting the last of its findings on the science roadmap for European astronomy (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3), and with the US Decadal Review recently completed (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2), there does not appear to be an obviously compelling case for a UK-specific review of all astronomy, because it would duplicate at least some of this recent and ongoing effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Furthermore, the reduced survey return rate in the AAP consultation may indicate a decreased UK astronomy community capacity or appetite for further consultation activities, though the global health crisis may also have been a factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Nevertheless, there are some focussed areas where further consultations may be useful, such as an overview of community aspirations and priorities in sub-mm/mm-wave science and technology (Recommendation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1), and an STFC review of the career structure of instrumentation and technical roles both within and beyond astronomy (Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The UK CMB community have already organised their own bottom-up community roadmap, updated in the CMB community white paper submitted to AAP and also addressed in the PAAP roadmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The UK exoplanet community last had an STFC community consultation in 2015, and our white paper submissions clearly evidence a desire of the UK exoplanet 56 community to develop a stellar variability and extreme precision radial velocity roadmap in liaison with other international bodies such as ESA, ESO, NASA, NSF etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' (Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' such a roadmap would cover areas of interest of both AAP and SSAP, as well as UKSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Radio astronomy was last reviewed by STFC in 2017 following the 2016 STFC Balance of Programmes Review, and 57 among several conclusions recommended a review of e-MERLIN from 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' the radio community has since created a vision for a UKNRAO (section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='6) so the landscape has changed since the 2017 review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Furthermore, the JIVE ERIC, linked to the pursuance of e-MERLIN, is currently three years into its five-year window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' e-MERLIN is nominally operational at least until SKA is operational and the role for other radio contributions can be reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Summary list of recommendations Recommendation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1: The 2022 AAP Science and Technology Roadmaps should not be used in isolation to evidence current community support without further reference to AAP, because astronomy is a fast-paced and rapidly changing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Nor should these roadmaps be used to draw a funding threshold line in the event of changes to the core programme funding, because the Balance of Programmes Review (or a hypothetical Astronomy-specific near-term tensioning exercise) is the appropriate process for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1: STFC and UKSA should support the UK exoplanet community in developing a stellar variability and extreme precision radial velocity roadmap in liaison with other international bodies such as ESA, ESO, NASA, NSF etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2: STFC should prioritise increasing the exploitation line, which is the community’s top priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 57 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ukri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/about-us/stfc/planning-strategy-reviews/2017-uk-radio-astronomy-strategic-review/ 56 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ukri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/about-us/stfc/planning-strategy-reviews/exoplanets-uk-research-review/ Page 30 of 37 Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3: UK association to the EU Horizon funding programme is critical for astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Replacing the funding alone would alleviate the exploitation pressure but would disconnect the UK from continental-sized research networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4: STFC should review the career structure for instrumentation and technical roles, both within and beyond astronomy and discuss with Universities as to how to implement that within their structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='5: STFC must maintain the UK role in the SKA and support the development of the UK SKA Regional Centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='6: The UK must remain a member of the European Southern Observatory and play leading roles in its development of its world-class instrumentation, including the second- and third-generation instrument suite of the ELT and the development of ALMA instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='7: UK HPC capabilities such as DiRAC & IRIS underpin a wide range of world-leading UK theoretical astrophysics and data science that must be continually supported and upgraded to remain competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1: STFC (via AAP) should commission a review of UK submm/mm-wave science and technology, covering UK aspirations for current and future large single-dish facilities that feed the major international interferometers, and the underpinning aspirations for next-generation instrumentation, identifying areas of international excellence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2: There should be increased support for digital infrastructure in UK astronomy, including the creation of value-added data and software products, open science, and the development and implementation of machine learning technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' See also Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='1: There must be strong UK investment in a broad programme of advanced instrumentation and its supporting technologies, including (but not limited to) precision optomechanics, (astro)photonic technology, nm-resolution metrology, sensor development, kinetic inductance detectors and infrared detectors in general, CCDs and CMOS detectors, receiver development, low noise high-electron-mobility transistors, device fabrication capabilities, software and computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='2: The ESA mandatory science programme covers projects supporting almost the entirety of the UK astronomy science and technology communities, with missions comparable (or greater) in cost, impact and community support to our highest-rated ground-based projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' The UK must continue ESA membership, and there is widespread astronomy community appetite for exploring bilateral space agreements and other space opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Recommendation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='3: Maintaining UK access to the top priority facilities is not sufficient to ensure returns on these major investments, so STFC should maintain and develop a broad portfolio of high/very high priority science facilities, including (but not limited to) WEAVE, LOFAR, JCMT, GOTO, LT/NRT, WASP, NGTS, DIRAC, IRIS, e-MERLIN, Lovell Telescope, among current facilities, and UK roles in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' DESI, CCAT, 4MOST, BlackGEM, as well as fostering roles in currently non-STFC-supported projects such as REACH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' looking further ahead, HARPS-3, Rubin LSST, Simons Observatory, AtLAST, MSE, CTA, VLT/MOONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' As Page 31 of 37 part of this broad portfolio supporting the highest priority science, AAP further recommends support for laboratory astrochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Appendix: List of acronyms 4MOST: 4-metre Multi-Object Spectroscopic Telescope AAP: Astronomy Advisory Panel AGP: Astronomy Grants Panel ALMA: Atacama Large Millimetre Array ANDES: ArmazoNes high Dispersion Echelle Spectrograph (formally known as HIRES) AO: Adaptive Optics APEX: Atacama Pathfinder Experiment APPEC: Astroparticle Physics European Consortium ASTRONET: a network of European funding agencies and research organisations in astronomy (not an acronym,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' but capitalised) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ASTRI: Astrofisica con Specchi a Tecnologia replicante Italiana ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ATC: Astronomy Technology Centre ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='AtLAST: Atacama Large Aperture Submillimeter Telescope ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='BINGO: Baryon Acoustic Oscillations from Integrated Neutral Gas Observations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='BiSON: Birmingham Solar Oscillations Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='BISOU: Balloon Interferometer for Spectral Observations of the Universe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='BlackGEM: a wide-field array of optical telescopes to be located at ESO’s La Silla Observatory in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='Chile’s Atacama desert (not an acronym ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='BlueMUSE: Blue Multi Unit Spectroscopic Explorer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='BoP: Balance of Programmes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='CAHA: Centro Astronómico Hispano-Alemán ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='CARMENES: Calar Alto high-Resolution search for M dwarfs with Exoearths with Near-infrared and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='optical Échelle Spectrographs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='CBASS: C-Band All Sky Survey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='CCAT: Cerro Chajnantor Atacama Telescope ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='CCD: Charge Coupled Device ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='CFHT: Canada-France-Hawaii Telescope ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='CHARA: Center for High Angular Resolution Astronomy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='CMB: Cosmic Microwave Background ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='58 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/public/about-eso/acronyms/ Page 32 of 37 CMB-S4: Stage Four Cosmic Microwave Background experiment CMOS: Complementary metal–oxide–semiconductor Co-I: Co-investigator CONCERTO: Carbon CII line in post-reionisation and reionisation epoch CRIRES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' CRIRES+: Cryogenic high-resolution cross-dispersed infrared echelle spectrograph CSA: Canadian Space Agency CTA: Cherenkov Telescope Array CUBES: Cassegrain U-Band Efficient Spectrograph CUGAS: Consortium of Universities for Goonhilly Astronomy and Space DESI: Dark Energy Spectroscopic Instrument DiRAC: Distributed Research using Advanced Computing DKIST: Daniel K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Inouye Solar Telescope EAGLE: Evolution and Assembly of GaLaxies and their Environments EC: European Commission EDI: Equity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Diversity and Inclusion EHT: Event Horizon Telescope ELT: Extremely Large Telescope e-MERLIN: enhanced Multi Element Remotely Linked Interferometer Network EOSC: European Open Science Cloud ERIC: European Research Infrastructure Consortium ERIS: Enhanced Resolution Imager and Spectrograph ESA: European Space Agency ESO: European Southern Observatory ESPRESSO: Echelle SPectrograph for Rocky Exoplanets and Stable Spectroscopic Observations EVN: European VLBI Network FAIR: Findable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Accessible,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Interoperable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Reusable FRB: Fast Radio Burst GMRT: Giant Metrewave Radio Telescope GNOSIS: Global Network on Sustainability in Space GOTO: Gravitational-wave Optical Transient Observer GRAND: Giant Radio Array for Neutrino Detection GranTeCan: Gran Telescopio Canarias Page 33 of 37 GRAVITY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' GRAVITY+: a 4-beam interferometric combiner at VLTI operating in K-band (not an acronym ) 59 GRST: Fermi Gamma-Ray Space Telescope GTM: Gran Telescopio Milimétrico Alfonso Serrano GW: Gravitational Wave HARMONI: High Angular Resolution Monolithic Optical and Near-infrared Integral field spectrograph HARPS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' HARPS-3: High Accuracy Radial velocity Planet Searcher HARPS-N: High Accuracy Radial velocity Planet Searcher,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' North ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='HERA: Hydrogen Epoch of Reionization Array ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='HESS: High Energy Stereoscopic System ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='HIRES: high-resolution spectrograph (ELT instrument now known as ANDES) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='HiRISE: High-Resolution Imaging and Spectroscopy of Exoplanets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='HPC: High Performance Computing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='IMF: Initial Mass Function ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ING: Isaac Newton Group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='INT: Isaac Newton Telescope ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='IRIS: eInfrastructure for Research and Innovation in STFC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='JCMT: James Clerk Maxwell Telescope ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='JIVE: Joint Institute for VLBI ERIC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='JWST: NASA/ESA/CSA next-generation infrared space telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' (Following the provisional best practice outlined by the Royal Astronomical Society , we do not expand the acronym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=') 60 JVLA: Jansky Very Large Array KAGRA: Kamioka Gravitational Wave Detector KIDS: Kinetic Inductance Detectors LGBTQI+: Lesbian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Gay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Bisexual,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Transgender,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Queer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Intersex LHC: Large Hadron Collider LIGO: Laser Interferometer Gravitational-Wave Observatory LISA: Laser Interferometer Space Antenna LMT: Large Millimetre Telescope (in Spanish: GTM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' Gran Telescopio Milimétrico Alfonso Serrano) LOFAR: Low Frequency Array LSPE-strip: Large-Scale Polarization Explorer/Survey Tenerife Polarimeter 60 See https://ras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='uk/news-and-press/news/ras-and-jwst 59 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/public/about-eso/acronyms/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='Page 34 of 37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='LSPE-swipe: Large-Scale Polarization Explorer/Short Wavelength Instrument for the Polarization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='Explorer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='LSST: Legacy Survey of Space and Time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='LT: Liverpool Telescope ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='MAGIC: Major Atmospheric Gamma Imaging Cherenkov telescopes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='MATISSE: Multi AperTure mid-Infrared SpectroScopic Experiment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='MAVIS: MCAO-Assisted Visible Imager and Spectrograph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='MCAO: Multi-Conjugate Adaptive Optics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='MeerKAT: originally the Karoo Array Telescope (not an acronym) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='MeerLICHT: the Dutch translation for ‘more light’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' is an astronomical project which aims to provide a simultaneous,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' real-time optical view of the radio (transient) sky as observed by MeerKAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' (Not an ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='acronym) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='MeerTime: a five-year pulsar timing program on the MeerKAT array (not an acronym) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='METIS: Mid-Infrared ELT Imager and Spectrograph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='MICADO: Multi-AO Imaging Camera for Deep Observations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='MIDI: Mid-infrared interferometric instrument ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='MKIDS: Microwave Kinetic Inductance Detectors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='MOSAIC: Multi-Object Spectrograph (for ELT) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='MOONS: Multi-Object Optical and Near-infrared Spectrograph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='MROI: Magdalena Ridge Observatory Interferometer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='MSE: Mauna Kea Spectroscopic Explorer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='MUSE: Multi Unit Spectroscopic Explorer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='MWA: Murchison Widefield Array ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='NASA: National Aeronautics and Space Administration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='NEID: NN-explore Exoplanet Investigations with Doppler spectroscopy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ngEHT: Next generation Event Horizon Telescope ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='NGTS: Next-Generation Transit Survey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='ngVLA: Next Generation Very Large Array ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='NIR: Near Infra-Red ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='NIRPS: Near Infra Red Planet Searcher ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='NN-Explore: NASA-NSF Explore,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content='org/record/3356296 Page 36 of 37 WHT: William Herschel Telescope WIYN: Telescope operated by the University of Wisconsin–Madison (W), Indiana University (I), Yale University (Y), and the National Optical Astronomy Observatories (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} +page_content=' VISTA: Visible and Infrared Survey Telescope for Astronomy VLA: Very Large Array VLBI: Very Long Baseline Interferometry VLT: Very Large Telescope VLTI: Very Large Telescope Interferometer Page 37 of 37' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE5T4oBgHgl3EQfJQ70/content/2301.05457v1.pdf'} diff --git a/qtAyT4oBgHgl3EQfZfdb/content/tmp_files/2301.00224v1.pdf.txt b/qtAyT4oBgHgl3EQfZfdb/content/tmp_files/2301.00224v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0fb0e38e242c5317dc7834d5dd27c8dcb9de647c --- /dev/null +++ b/qtAyT4oBgHgl3EQfZfdb/content/tmp_files/2301.00224v1.pdf.txt @@ -0,0 +1,3570 @@ +arXiv:2301.00224v1 [math.OC] 31 Dec 2022 +ERGODIC RISK-SENSITIVE CONTROL—A SURVEY +ANUP BISWAS† AND VIVEK S. BORKAR‡ +Abstract. Risk-sensitive control has received considerable interest since the seminal work of +Howard and Matheson [120] because of its ability to account for fluctuations about the mean, +its connection with H∞ control, and its application to financial mathematics. In this article we at- +tempt to put together a comprehensive survey on the research done on ergodic risk-sensitive control +over the last four decades. +Contents +1. +Introduction +2 +2. +Risk-sensitive control of Discrete time Markov chains +4 +2.1. +Finite state space +5 +2.2. +Countable state space +7 +2.3. +General state space +9 +3. +Risk-sensitive control of diffusions +11 +3.1. +Generalized principal eigenvalue +11 +3.2. +ERSC for controlled diffusions +13 +3.3. +Generalized Collatz-Wielandt formula +19 +4. +Risk-sensitive control of continuous time Markov chains +23 +5. +Risk-sensitive maximization problems and beyond +28 +5.1. +Risk-reward problems +28 +5.2. +Risk-sensitive games +29 +6. +Algorithms +29 +6.1. +Policy iteration +29 +6.2. +Relative value iteration +33 +6.3. +Linear programming +35 +6.4. +Reinforcement learning +39 +Acknowledgments +41 +References +41 +† Department of Mathematics, Indian Institute of Science Education and Research Pune, Dr. Homi +Bhabha Road, Pune 411008, India +Department of Electrical Engineering, Indian Institute of Technology, Powai, Mumbai 400076, +India +E-mail addresses: anup@iiserpune.ac.in, borkar@ee.iitb.ac.in. +Date: January 3, 2023. +2010 Mathematics Subject Classification. Primary: 90C40, 91B06, 93E03 Secondary: 93B36, 93B52. +Key words and phrases. Risk-sensitive control, Bellman equation, generalized principal eigenvalue, multiplicative +dynamic programming, verification theorem, Markov decision process. +1 + +2 +ANUP BISWAS AND VIVEK S. BORKAR +1. Introduction +Given a controlled stochastic process X = {Xt} on a state space S, controlled by the process ζ, +the ergodic risk-sensitive cost is defined as +Ex(c, ζ) := lim sup +T→∞ +1 +γT log Ex +� +e +� T +0 γc(Xt,ζt)dt� +, +x ∈ S, +c being the running cost and γ ̸= 0 being the risk-parameter. The ergodic risk-sensitive control +(ERSC) problem is about studying the minimization problem +λ∗ = inf +x∈S inf +ζ Ex(c, ζ). +Replacing ‘inf’ with ‘sup’ leads to the corresponding reward maximization problem which we discuss +briefly later. Observe , however, that unlike the classical cost functionals, the reward maximization +problem is not equivalent to the cost minimization problem obtained by flipping the sign of the +instantaneous reward. +Suppose c is non-negative (more generally, bounded from below). The decision maker is supposed +to be risk-averse or risk-sensitive for γ > 0, risk-neutral for γ = 0 and risk-seeking for γ < 0. +The risk-neutral case, in a suitable limiting sense, corresponds to the the classical ergodic control +problem which has already been studied extensively (see [6, 17] and references therein). The goal +of this article is to review the development of ERSC problems when γ ̸= 0. The study of ERSC +can be traced back to the seminal work of Howard and Matheson [120] where the problem was +studied for controlled Markov chains with finite state and action sets. Since then this area has +been developed intensively in the past forty years. +A major fillip came from a series of works +by Peter Whittle in the eighties, culminating in [159]. One major motivation was the strongly +felt need for criteria going beyond those based purely on mean rewards, that did not put any +weight whatsoever on fluctuations around the mean. The obvious extensions such as considering +a weighted sum of mean and variance in some form (the Markowitz model in finance being the +prime example) faced problems such as non-availability of the ‘principle of time-consistency’ or the +dynamic programming principle. As the exponential function can be viewed as the weighted sum +of all powers, its expectation is a weighted sum of all moments. Thus its expctation does account +for higher moments. In addition, by facilitating a multiplicative form of dynamic programming (as +opposed to the additive form for classical criteria), it does obey the principle of time consistency. +This made risk-sensitive control an attractive proposition. +Two classical applications of ergodic risk-sensitive control problem motivated by such consider- +ations come from robust control theory and portfolio optimization problems. +– (Robust control theory) Since it is often almost impossible to find a true model of a system, +robust control theory seeks criteria that could deal with model uncertainty. The connection +between risk-sensitive control and robust control started with the work of Glover and Doyle +[108] (see also Whittle [158,159]). Risk-sensitive minimization problems naturally give rise +to two person zero-sum differential games (first found in the work of Jacobson [123]) which +are of interest in the robust control theory. In the differential game formulation there are +two players, one representing the disturbance entering the system which will attempt to +degrade system performance, and the other representing the actual control for the system. +Readers may also consult [84] for more on power gain inequality and its connection to the +ergodic risk-sensitive value. We shall briefly discuss this and its connection to H∞ control +in Section 3.2.3. +Another way to deal with the model uncertainty is to consider partially observed or Hid- +den Markov chain models. ERSC has been studied in these frameworks as well. It should +be note that under fairly general assumptions and suitable change of measures, partially + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +3 +observed models can be changed into a fully observed control problems (cf. [124]). Inter- +ested readers may consult [22,94,117,124,136] for more details in this direction. +– (Portfolio optimization) The risk-sensitive formulation of the portfolio optimization problem +was introduced in the seminal works of Bielecki and Pliska [41], Fleming and Sheu [99]. Since +then this area has grown substantially (see [43,91,93,141,143] and references therein). In +addition to the considerations already discussed, multiplicative / exponential models arise +naturally in finance due to ‘compounding’ effects. Suppose that there are N risky assets +and the investor allocates fraction ui +t of its wealth to the i-th risky asset, i = 1, 2, . . . , N. +The total wealth Vt, at time t, of the investor is then given by +dVt = Vt +� +rt +� +1 − +N +� +i=1 +ui +t +� +dt + +N +� +i=1 +ui +t +dSi +t +Si +t +� +, +(1.1) +where r denotes the risk-free interest rate, Si the share price of the i-th risky asset. Let +U ⊂ RN be a constrain set and ζt = (u1 +t , . . . , uN +t ) ∈ U for all t. In portfolio optimization, +one wishes to maximize the long term value of γ−1 E[V γ +T ], for some γ ∈ (−∞, 1) \ {0}, over +all possible investment allocations. Now suppose that there are d economic factors given +by the vector ˜Xt ∈ Rd that governs the market performance and evolves according to the +stochastic differential equation +d ˜Xt = ˜b( ˜Xt)dt + dWt, +where W is a d-dimensional standard Brownian motion. The share price dynamics is given +by +dSi +t +Si +t += µi( ˜Xt)dt + σi +D · dWt + σi +I · d ˜W t, +i = 1, . . . , N, +where ˜W is an N -dimensional standard Brownian motion independent of W. Assume that +σi +D, σi +I, r are constant vectors. Applying Itˆo’s formula one can easily find from (1.1) the +differential equation satisfied by log Vt. Then defining +b(x, u) := ˜b(x) + γ +N +� +i=1 +uiσi +D, +¯µi(x) := µi(x) − r, +σi = [σi +D, σi +I] ∈ Rd+N, +ℓ(x, u) := −1 +2(1 − γ)| +N +� +i=1 +uiσi|2 + +N +� +i=1 +ui¯µi(x) + r, +one can check that above maximization problem is equivalent to maximizing (see [93]) +log E +� +eγ � T +0 ℓ(Xt,ζt)dt� +, +where ζt ∈ U and +dXt = b(Xt, ζt)dt + dWt. +Thus the long-term asymptotics (that is, as T → ∞) corresponds to the ergodic risk- +sensitive control problems. +It is worth noting that some of the early work in this direction came from information +theorists, notably Thomas Cover and his associates. See [76], Chapter 16, and its biblio- +graphical note. +We mention in passing another application, viz. to minimizing or maximizing the asymptotic +rate of exit of a controlled Markov process from a prescribed subset of its state space. This can be +reduced to a risk-sensitive control problem [49], [56]. + +4 +ANUP BISWAS AND VIVEK S. BORKAR +The main focus of this article is on ERSC problems. There is also an enormous amount of work +done on the finite horizon version of risk-sensitive control problems which we do not discuss in this +article. Interested readers may look at [2,26–28,31,35, 36,127, 158]. Broadly speaking, the ERSC +problems are treated in three different ways. The first one corresponds to the variational repre- +sentation of the moment generating function. This helps us to transform the above minimization +problem to an ergodic zero-sum game problem (see [70,88,96,97]). The second approach for solving +ERSC problem is an approximation method based on the discounted risk-sensitive problem (see +[69, 80, 138]). Discounted risk-sensitive control is not amenable to dynamic programming, but by +treating the risk-snsitivity parameter as a variable, one manages to make the problem analytically +tractable. The dynamic programming equation of the risk-sensitive control problem is a nonlinear +eigenvalue problem. The third approach is more direct where the nonlinear eigenvalue problem is +analyzed using Krein-Rutman theorem (see [5,14,44]). We divide the review of ERSC problems in +three major parts, namely, discrete time set up, controlled diffusions and continuous time Markov +chains, wherein we touch upon all three approaches above. +The following is a list of the abbreviations used in this paper +DTCMC +discrete time controlled Markov chain +ERSC +ergodic risk-sensitive control +CTCMC +continuous time controlled Markov chain +PIA +policy iteration algorithm +RVI +relative value iteration +We also summarize key notations used in this article +B(X) +Borel σ algebra on the topological space X +Cb(X) +set of all real-valued bounded, continuous functions on X +Ck(X) +set of all k-times continuous differentiable functions on X ⊂ Rd +Ck ++(X) +subset of functions of Ck(X) that are positive on X +λ∗,m +optimal ergodic risk-sensitive value for DTCMC +λ∗,d +optimal ergodic risk-sensitive value for controlled diffusion +λ∗,c +optimal ergodic risk-sensitive value for CTCMC +2. Risk-sensitive control of Discrete time Markov chains +We begin by introducing the general setting of a controlled discrete time Markov chain. Consider +a controlled Markov process X := {X0, X1, . . . } on a Borel space S controlled by a control process +ζ := {ζ0, ζ1, . . . } taking values in U. Here U is a Borel space endowed with the Borel σ algebra +B(U). For every x ∈ S, U(x) ∈ B(U) stands for the nonempty compact set of all admissible actions +when the system is at the state x. The space of all admissible state action pairs is given by K := +{(x, u) : x ∈ S, u ∈ U(x)}. For each A ∈ B(S) the controlled stochastic kernel P(A|·) : K → [0, 1] +is Borel measurable. We denote by c : K → R+ the one-stage cost function. For each t ∈ N, the +space Ht denotes the admissible histories up to time t, where H0 := S, Ht = K × Ht−1. A generic +element ht of Ht is a vector of the form +ht = (x0, u0, x1, u1, . . . , xt−1, ut−1, xt), +with (xs, us) ∈ K, +0 ≤ s ≤ t − 1, x0 ∈ S, +denotes the observable history of the process up to time t. Let us also denote by Fn = B(Hn) := +the Borel σ-field of Hn. An admissible control is a sequence ζ = {ζ0, ζ1, . . . } where for each t ∈ N , +ζt : Ht → U is a measurable map satisfying ζt(ht) ∈ U(xt), for all ht ∈ Ht. The set of all admissible +policies is denoted by U. It is well known that for a given initial state x ∈ S and policy ζ ∈ U +there exists a unique probability measure Pζ +x on (Ω, B(Ω)), where Ω = (S × U)∞, (see [118, p.4], + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +5 +[6]) satisfying the following +Pζ +x(X0 = x) = 1, +and +Pζ +x(Xt+1 ∈ A|Ht, ζt) = P(A|Xt, ζt) +∀ A ∈ B(S) . +(2.1) +The corresponding expectation operator is denoted by Eζ +x. A policy ζ ∈ U is said to be a Markov +policy if ζt(ht) = vt(xt) for all ht ∈ Ht, for some measurable map vt : S → U such that vt(x) ∈ U(x) +for all x ∈ S. The set of all Markov policies is denoted by Um. If the map vt does not have any +explicit time dependence, that is, ζt(ht) = v(xt) for all ht ∈ Ht, then ζ is called a stationary +Markov strategy and we denote the set of all stationary Markov strategies by Usm.(We use the +words ‘strategy’ and ‘policy’ interchangeably.) From [118, p.6] (also see [6]), it is easy to see that +under any Markov policy ζ ∈ Um, the corresponding stochastic process X is strong Markov. For +each ζ ∈ U, the ergodic risk-sensitive cost is given by +Ex(c, ζ) := lim sup +T→∞ +1 +γT log Eζ +x +� +e +�T −1 +t=0 γc(Xt,ζt)� +, +(2.2) +where γ ̸= 0 and X is the discrete time controlled Markov chain (DTCMC) corresponding to the +control ζ ∈ U, with initial state x. Our aim is to minimize (2.2) over all admissible policies U. In +other words, we are interested in the quantity +λ∗,m = inf +x∈S inf +ζ∈U Ex(c, ζ). +(2.3) +We refer to this as an ergodic risk-sensitive control (ERSC) problem. A policy ζ∗ ∈ U is said to be +optimal if for all x ∈ S +Ex(c, ζ∗) = inf +x∈S inf +ζ∈U Ex(c, ζ). +Note that in general, Ex(c, ζ) is not independent of x for ζ ∈ Usm. Let us also mention the optimality +equation which will be important for characterizing the optimal stationary Markov controls. +Definition 2.1. A positive function ψ : S → (0, ∞) and a real number λ are said to form an +eigen-pair (ψ, λ) if +sign(γ)eγλψ(x) = min +u∈U(x) +� +sign(γ)eγc(x,u) +� +S +ψ(y)P(dy|x, u) +� +for x ∈ S. +(2.4) +We call ψ an eigenfunction corresponding to the eigenvalue γλ. +We impose the following standard assumption on our model. +Assumption 2.1. The following hold. +(i) The transition kernel P(·|x, u) is weakly continuous in (x, a), that is, for every f ∈ Cb(S) +we have +� +S f(y)P(dy|x, u) continuous in K. +(ii) u �→ c(x, u) is continuous in U(x) for all x ∈ S. +2.1. Finite state space. Suppose that S is a finite set. The very first ERSC control problem +appeared in the work of Howard and Matheson [120] where the authors studied an ergodic risk- +reward problem under the assumption that X is irreducible and aperiodic under every stationary +Markov policy. Since then the finite state situation has been studied in several works [4, 58, 60, +63, 66, 67, 67–69, 94, 117, 149]. For instance, for a (uncontrolled) Markov chain X with transition +matrix P, it is well-known that +x �→ lim sup +T→∞ +1 +γT log Ex +� +e +�T −1 +t=0 γc(Xt)� +is constant on each communicating class (cf. [58, Lemma 1]). Moreover, if X is irreducible, then +lim sup +T→∞ +1 +γT log Ex +� +e +�T −1 +t=0 γc(Xt)� += 1 +γ log ρ( ˜P) + +6 +ANUP BISWAS AND VIVEK S. BORKAR +where ˜Pij := Pijeγc(i) and ρ( ˜P) denotes the spectral radius of ˜P. Furthermore, the existence of +such an eigen-pair can be characterized by the following result. +Theorem 2.1 ([68]). Let Pxy = P(y|x) for x, y ∈ S. Then the following are equivalent. +(i) For each c : S → R there exists an eigen-pair (ψ, λ) satisfying +eγλψ(x) = eγc(x) Ex[ψ(X1)] = eγc(x) � +y∈S +ψ(y)Pxy, +for all x ∈ S. +(ii) For every cost function c the mapping +x �→ lim sup +T→∞ +1 +γT log Ex +� +e +�T −1 +t=0 γc(Xt)� +is constant. +(iii) The transition matrix P has a unique recurrent class C ⊂ S and there exists a constant m +such that +Px(˘τC ≤ m) = 1 +for all x ∈ S +where ˘τC denotes the return time to the set C, that is, +˘τC := inf{n ≥ 1 : Xn ∈ C}. +Moreover, if one of the above conditions holds, then the eigenfunction ψ can be represented as +follows [66,68] +ψ(x) = Ex +� +eγ �˘τz−1 +i=0 (c(Xi)−λ)� +∀ x ∈ S \ {z}, ψ(z) = 1, +(2.5) +where ˘τz = ˘τ{z}. +λ is the value of the average risk-sensitive cost. +This representation of the +eigenfunction will be crucial in our study and will appear in several places below. The following +result on the ERSC problems can be found in [66, Theorem 3.1] (see also [64,89]) +Theorem 2.2. Suppose that under every stationary policy ζ ∈ Usm, each pair of states in S +communicates under X. Then the following hold for every γ ̸= 0. +(i) There exists an eigen-pair (Ψ, λ), Ψ > 0, satisfying +sign(γ)eγλΨ(x) = min +u∈U(x) + +sign(γ)eγc(x,u) � +y∈S +Ψ(x)P(y|x, u) + + +for x ∈ S. +(2.6) +(ii) infζ∈U Ex(c, ζ) = λ∗,m = λ for each x ∈ S. +(iii) Every minimizing selector of (2.6) is an optimal policy. +(iv) (Ψ, λ) satisfying (2.6) is unique provided we set Ψ(z) = 1 for a prescribed state z ∈ S. +Note that the above result requires the DTCMC to be communicating under every stationary +policy. Theorem 2.2 also appears in [42] where it is proved under an additional assumption that +P(x|x, u) > 0 for all (x, u) ∈ K. In [59] the author shows that given any two states x, y ∈ S, if we +can find a stationary policy under which y is accessible from x, then there exists Λ0 > 0 such that +an eigen-pair satisfying (2.6) exists for γ satisfying γ∥c∥sp < Λ0 (∥·∥sp denotes the span semi-norm +defined as ∥c∥sp = supx,u c − infx,u c). Also, note that the hypothesis of a single communicating +class for every stationary control is important to ensure that infζ∈U Ex(c, ζ) is independent of x (a +specific example can be found in [64, Proposition 3.1]). +Consider the assumption: +Assumption 2.2 (Simultaneous Doeblin Condition). There exists a state z ∈ S and a positive +integer K such that +Eζ +x[˘τz] ≤ K +for all x ∈ S, +and ζ ∈ Usm. +A general characterization of the optimal value is then obtained in [67, Theorem 3.5]. + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +7 +Theorem 2.3. Suppose that U is a finite set and Assumption 2.2 holds. Then for every x ∈ S we +have +inf +ζ∈Usm Ex(c, ζ) = inf +g∈G g(x), +where G denotes the collection all functions g : S → R satisfying +(i) For each x ∈ S +g(x) = min +u∈U(x) max{g(y) : P(y|x, u) > 0}. +(ii) There exists a positive function h such that +eγg(x)h(x) ≥ min +Bg(x) + +eγc(x,u) � +y∈S +h(y)P(y|x, u) + + +x ∈ S, +where +Bg(x) := +� +u ∈ U(x) : g(x) = max{g(y) : P(y|x, u) > 0} +� +. +A generalization of the above result for DTCMC with a general state space can be found in [73]. +2.2. Countable state space. Now suppose that S is countable. Without any loss of generality, +assume that S = {0, 1, 2, . . .}. The analysis of ERSC problem becomes more involved due to non- +compactness of S. If the running cost c is bounded, then a result analogous to Theorem 2.2 is +possible, provided γ is small. +Theorem 2.4 ([64]). Let Assumption 2.2 hold. Define +µ = log(K + 1) − log K +K + 1 +, +∥c∥ = sup +K +|c(x, u)|. +Then for each 0 ̸= γ ∈ (− +µ +2∥c∥, +µ +2∥c∥) there exists an eigen-pair (Ψ, λ) with bounded Ψ that satisfies +(2.6). Furthermore, the conclusions of Theorem 2.2 (ii)-(iv) hold in this case. +Assumption 2.2 in the above theorem can be relaxed provided the state space S is communicating +under every stationary policy and the cost function c is supported on a finite set. For more details, +see [61]. Some other works that also study ERSC problem with bounded cost functions are [65,116]. +Since the simultaneous Doeblin condition in Assumption 2.2 is quite restrictive, we are going to +impose some structural condition on the cost function, known as near-monotonicity, which also +allows unbounded cost functions. +Definition 2.2. We say that the one-step cost function c is near-monotone with respect to ρ if +lim inf +x→∞ +min +u∈U(x) c(x, u) > ρ. +Suppose that for some stationary Markov control ˜ζ, we have Ex(c, ˜ζ) independent of x ∈ S and +c is near-monotone with respect to Ex(c, ˜ζ). It is then shown in [115] that, for γ > 0, there exists +a positive ψ : S → (0, ∞] satisfying +eγλ∗,m +m +Ψ(x) ≥ +inf +u∈U(x) + +ec(x,u) � +y∈S +Ψ(y)P(y|x, u) + + +for all x ∈ S, +(2.7) +where λ∗,m +m +is given by +λ∗,m +m += inf +x∈S inf +ζ∈Um Ex(c, ζ). +(2.8) +Furthermore, if ζ∗ is a minimizing selector of (2.7), then λ∗,m +m += Ex(c, ζ∗) for all x ∈ {Ψ < ∞}. The +main idea in [115] (motivated from [89]) is to transform the risk-sensitive minimization problem to +a risk-neutral game problem using a change of variables (a ‘logarithmic transformation’ that we see + +8 +ANUP BISWAS AND VIVEK S. BORKAR +later) and then use the approach of discounted-control problems for the ergodic risk-neutral game +to construct a solution for (2.7). +Definition 2.3. We say a function F : S → R is norm-like if for each integer n the set {F ≤ n} +is either empty or finite. +Around the same time multiplicative ergodic theorems with norm-like potential functions F are +studied in [21]. The ideas of [21] are extended to study ERSC problems for norm-like cost function +c in [57]. To explain the result of [57] we introduce some additional notations. Fix a state z ∈ S. +For a Markov policy ζ ∈ Um, define +Λ(ζ) = inf +� +Λ : Eζ +z +� +e +�˘τz−1 +t=0 +γ(c(Xt,ζt)−Λ) ≤ 1 +�� +, +and +Λ∗ = inf +ζ∈Um Λ(ζ). +(2.9) +The first entrance time to the state z is defined as σz = inf{n ≥ 0 : Xn = z}. Let us also define, +for x ∈ S, +Ψ∗(x) := inf +ζ∈Um Eζ +x +� +e +�σz +t=0 γ(c(Xt,ζt)−Λ∗)� +, +w∗(x) := Arg min +u∈U(x) + +ec(x,u) � +y∈S +Ψ∗(y)P(y|x, u) + + . +The following result is proved in [57, Theorem 3.6] +Theorem 2.5. Let γ > 0. Suppose that U(x) is finite for all x and c(·, u) is norm-like for all +u ∈ U. Also assume that the chain X is communicating under every Markov policy and aperiodic +under any stationary Markov policy. Then, provided Λ∗ is finite, the following hold. +(i) Λ∗ = λ∗,m +m += Ex(c, w∗) for all x where λ∗,m +m +is given by (2.8). +(i) Ψ∗ is finite on S and +eγλ∗,m +m +Ψ∗(x) ≥ +inf +u∈U(x) + +eγc(x,u) � +y∈S +Ψ∗(y)P(y|x, u) + + +for all x ∈ S. +The above result requires Λ∗ to be finite and the chain to be aperiodic under each stationary +Markov control. Another result of similar flavor is recently obtained in [50], which we state below. +Theorem 2.6. In addition to Assumption 2.1 let us also assume the following to hold. +(i) There exists a state i0 ∈ S such that +min +u∈U(i0) P(j|i0, u) > 0 +for all j ̸= i0. +(ii) X is recurrent under each stationary Markov control. +(iii) infζ∈Usm Ex(c, ζ) < ∞ for all x ∈ S and c is near-monotone with respect to λ∗,m +m +in the sense +of Definition 2.2. +Then there exists a positive Ψ satisfying +eγλ∗,m Ψ(x) ≥ +inf +u∈U(x) + +eγc(x,u) � +y∈S +Ψ(y)P(y|x, u) + + +for all x ∈ S, +and every minimizing selector is an optimal stationary Markov control. Moreover, λ∗,m +m += λ∗,m. +[50] also considers the ERSC problem under a blanket stability hypothesis but without the +near-monotone condition. + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +9 +Assumption 2.3. Let X be irreducible under any stationary Markov control. In (i) and (ii) below +the function V on S takes values in [1, ∞) and �C is a positive constant. Assume that one of the +following holds. +(i) For some positive constant β ∈ (0, 1) and a finite set C it holds that +sup +u∈U(x) +� +y∈S +V(y)P(y|x, u) ≤ (1 − β)V(x) + �C1C(x) +x ∈ S, +and γ supK c < θ where θ = log( +1 +1−β). +(ii) For a finite set C and a norm-like function ℓ : S → R+ it holds that +sup +u∈U(x) +� +y∈S +V(y)P(y|x, u) ≤ (1 − β(x))V(x) + �C1C(x) +x ∈ S, +where 1 − e−ℓ(x) = β(x). Moreover, the function ℓ − γ maxu∈U(·) c(·, u) is norm-like. +Condition (ii) above is useful for treating ERSC problems with an unbounded cost function c. +The following result is obtained in [50]. +Theorem 2.7. Suppose that Assumption 2.1 and 2.3 hold. Also assume the condition (i) of The- +orem 2.6. Then we have the following +(i) There exists a unique, positive Ψ, with Ψ(i0) = 1, satisfying +eγλ∗,m Ψ(x) = +inf +u∈U(x) + +eγc(x,u) � +y∈S +Ψ(y)P(y|x, u) + + +for all x ∈ S. +(2.10) +(ii) A stationary Markov control is optimal if and only if it is a minimizing selector of (2.10). +Theorem 2.6 and 2.7 are proved using a different approach. The authors first solve a nonlinear +eigenvalue problem on finite sets containing i0 and then increase the sets to S. The condition (i) +in Theorem 2.6 ensures that the limiting eigenfunction Ψ is positive. This condition is recently +removed in [74] where the authors used the approach of [57] (see Theorem 2.5) to define the +eigenfunction. +2.3. General state space. Next we describe the results known for the general state space. Some +of the important works in this direction are [79–81, 125]. It is natural that one needs to impose +additional conditions to ensure existence of an eigen-pair. We begin by recalling the following result +from [79]. +Theorem 2.8. Let S be a complete separable metric space, U(x) = U for all x and γ > 0. We +assume the following to hold. +(A1) There exists δ < 1 such that for all x, x′ ∈ S, B ∈ B(S) and u, u′ ∈ U we have P(B|x, u) − +P(B|x′, u′) ≤ δ; +(A2) δeγ∥c∥sp < 1 where ∥c∥sp denotes the span semi-norm of c. +Then there exists a bounded, positive continuous function Ψ satisfying +eγλ∗,m Ψ(x) = min +u∈U +� +eγc(x,u) +� +S +Ψ(y)P(dy|x, u) +� +. +(2.11) +Furthermore, any minimizing selector of the above equation is an optimal stationary Markov control +for the ERSC problem, and Ψ is unique, up to a positive multiplicative constant, in the class Cb(S). +Note that condition (A2) above requires γ to be small. Writing ϕ = log Ψ we see from above +that +γλ∗,m + ϕ(x) = min +u∈U +� +γc(x, u) + log +� +S +eϕ(y)P(dy|x, u) +� +. + +10 +ANUP BISWAS AND VIVEK S. BORKAR +Letting +Tg(x) = min +u∈U +� +γc(x, u) + log +� +S +eg(y)P(dy|x, u) +� +, +it is shown in [79] that {Tn0, n ≥ 0}, where 0 := the function identically equal to zero, converges +in the space Cb(S) with respect to the span semi-norm. The limit of this sequence gives a fixed +point (up to a positive scalar multiplier) which solves (2.11). It can be easily checked that by (A1) +, T is a local contraction [80, Proposition 2.2] and therefore, uniqueness is immediate. Condition +(A2) above was replaced by a more technical condition in [80] to obtain (2.11). Conditions (A1)- +(A2) were replaced by a minorization condition and certain exponential moment bounds on the +hitting time to a certain compact set in [81] in order to study the optimality equation (2.11). These +results are further extended in [73,125,126] to Borel state spaces and for unbounded cost functions. +These works study the ERSC control problem using discounted approximation approach which was +initiated in [80]. For β ∈ (0, 1), let Vβ be a positive solution to the dynamic programming equation +Vβ(x) = min +u∈U(x) +� +eγc(x,u) +� +S +(Vβ(y))βP(dy|x, u) +� +x ∈ S. +(2.12) +Vβ is basically the discounted value function associated with a certain dynamic game [125, Lemma 1]. +Under Assumption 2.1, there exists a unique, bounded solution to (2.12) whenever c is bounded (cf. +[80, Proposition 4.1]). Under some additional assumptions on the transition kernels (cf. [80, The- +orem 4.2]), it can be shown that +Vβ +Vβ(z), z ∈ S is a fixed point, that converges as β ↑ 1 to some Ψ +satisfying (2.11) and +γ−1 lim +β→1(1 − β) log Vβ(z) = λ∗,m. +The above analysis served as the starting point for [73,125] where the authors allow the cost to be +unbounded. Suppose that +x �→ U(x) is upper-semicontinuous. +(2.13) +Consider c ≥ 0 and possibly unbounded and γ > 0. Then one can solve (2.12) for cN = min{N, c} to +obtain a sequence of Vβ,N for each β ∈ (0, 1). Letting N → ∞, it is then shown that limN→∞ Vβ,N = +Vβ, and (cf. [73, Lemma 3.1], [125, Lemma 2]) +Vβ(x) = min +u∈U(x) +� +eγc(x,u) +� +S +(Vβ(y))βP(dy|x, u) +� +x ∈ S. +(2.14) +Define mβ := infS Vβ. Letting +˜Vβ := +1 +mβ +Vβ, +provided mβ > 0, in (2.14) gives +e(1−β) log mβ ˜Vβ(x) = min +u∈U(x) +� +eγc(x,u) +� +S +( ˜Vβ(y))βP(dy|x, u) +� +x ∈ S. +In [125], under the assumption (2.13) and supβ∈(0,1) ˜Vβ < ∞, it is shown that, for any sequence +βn → 1, +λ∗,m = 1 +γ lim +n→∞(1 − βn) log mβn, +and +Ψ := lim inf +n→∞ +˜Vβ, +satisfy +eγλ∗,mΨ(x) ≥ +inf +u∈U(x) +� +ec(x,u) +� +S +Ψ(y)P(dy|x, u) +� +for all x ∈ S, + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +11 +and every minimizing selector is an optimal stationary Markov control. A similar result is also +obtained in [73, Theorem 5.2] under a milder hypothesis that requires mβ to be finite for all +β ∈ (0, 1), but it also assumes (compare it with Definition 2.2) that +{x ∈ S : +min +u∈U(x) c(x, u) ≤ λ∗,m + δ} +to be compact for some δ > 0. +3. Risk-sensitive control of diffusions +In this section we review some recent progress on the ERSC problem for controlled diffusions. +To begin with, consider the problem for uncontrolled diffusion. +3.1. Generalized principal eigenvalue. Let X = {Xt} be a diffusion process in Rd given by +dXt = b(Xt)dt + σ(Xt)dWt, +(3.1) +where b : Rd → Rd is the drift vector, σ : Rd → Rd×d is the diffusion matrix and W is a d- +dimensional standard Wiener process on a complete filtered probability space (Ω, F, P). +There +exists a unique strong solution of (3.1) (see [114,152,161]) for every initial data X0 = x ∈ Rd, b is +Borel measurable and σ is locally Lipschitz and locally non-degenerate , provided b, σ have at-most +linear growth. Let a(x) = 1 +2σσT(x). Given a continuous function c : Rd → R let us define +Ex(c) = lim sup +T→∞ +1 +T Ex +� +e +� T +0 c(Xs)ds� +. +As in Section 2, the above quantity is related to an eigen-equation which we describe below. We +define the extended generator of (3.1) as +Lf(x) = trace(a(x)∇2f) + b(x) · ∇f(x). +(3.2) +Definition 3.1. We say a pair (ψ, λ) ∈ C2(Rd) × R is an eigen-pair of L + c if ψ > 0 in Rd and +Lψ(x) + c(x)ψ(x) = λψ(x) +in Rd. +To understand the relation between Ex and an eigen-pair, let us consider the problem in a smooth +bounded domain D. More precisely, let τ(D) be the first exit time from D, that is, +τ(D) = inf{t > 0 : Xt /∈ D}, +and we define +Ex(c, D) := lim sup +T→∞ +1 +T Ex +� +e +� T +0 c(Xs)ds1{T<τ(D)} +� +x ∈ D. +It is then well-known that there exists a λD ∈ R such that Ex(c, D) = λD for all x ∈ D [82] and +for some ψD ∈ C2(D) ∩ C( ¯D) we have +LψD + c(x)ψD(x) = λDψD(x) +in D +ψ > 0 +in D, +ψ = 0 +on ∂D. +(3.3) +Thus (ψD, λD) forms a Dirichlet eigen-pair for L + c in D. Furthermore, λD is the generalized +principal eigenvalue in the sense of [37,144,147], that is, +λD = inf{λ : ∃ ψ ∈ C2 ++(D) ∩ C( ¯D) satisfying Lψ + c(x)ψ ≤ λψ in D}, +(3.4) +where C2 ++(D) denotes the subset of C2(D) containing functions that are positive in the interior of +D. (We define λD likewise for unbounded D.) In addition, it can be easily shown that the principal +eigenfunction ψD in (3.3) is unique up to a multiplicative constant. So we may want to ask whether +Ex(c) = λRd. The answer to this question is negative in general. In fact, [14, Example 3.1] shows +that λRd < infx Ex(c). Thus the risk-sensitive problem in the whole space becomes quite delicate. + +12 +ANUP BISWAS AND VIVEK S. BORKAR +Let us now recall the definition of principal eigenvalue in Rd from [38]. The generalized principal +eigenvalue of L + c in Rd is defined as follows +λRd = inf{λ : ∃ positive ψ ∈ C2(Rd) satisfying Lψ + c(x)ψ ≤ λψ in Rd}. +(3.5) +To illustrate explicit dependence on the potential c we would also use the notation λRd(c). Let us +also recall the following result from [38, Theorem 1.4]. +Theorem 3.1. For every λ ∈ [λRd, ∞) there exists a positive ψ ∈ C2(Rd) satisfying +Lψ(x) + c(x)ψ(x) = λψ(x) +in Rd. +In particular, there are infinitely many eigen-pairs for L + c in Rd. +To equate Ex(c) with +the generalized principal eigenvalue λRd, we must impose additional conditions on the diffusion +coefficients. More discussion in this direction can be found in [14]. Another important concern is +the simplicity of the principal eigenvalue λRd. We need following definition for this purpose. +Definition 3.2 (Minimal growth at infinity). An eigen-pair (ψ, λ) is said to have a minimal growth +at infinity if for any compact set K and any positive v ∈ C2(Kc) ∩ C(Rd) satisfying +Lv + (c − λ)v ≤ 0 +in Kc, +we have v ≥ κψ for some κ > 0. +The above criterion was introduced by Agmon in [3] and is very useful to establish simplicity of +eigenvalues. Let C+ +◦ (Rd) denote the collection of all non-trivial, non-negative, continuous functions +which vanish at infinity. The following notions of monotonicity are introduced in [14]. +Definition 3.3. We say the generalized principal eigenvalue λRd is strictly monotone at c if for +some h ∈ C+ +◦ (Rd) we have λRd(c − h) < λRd(c). We say λRd is monotone on the right at c if for all +h ∈ C+ +◦ (Rd) we have λRd(c) < λRd(c + h). +It is shown in [14] that strict monotonicity at c implies λRd(c − h) < λRd(c) for all h ∈ C+ +◦ (Rd) +and therefore, by convexity, it also implies monotonicity on the right at c. The following equivalence +criterion is proved in [11,14] (see also [13] for its generalization to weakly-coupled systems). +Theorem 3.2. Suppose that λRd(c) is finite. Then the following are equivalent. +(i) Eigen-pair (ψ, λRd(c)) has a minimal growth at infinity. +(ii) λRd is monotone on the right at c. +(iii) For some compact ball B, we have +ψ(x) = Ex +� +e +� ˘τ +0 (c(Xt)−λRd)dtψ(X˘τ) +� +x ∈ Bc, +(3.6) +where ˘τ = τ(Bc), the first hitting time to B. +Furthermore, if one of the above holds, then λRd(c) is simple. +The analogy between (2.5) and (3.6) should be noted. +To characterize the notion of strict +monotonicity we need to introduce the twisted diffusion. Given an eigen-pair (ψ, λ) of L + c, the +twisted diffusion is given by +dYt = b(Yt)dt + 2a(Yt)∇ log ψ(Yt)dt + σ(Yt)dWt. +The twisted process corresponding to a principal eigen-pair is said to be a ground state process due +to its interpretation in physics. The following result can be found in [14, Theorem 2.1] (see also +[121,122,130]). +Theorem 3.3. Suppose that λRd(c) is finite. Then +(i) For every λ > λRd(c), the twisted process corresponding to any eigen-pair (ψ, λ) is transient. +(ii) The following are equivalent. + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +13 +(a) λRd is strictly monotone at c. +(b) The ground state process is exponentially ergodic. +Let us remark that [14] requires c to non-negative for Theorem 3.2 and 3.3 to hold, but this +restriction on c is removed in [13]. +3.2. ERSC for controlled diffusions. In this section we review ERSC problem for controlled +diffusions. We begin with the exponential linear-quadratic model. +3.2.1. Exponential Linear-quadratic model. Exponential linear quadratic model is a risk-sensitive +generalization of the classical linear-quadratic model. Such problems are quite central to the optimal +investment models, appearing in mathematical finance (see [92, 141–143] and references therein). +More precisely, the controlled diffusion is given by (we consider a slightly more general form) +dXt = b(Xt)dt + g(Xt, ζt)dt + σ(Xt)dWt +(3.7) +where ζt is a progressively measurable process that is non-anticipative in the sense that for s < t, +Wt −Ws is independent of the completion of the sigma-field generated by {X0, ζr, Wr : r ≤ s} with +respect to (F, P). The control process ζ is generally assumed take values in some Euclidean space +Rm. As before, we denote the set of all admissible controls by U. Implicitly, we assume that under +every admissible control there exists a unique strong solution to (3.7) in the sense that, given ζ, W +as above on a probability space, there exists an a.s. unique X satisfying (3.7). Let V : Rd → [0, ∞) +and φ : Rd × Rm → [0, ∞) be two given functions. We define +λ∗,d = inf +x∈Rd lim sup +T→∞ +1 +γT log J(x, T) +where +J(x, T) = inf +ζ∈U Ex +� +eγ � T +0 (V (Xt)+φ(Xt,ζt))dt� +. +(3.8) +It should be noted that for a given γ > 0, J(x, T) might not be finite for all T. This is known as +the breakdown phenomenon. In fact, [34, Example 1] shows that breakdown can actually happen +for some large values of T. Thus we need to impose conditions on the coefficients to ensure no +breakdown [32,140]. As mentioned before, the above ERSC problem (3.8) is related to the nonlinear +eigenvalue problem given by +γλ∗,dΨ(x) = trace(a(x)∇2Ψ(x)) + b(x) · ∇Ψ(x) + min +u∈Rm{g(x, u) · ∇Ψ + γφ(x, z)Ψ(x)} + γV (x)Ψ(x) +for x ∈ Rd. Assume γ > 0. Letting w(x) = 1 +γ log Ψ(x) in the above, we obtain +λ∗,d = trace(a(x)∇2w(x)) + b(x) · ∇Ψ(x) + Q0(x, ∇w) + V (x), +(3.9) +where +Q0(x, ξ) = γξa(x) · ξ + min +u∈Rm{g(x, u) · ξ + φ(x, z)}. +If we choose a, g, φ in such a way that +− κ1|ξ|2 ≤ Q0(x, ξ) ≤ −κ2|ξ|2 +x, ξ ∈ Rd, +(3.10) +for some positive constants κ1, κ2, and +���� +∂Q0(x, ξ) +∂ξ +���� ≤ κ3|ξ| + κ4, +���� +∂Q0(x, ξ) +∂x +���� ≤ κ3|x|2 + κ4, +(3.11) +for some κ3, κ4 > 0, we are in the framework of the Hamilton-Jacobi-Isaacs equation of the ergodic +type [30]. More precisely, if V is coercive, the existence and uniqueness of solution to (3.9) can be +obtained from [30]. The following result is proved in [140, Theorem 3.4]. +Theorem 3.4. We impose the following conditions. +(i) σ, b, g, V, φ are smooth and σ, b are Lipschitz. Also, all the derivatives of σ, b, V are bounded +by M(1 + |x|k) for some M, k > 0; +(ii) |g(x, z)| ≤ κ˜g(z) for some locally bounded ˜g and a constant κ; + +14 +ANUP BISWAS AND VIVEK S. BORKAR +(iii) For some constant κ◦ > 0 we have +ξa(x) · ξ ≥ κ◦|ξ|2 +ξ, x ∈ Rd. +(iii) V is coercive and +lim +|z|→∞ φ(x, z) = ∞, +lim +|z|→∞ +|g(x, z)| +φ(x, z) = 0 +uniformly in x. +Also, assume that (3.10)-(3.11) hold and +Q0(x, βξ) ≥ β2Q0(x, ξ) − κβ(1 − β)ξa(x) · ξ − β(1 − β)L(x), +β ∈ (0, 1), +for κ < κ2 and some locally bounded function L satisfying +V (x) − L(x) → ∞ +as +|x| → ∞. +Then there exists a unique eigen-pair (w, λ) ∈ C2(Rd) × R, w coercive in nature, satisfying (3.9). +Furthermore, when Q0(x, ξ) = −κ ξa(x)ξT for some κ > 0, we have λ = λ∗,d, given by (3.8). +Similar result can also be found in [129, Theorem 3.3] where the authors imposed some structural +assumptions on g. +3.2.2. ERSC with a compact action set. In this section we review the result on ERSC problem +when the action set U is compact. Let X = {Xt} be a controlled diffusion in Rd governed by the +Itˆo equation +dXt = b(Xt, ζt)dt + σ(Xt)dWt +(3.12) +where ζt is an admissible control in the sense of Section 3.2.1, taking values in a compact metric +space U. We impose the following conditions on the coefficients to guarantee the existence and +uniqueness of solution to (3.12). +(B1) Local Lipschitz continuity: The functions b : Rd × U → Rd and σ : Rd → Rd×d are +continuous and satisfy +|b(x, u) − b(y, u)| + ∥σ(x) − σ(y)∥ ≤ CR|x − y| +∀ x, y ∈ BR, ∀ u ∈ U, +for some constant CR, depending on R > 0, where BR denotes the ball of radius R centered +at 0. +(B2) Affine growth condition: There exists a constant C0 such that +max +u∈U [b(x, u) · x]+ + ∥σ(x)∥2 ≤ C0(1 + |x|2) +x ∈ Rd. +(B3) Local non-degeneracy: For each R > 0, there exists CR satisfying +ξa(x) · ξ ≥ C−1 +R |ξ|2 +∀ ξ ∈ Rd, x ∈ BR, +where a(x) = 1 +2σσT(x). +It is well known that under (B1)-(B3), for any admissible control ζ ∈ U there exists a unique +solution of (3.12) [17, Theorem 2.2.4]. As before, a stationary Markov control would correspond to +a Borel measurable map from Rd to U and the class of all stationary Markov controls is denoted +by Usm. It is also well known that for every stationary Markov control in Usm there exists a unique +strong solution to (3.12) which is also a strong Markov process [114, 152, 161]. Now consider a +continuous function c : Rd × U → [0, ∞) which is locally Lipschitz in x uniformly with respect to +u ∈ U. As before, we define the ERSC problem as follows. +λ∗,d = inf +x∈Rd inf +ζ∈U Ex(c, ζ) +where +Ex(c, ζ) := lim sup +T→∞ +1 +γT log Eζ +x +� +e +� T +0 γc(Xt,ζt)dt� +. +(3.13) +For the remaining part of this section, we discuss the risk-averse problem and therefore, we shall +consider γ to be positive. As discussed before, the above ERSC problem corresponds to a nonlinear + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +15 +eigenvalue problem. +For this purpose we introduce a family of operators Lu, parametrized by +u ∈ U, defined as follows +Luf(x) = trace(a(x)∇2f(x)) + b(x, u) · ∇f. +We shall be interested in an eigenfunction Ψ ∈ C2(Rd), Ψ > 0, satisfying +min +u∈U{LuΨ(x) + γc(x, u)Ψ(x)} = γλ∗,d Ψ(x) +in Rd. +(3.14) +The first major contribution for the ERSC of diffusion came from Fleming and McEneaney [97] +(see also [96]). They prove the following in [97, Theorem 7.2 and 7.3]. +Theorem 3.5. Suppose that b(·, u), c(·, u) are C1 for each u ∈ U, σ is constant, and the following +hold. +(i) c, ∇xc are bounded. γ > 0. +(ii) ∇xb is bounded in Rd. +(iii) For some κ > 0 we have +(x − y) · (b(x, u) − b(y, u)) ≤ −κ|x − y|2 +∀ x, y ∈ Rd, u ∈ U. +There there exists a Ψ ∈ C2(Rd), Ψ > 0, satisfying +min +u∈U{LuΨ(x) + γc(x, u)Ψ(x)} = γλ∗,d Ψ(x) +in Rd. +Furthermore, any measurable selector of the above equation is an optimal stationary Markov control. +Apart from the condition (iii) above, the constant diffusion matrix σ also plays a key role in the +above result. These two conditions together render Lipschitz regularity to log Ψ. More precisely, +the authors use a logarithmic transformation to change the risk-sensitive minimization problem to +an ergodic game problem. Then using the standard method of vanishing discount, they establish +the existence of solution to the Hamilton-Jacobi-Isaacs equation for the ergodic game problem. In +order to extend the result to a more general class of b and σ, [138] considers ERSC problem under +a periodic setting. This is the content of our next result. +Theorem 3.6. Suppose that σ = +√ +2I and b, c are periodic in x variable with period 1. Also, assume +that b, c are Lipschitz in the x variable. Then there exists a unique, periodic Ψ ∈ C2(Rd), Ψ > 0, +satisfying +∆Ψ(x) + min +u∈U{b(x, u) · ∇Ψ(x) + γc(x, u)Ψ(x)} = γλ∗,d Ψ(x) +in Rd. +(3.15) +The main idea of the proof goes as follows: one starts with an exponential of the discounted cost +defined as (this is actually the continuous version of the approach that appeared in [65,80]) +wα(γ, x) = inf +ζ∈U Ex +� +exp +� +γ +� ∞ +0 +e−αtc(Xt, ζt)dt +�� +, +α ∈ (0, 1), +and shows that +− αγ ∂wα +∂γ + ∆wα + min +u∈U{b(x, u) · ∇wα + γc(x, u)wα} = 0, +(3.16) +and wα(0, x) = 1. Note that this is a parabolic equation when we treat γ as a variable. Defining +uα = γ−1 log wα, it is then shown that αuα, ∇xuα are globally bounded, uniformly in α. This helps +us to pass the limit in (3.16) to obtain (3.15). This idea of [138] was then pushed in [46–48] to +solve (3.15) beyond the periodic setting and under near-monotone hypothesis. +Definition 3.4. We say that c is near-monotone with respect to ρ ∈ R if it satisfies +lim inf +|x|→∞ min +u∈U c(x, u) > ρ. + +16 +ANUP BISWAS AND VIVEK S. BORKAR +In particular, it was proved in [48] that if c is near-monotone with respect to λ∗,d and the diffusion +(3.12) is recurrent under each stationary Markov control, then there exists a positive Ψ satisfying +(3.15) and every measurable selector is an optimal stationary Markov control. But the uniqueness +of Ψ remains an issue. The approach of [48, 138] establishes γλ∗,d as an eigenvalue of nonlinear +operator in (3.15), but in view of Theorem 3.1 (for nonlinear operators, see [51, Theorem 2.1]), it +is hard to identify γλ∗,d as the principal eigenvalue. Thus it is important to establish uniqueness of +Ψ (up to a positive multiplicative constant) for the verification result of optimal stationary Markov +controls. Define the nonlinear operator G as +Gf(x) = min +u∈U(Luf(x) + γc(x, u)f(x)). +(3.17) +The generalized principal eigenvalue λ1(G) of G is defined as before (along the lines on [38,51]) : +λ1(G) = inf{λ : ∃ positive ψ ∈ C2(Rd) satisfying Gψ ≤ λψ in Rd}. +A natural question is: under what condition can we show that λ1(G) = γλ∗,d? If we start with +the Dirichlet generalized eigenvalue problem for G on a sequence of increasing, smooth bounded +domains and let the domains increase to Rd, then applying Harnack’s inequality and monotonicity +of generalized principal eigenvalues, it can be shown that the Dirichlet principal eigenvalues con- +verges to λ1(G). In [44], the author applies this idea to show that λ1(G) ≤ γλ∗,d, in general, and +furthermore, if c is near-monotone with respect to λ∗,d and the diffusion (3.12) is recurrent under +each stationary Markov control, then there exists a positive Ψ satisfying +GΨ = γλ∗,d Ψ +in Rd, +and every minimizing selector is an optimal stationary Markov control. +Note that the near- +monotone criterion penalizes instability of the process X. Thus it is expected that an optimal +stationary Markov control would stabilize the process, that is, keep it positive recurrent. Using +this fact, the blanket stability hypothesis was removed in [7], proving the the following. +Theorem 3.7. Assume (B1)-(B3) and also suppose that c is bounded and is near-monotone with +respect to λ∗,d. In addition, suppose that b, σ are bounded, σ is Lipschitz, a is uniformly elliptic +and +max +u∈U +[b(x, u) · x]+ +|x| +→ 0 +as |x| → ∞. +(3.18) +Then there exists a Ψ ∈ C2(Rd) satisfying infRd Ψ > 0 and +min +u∈U{LuΨ(x) + γc(x, u)Ψ(x)} = γλ∗,d Ψ(x) +in Rd. +(3.19) +Moreover, the following hold. +(i) λ1(G) = λ∗,d = infζ∈U Ex(c, ζ) for all x. +(ii) If v ∈ Usm is a minimizing selector of (3.19), then v is stable and is an optimal stationary +Markov control. +(iii) In addition, if we have λ∗,d +m (c) < λ∗,d +m (c + h) for all h ∈ C+ +◦ (Rd) where +λ∗,d +m (c) := inf +x∈Rd +inf +ζ∈Usm Ex(c, ζ), +then Ψ in (3.19) is unique up to a positive scalar multiple and any optimal stationary +Markov control is given by a measurable selector of (3.19). +Note that Theorem 3.7 does not impose any stability assumption. Condition (3.18) is used to +show that any minimizing selector is in fact stable. The boundedness assumption on b and c was +relaxed in [10, Proposition 5.2] where the authors allowed polynomial growth of b and c. +The +condition of monotonicity on the right in Theorem 3.7(iii) is not easy to verify. To tackle this +difficulty, an alternative set of conditions has also been used for the ERSC problems as follows. + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +17 +Assumption 3.1. There exists a positive V ∈ C2(Rd) with infRd V > 0 such that one of the +following holds: +(i) There exists an inf-compact, positive ℓ ∈ C(Rd) and a compact set K satisfying +sup +u∈U +LuV ≤ ¯κ1K − ℓV +in Rd, +(3.20) +for some constant ¯κ, and ℓ − maxu∈U γc(·, u) is inf-compact. +(ii) For some positive constants ¯κ, θ and a compact set K we have +sup +u∈U +LuV ≤ ¯κ1K − θV +in Rd, +(3.21) +and +lim sup +|x|→∞ +max +u∈U γc(x, u) < θ. +We remark here that (3.20) is not possible when a, b are bounded [38, Proposition 2.6]. This +is the reason for introducing (3.21). +Also, note that Assumption 3.1 does not require c to be +near-monotone, but imposes a blanket stability hypothesis on the stationary Markov controls. +Assumption 3.1(ii) was also used in [45] to prove the existence of a solution Ψ of (3.19) and the +existence of an optimal stationary Markov control. Uniqueness and verification results are settled +in [14] where the authors prove the following. +Theorem 3.8. Assume that (B1)-(B3) and Assumption 3.1 hold. Then there exists a Ψ ∈ C2(Rd) +satisfying Ψ > 0 and +min +u∈U{LuΨ(x) + γc(x, u)Ψ(x)} = γλ∗,d Ψ(x) +in Rd. +(3.22) +Moreover, the following hold. +(i) λ1(G) = λ∗,d = infζ∈U Ex(c, ζ) for all x. +(ii) If v ∈ Usm is a minimizing selector of (3.22), then it is an optimal stationary Markov +control. +(iii) Ψ in (3.19) is unique up to a positive scalar multiple and any optimal stationary Markov +control is given by a measurable selector of (3.22). +Incidentally, the approach of [14] does not extend to jump diffusions. The eigenvalue approach +in [7,14] crucially uses the Harnack inequality to establish the existence of principal eigenfunction +of G and the Harnack inequality does not hold for the nonlocal equation with rough kernels (cf. +[19, Example 1.1]). To tackle this problem, [9] used the Lyapunov function in Assumption 3.1 +as a barrier function to bound the Dirichlet principal eigenfunctions. +More precisely, under a +stability assumption analogous to Assumption 3.1, [9] studies the ERSC problem for a class for +jump diffusions with jump-kernels having finite measure and establishes a result analogous to +Theorem 3.8. +3.2.3. Connection to H∞ control. In this section, we briefly touch upon the connection between H∞ +control and the small noise asymptotics of ERSC problem. Readers are encouraged to consult the +book [20] to find out more on H∞ control. Let us start with a (deterministic) nonlinear, controlled +dynamical system +dyt = g(yt, ζt, ξt)dt +where ζt and ξt are two control process, taking values in some subsets U ⊂ Rm and V ⊂ Rn, +respectively. Define +A := L2 +loc(R+, U), +B := L2 +loc(R+, V). +We choose ζ as a causal feedback to ξ, that is, ζ = α(ξ) for some α : B → A satisfying +if for some t > 0 we have ξ = ¯ξ in [0, t], then α(ξ) = α(¯ξ) on [0, t]. + +18 +ANUP BISWAS AND VIVEK S. BORKAR +The class of such causal feedback controls is denoted by Ucausal. The H∞ control problem can be +described as follows. Assume that the dynamical system is stable under the control ζ ≡ 0 and given +a response function h : Rd × U × V → R+, we have a γ > 0 and a strategy α ∈ Ucausal satisfying, +for some starting point y0 ∈ Rd, +� T +0 +h(yt, α(ξ)(t), ξt)dt ≤ γ2 +� T +0 +|ξt|2dt +for all T > 0, ξ ∈ B. +(3.23) +The least γ satisfying (3.23) is called the H∞ norm. When existence of α is possible, we say that +the H∞ suboptimal control problem is solvable with disturbance attenuation level γ. Note that the +above problem can also be studied by considering the value function +Vγ(x) = +inf +α∈Ucausal +sup +ξ∈B +sup +T≥0 +� T +0 +[h(yt, α(ξ)(t), ξt) − γ2|ξt|2]dt, +where y0 = x. Note that Vγ ≥ 0. The points where Vγ vanishes correspond to the points from +where the H∞ problem is solvable. As shown in [150], the value function Vγ is a viscosity solution +to +sup +v∈V +inf +u∈U{g(x, u, v) · ∇Vγ + h(x, u, v) − γ2|v|2} = 0 +in Rd. +(3.24) +Thus the H∞ control problem is related to the study of non-negative viscosity solution to (3.24). +In order to understand the connection of (3.24) with the ERSC problem, consider the controlled +diffusion +dXt = b(Xt, ζt)dt + +� ε +2γ2 +�1/2 +dWt, +(3.25) +where ε > 0, and ζ is an admissible control process taking values in U. Also, letting γ = ε−1 in +(3.13), we define +Λε = inf +x∈Rd inf +ζ∈U Ex(c, ζ), +where the controlled diffusion is given by (3.25). As we have seen before, the above ERSC problem +corresponds to the eigen-equation +ε−1ΛεΨε(x) = +ε +4γ2 ∆Ψε + min +u∈U{b(x, u) · ∇Ψε + ε−1c(x, u)Ψε(x)} +in Rd. +Letting Wε = ε log Ψε we obtain +Λε = +ε +2γ2 ∆Wε + max +v∈Rd min +u∈U +� +(b(x, u) + v) · ∇Wε + c(x, u) − γ2|v|2� +. +(3.26) +Thus, if we could show that the family {Wε} is locally equicontinuous and Λε → Λ0 as ε → 0 (along +some subsequence), then using the stability of viscosity solutions, it can be shown from (3.26) that +max +v∈Rd min +u∈U +� +(b(x, u) + v) · ∇W0 + c(x, u) − γ2|v|2� += Λ0, +(3.27) +where W0 is a limit of Wε in the viscosity sense as ε → 0. If we set g(x, u, v) = b(x, u) + v and +h(x, u, v) = c(x, u), then (3.27) is same as (3.24) when Λ0 = 0 and V = Rd. In fact, the following +result was proved in [90, Theorem 2.10] (the control process ξ does not play any role in this result) +Theorem 3.9. Suppose that g(x, u, v) = b(x) + v where b satisfies the conditions in Theorem 3.5 +and b(0) = 0, h(x, u, v) = |h1(x)|2 for some C1 function h1 : Rd → Rm with h1(0) = 0 and h1, ∂xih1 +are bounded for all i = 1, 2, . . . , d. Then the H∞ suboptimal control problem is solvable, starting at +the point 0, at the level γ, if and only if limε→0 Λε = Λ0 = 0. +The existence of solution to the more general equation (3.27) and a discussion of H∞ control can +be found in [97,98], whereas uniqueness is discussed in [137]. In the linear-quadratic setting, similar +problems are also studied in [33,128,129]. Let us also mention two interesting works [95,131] where +(3.27) is studied in the framework of max-plus calculus. + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +19 +3.3. Generalized Collatz-Wielandt formula. Consider a non-negative, irreducible matrix A ∈ +Rd×d. Then the celebrated Collatz-Wielandt [75,160] formula suggests +λ(A) = +max +0≤x=(x1,...,xd) min +i:xi>0 +(Ax)i +xi += +min +0≤x=(x1,...,xd) max +i:xi>0 +(Ax)i +xi +, +(3.28) +where λ(A) denotes the Perron-Frobenius eigenvalue of A. An alternate characterization of λ(A) +can also be given as follows. Write A = (aij) = DR where +D = diag[κ1, . . . , κd], +κi := +� +j +aij +R := (p(j|i)), +p(j|i) := aij +κi +. +Let +G = {(π, ˜P) : π is the stationary probability of the stochastic matrix ˜P = (˜p(j|i))}. +Then the following representation can be found in [77] +log λ(A) = +sup +(π, ˜P )∈G +�� +i +π(i) +� +κi − DKL(˜p(·|i)||p(·|i)) +� +� +, +(3.29) +where DKL(·||·) denotes the Kullback-Leibler divergence defined as +DKL(˜p(·|i)∥p(·|i, u)) = +�� +j∈S ˜p(j|i) log +� +˜p(j|i) +p(j|i,u) +� +if ˜p(·|i) ≪ p(·|i, u), +∞ +otherwise. +Given the connection between Perron-Frobenius eigenvalue and the risk-sensitive limits, it is natural +to expect a similar representation for λ∗,m or λ∗,d. Let us first consider a DTCMC taking values +in a finite set S and the action set U is also finite. By P(U) we denote the set of all probability +vectors on U. The cost function c and transition probability can be extended to P(U) in an obvious +fashion. In particular, for v ∈ P(U), we define +c(x, v) = +� +u∈U +c(x, u)v(u), +P(·|i, v) = +� +u∈U +P(·|i, u)v(u). +Also, extend the set Usm by allowing the controls to take values in P(U). In [16], the authors +consider the ERSC problem (fix γ = 1, for simplicity) +¯λ∗,m = max +i∈S +inf +ζ∈Usm Ei(c, ζ). +Then a generalization of (3.29) is obtained in [16] for the controlled problem. In order to state this +result, we denote by Q the set of all stochastic matrices q = (qij) satisfying +qij = 0 +if max +u∈U P(j|i, u) = 0. +Let Mq denote the set of all stationary probability vectors of q ∈ Q. +Theorem 3.10 ([16]). Define +˜c(i, q, u) = c(i, u) − DKL(q(·|i)||P(·|i, u)) +where +q(j|i) := qij, +˜c(i, q, v) = +� +u∈U +˜c(i, q, u)v(u), +v ∈ P(U), +�Φ(q, v) = sup +π∈Mq +� +i∈S +π(i)˜c(i, q, v). +(3.30) +Then we have +¯λ∗,m = min +v∈Usm max +q∈Q +�Φ(q, v) = max +q∈Q min +v∈Usm +�Φ(q, v), + +20 +ANUP BISWAS AND VIVEK S. BORKAR +and there exists a saddle point equilibrium point (q∗, v∗) for the above zero-sum game. +The readers must have noticed the analogy between (3.29) and (3.30). In fact, (3.30) can be +seen as an ergodic average of ˜c with respect to a suitable Markov chain dictated by the transition +probability matrix q ∈ Q. Thus the control v has no effect on the dynamics but only on the cost +function. This forms a single controller zero-sum ergodic game [86]. We revisit this theme later. +For a general state space, Donsker and Varadhan [83] proved the following min-max formula for +diffusions. +Theorem 3.11. Let X be a compact metric space and {Tt} be a strongly continuous, positive +semigroup on C(X) satisfying Tt1 = 1 for all t ≥ 0. Let L be the generator of T and c be any +continuous function on X. Then +λ(c) = inf +ψ∈D+ sup +x∈X +Lψ(x) + c(x)ψ(x) +ψ(x) += +sup +µ∈P(X) +inf +ψ∈D+ +� +X +Lψ(x) + c(x)ψ(x) +ψ(x) +dµ, +(3.31) +where D+ is the subset of the domain of L containing all positive functions, P(X) denotes the set +of all Borel probability measures on X and +λ(c) = lim +t→∞ +1 +T log∥T c +t ∥, +where {T c +t } denotes the semigroup generated by L + c. +If we associate the semigroup {Tt} with a Markov process {Xt} taking values in X, that is , +Ttf(x) = Ex[f(Xt)], +then λ(c) is nothing but +λ(c) = lim +T→∞ +1 +T log sup +x∈X +Ex +� +e +� T +0 c(Xt)dt� +. +Thus Theorem 3.11 gives a Collatz-Wielandt representation to the risk-sensitive value. +In the context of discrete time Markov chains, the following representation is proved in [5, The- +orem 2.2]. Let +G := {η(dx, du, dy) = η0(dx)η1(du|x)η2(dy|x, u) such that η0(dx) is invariant under the transition +kernel +� +η2(dy|x, u)η1(du|x)}. +Theorem 3.12. Let S be a compact metric space and X be a controlled Markov process with on +S with a compact metric action space U and a continuous transition kernel (x, u) ∈ S × U �→ +P(dy|x, u). Suppose that Assumption 2.1 holds and the support of P(·|x) is S for all x. Also, +consider a continuous function c on S × U × S. Then there exists a unique λ1 > 0 (the Perron- +Frobenius eigenvalue) and a positive Ψ ∈ C(S) such that TΨ = λ1Ψ where T is defined as follows. +Tf(x) = sup +ϕ +∈ P(U) +� +S +ec(x,u,y)f(y)ϕ(du)P(dy|x) +for f ∈ C(S). +Furthermore, the following representations hold for λ1 +λ1 = +inf +0<ψ∈C(S) +sup +µ∈P(S) +� +S Tψ(x)dµ +� +S ψdµ += +sup +0<ψ∈C(S) +inf +µ∈P(S) +� +S Tψ(x)dµ +� +S ψdµ +log λ1 = sup +η∈G +� � � � +η(dx, du, dy)c(x, u, y)− +� � +η0(dx)η1(du|x)D(η2(dy|x, u)∥p(dy|x, u)) +� +. + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +21 +The results of [5] go far beyond the above setting where the representation is proved for the +optimal value corresponding to a risk-reward problem. A similar representation is also possible for +λ∗,m, the optimal value of the ERSC problem. In fact, the following min-max formula is established +in [62, Theorem 3.1] for DTCMC. +Theorem 3.13. Let S be a denumerable state space and we consider the setting of Section 2. Let +c ≥ 0 and γ > 0. Suppose that Assumption 2.1 and 2.2 hold, and every state x is accessible from +z, under every stationary Markov policy. Then, if λ∗,m is finite, we have +λ∗,m = inf{λ : ∃ positive vector ψ satisfying eγλψ(x) ≥ min +u∈U(x) eγc(x,u) � +y∈S +ψ(y)P(y|x, u)}. +(3.32) +Furthermore, we have +eγλ∗,m = inf +ψ>0 sup +x∈S +minu∈U(x) eγc(x,u) � +y∈S ψ(y)P(y|x, u) +ψ(x) +. +(3.33) +To be precise, [62] established that for some positive vector ψ∗ one has +eγλ∗,m = sup +x∈S +minu∈U(x) eγc(x,u) � +y∈S ψ∗(y)P(y|x, u) +ψ∗(x) +≥ inf +ψ>0 sup +x∈S +minu∈U(x) eγc(x,u) � +y∈S ψ(y)P(y|x, u) +ψ(x) +. +But the above inequality cannot be strict. Otherwise, for some ε > 0 and ψ > 0 we would have +eγ(λ∗,m−ε) > sup +x∈S +minu∈U(x) eγc(x,u) � +y∈S ψ(y)P(y|x, u) +ψ(x) +, +which will contradict (3.32). This gives us (3.33). +For controlled diffusions, similar representation was studied in [18] with the help of the nonlinear +Krein-Rutman theorem. To present the result of [18], we consider a bounded domain D ⊂ Rd with +a C3 boundary. The reflected controlled diffusion on ¯D is given by +dXt = b(Xt, ζt)dt + σ(Xt)dWt − δ(Xt)dξt, +dξt = 1{Xt∈∂D}dξt, +ξ0 = 0, +(3.34) +where b, σ are as before (see (B1)-(B3)), ζ ∈ U, and δ : Rd → Rd is co-normal, that is, δ(x) = +2a(x)n(x) where n(x) denote the unit outward normal on ∂D. As before, we define the ERSC +problem as +λ∗,d = inf +x∈Rd inf +ζ∈U Ex(c, ζ) +where +Ex(c, ζ) := lim sup +T→∞ +1 +γT log Eζ +x +� +e +� T +0 γc(Xt,ζt)dt� +, +(3.35) +where (Xt, ζt) satisfies (3.34). Define +C2 +δ,+(D) = {ψ ∈ C2( ¯D) : ψ ≥ 0, ∇ψ · δ = 0 on ∂D}. +Also, recall the operator G from (3.17). +The following representation of λ∗,d can be found in +[18, Theorem 2.1]. +Theorem 3.14. There exists a unique pair (ρ, Ψ) ∈ R × C2 +δ,+(D) satisfying +GΨ = ρΨ +and +max +¯Q Ψ = 1. +Moreover, ρ = λ∗,d, given by (3.35), and the following hold. +λ∗,d = +inf +0<ψ∈C2 +δ,+(D) +sup +µ∈P( ¯D) +� +¯D +Gψ +ψ dµ += +sup +0<ψ∈C2 +δ,+(D) +inf +µ∈P( ¯D) +� +¯D +Gψ +ψ dµ. + +22 +ANUP BISWAS AND VIVEK S. BORKAR +Note that the ERSC problem in (3.35) is related to the Nisio semigroup given by +Stf(x) := inf +ζ∈U Ex +� +e +� t +0 c(Xs,ζs)dsf(Xt) +� +f ∈ C( ¯D). +Thus, Theorem 3.14 can be seen as a generalization of Theorem 3.11 to nonlinear semigroup. One +can also have a similar Collatz-Wielandt formula for the generalized Dirichlet principal eigenvalue +which is defined replacing L + c by G in (3.4). +Theorem 3.15 ([8]). Let D be a bounded, smooth domain and λD(G) denote the generalized Dirich- +let principal eigenvalue of G in D. Then we have +λD(G) = +inf +ψ∈C2 ++(D) +sup +µ∈P(D) +� +D +Gψ +ψ dµ += +sup +ψ∈C2 ++(D)∩C0(D) +inf +µ∈P(D) +� +D +Gψ +ψ dµ. +As pointed out in [8, Remark 2.2], the set C2 ++(D) ∩ C0(D) in the second equality cannot be +extended to C2 ++(D). By Theorem 3.7 and 3.8 we know that λ∗,d = λ1(G), the generalized principal +eigenvalue of G in Rd. So one might expect an analog of Theorem 3.15 for λ∗,d. It turns out that +for a linear operator L of the form (3.2), one has (cf. [8]) +λRd(c) = +inf +0<ψ∈C2(Rd) +sup +µ∈P(Rd) +� +Rd +�Lψ +ψ + c +� +dµ. +But by considering Lf = f ′′ − f ′ in R and c = 0, it is shown in [8, Example 2.3] that +λRd(c) < +sup +0<ψ∈C2 +b (Rd) +inf +µ∈P(Rd) +� +Rd +Lψ +ψ dµ ≤ +sup +0<ψ∈C2(Rd) +inf +µ∈P(Rd) +� +Rd +Lψ +ψ dµ. +Thus we need an additional condition on the operator in order to obtain a full Collatz-Wielandt +type formula. In [8], the following condition, which is slightly stronger than Assumption 3.1, is +used in order to obtain a Collatz-Wielandt type formula. +Assumption 3.2. There exists a positive V ∈ C2(Rd) with infRd V > 0 such that one of the +following hold: +(i) There exists an inf-compact, positive ℓ ∈ C(Rd) and a compact set K satisfying +sup +u∈U +LuV ≤ ¯κ1K − ℓV +in Rd, +for some constant ¯κ, and βℓ − maxu∈U γc(·, u) is inf-compact, for some β ∈ (0, 1). +(ii) For some positive constants ¯κ, θ and a compact set K, we have +sup +u∈U +LuV ≤ ¯κ1K − θV +in Rd, +and +lim sup +|x|→∞ +max +u∈U γc(x, u) < θ. +By o(V) we denote the class of functions growing slower than V, that is, f ∈ o(V) if and only if +lim sup +|x|→∞ +|f(x)| +V(x) = 0. +Theorem 3.16 ([8]). Suppose that (B1)-(B3) and Assumption 3.2 hold. Then we have +λ∗,d = λ1(G) = +inf +0<ψ∈C2(Rd) +sup +µ∈P(Rd) +� +Rd +Gψ +ψ dµ + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +23 += +sup +0<ψ∈C2(Rd)∩o(V) +inf +µ∈P(Rd) +� +Rd +Gψ +ψ dµ. +4. Risk-sensitive control of continuous time Markov chains +In this section we review the recent developments on ERSC for continuous time controlled Markov +chains. We consider a continuous time controlled Markov chain (CTCMC) X = {Xt ,t ≥ 0}, on +a denumerable state space S, controlled by the control process ζt , t ≥ 0 , taking values in U. As +before, U is the action space of the controller, which is assumed to be a Borel space with Borel +σ-algebra B(U). For each i ∈ S, let U(i) be the space of all admissible actions of the controller +when the system is at state i. Let K := {(i, u) : i ∈ S, u ∈ U(i)} be the set of all feasible state +action pairs. As before, we denote by c : K → R+ the running cost function. The transition rates +q(j|i, u), u ∈ U(i) , i, j ∈ S, satisfy the condition q(j|i, u) ≥ 0 for all u ∈ U(i), i, j ∈ S and j ̸= i. +In addition, we also impose the following: +Assumption 4.1. +(a) For each i ∈ S, the admissible action space U(i) is a nonempty compact +subset of U . +(b) The model is conservative: +� +j∈S +q(j|i, u) = 0 +∀ u ∈ U(i), i ∈ S . +(c) The model is stable: +q(i) := +sup +u∈U(i) +(−q(i|i, u)) = sup +u∈U(i) +� +j̸=i +q(j|i, u) < ∞ +∀ i ∈ S . +For each i, j ∈ S, q(j|i, u) is a measurable map on U(i) . Let c : S × U → R+ be the running +cost function. +Following [134] (see also [110, 112, 146]) we briefly describe the evolution of the CTCMC. Let +S∞ := S ∪ {i∞} for an isolated point i∞ /∈ S. Define the canonical sample space Ω := (S × +(0, ∞))∞ ∪ {(i0, θ1, i1, . . . , θm, im, ∞, i∞, ∞, i∞, . . . ) | θk ̸= ∞, ik ̸= i∞ +for all +0 ≤ k ≤ m, m ≥ +1} , with Borel σ-algebra B(Ω) . For each sample point ω = (i0, θ1, i1, . . . , θm, im, . . .) ∈ Ω, we set +T0(ω) = 0, Tk(ω) = θ1 + θ2 + · · · + θk, and define T∞(ω) = limk→∞ Tk(ω) . +Now we define a +controlled process {Xt}t≥0 on (Ω, B(Ω)) by +Xt = +� +k≥0 +1{Tk≤t 0 and C2 ≥ 0 +such that +(a) � +j∈S ˜V(j)q(j|i, u) ≤ C0˜V(i) + C2 for all (i, u) ∈ K ; +(b) q(i) ≤ C1˜V(i) for all i ∈ S . +For the rest of this section, we are going to assume that Assumption 4.2 holds. +Note that +Assumption 4.2 holds if supi∈S q(i) < ∞. In this case we can choose ˜V to be a suitable constant. +From [111, Theorem 3.1] (see also, [110, Proposition 2.2]) it also follows that, under Assumption 4.2, +Pζ +i (T∞ = ∞) = 1 for all i ∈ S and ζ ∈ U. +We also assume the following for our CTCMC (compare with Definition 2.1). +Assumption 4.3. +(a) For each i ∈ S, the map u �→ c(i, u) is continuous on U(i). +(b) For each i ∈ S and bounded measurable function f : S → R, the map u �→ Σj∈Sf(j)q(j|i, u) +is continuous on U(i). +For each admissible control ζ the ergodic risk-sensitive cost is given by +Ei(c, ζ) := lim sup +T→∞ +1 +γT log Eζ +i +� +eγ +� T +0 c(Xt,ζt)dt� +, +γ > 0, +(4.2) +where X is the CTCMC corresponding to ζ with initial state i. As before, our aim is to minimize +(4.2) over all admissible policies in U. A policy ζ∗ ∈ U is said to be optimal if for all i ∈ S +Ei(c, ζ∗) = inf +i∈S inf +ζ∈U Ei(c, ζ) := λ∗,c +for all i . +We also define +λ∗,c +m = inf +x∈S inf +ζ∈Um Ex(c, ζ). +(4.3) +Recall that a stationary Markov process X with rate matrix Q = [q(j|i)] is said to be irreducible +if for any i, j ∈ S, i ̸= j, there exists distinct i1, i2, . . . , ik ∈ S satisfying q(i1|i) · · · q(j|ik) > 0 (cf. +[109, p. 107]). The following result is proved in [155, Theorem 3.2] when S is finite. +Theorem 4.1. Let S be finite and Assumption 4.1, 4.3 hold. Also, assume that the CTCMC X +is irreducible under every stationary Markov control in Usm. Then there exists a positive vector Ψ +satisfying +γλ∗,cΨ(i) = min +u∈U(i) + +� +j∈S +Ψ(j)q(j|i, u) + γc(i, u)Ψ(i) + + +for i ∈ S . +(4.4) + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +25 +Moreover, any minimizing selector of (4.4) is an optimal stationary Markov control. +[155, Theorem 3.2] actually proves the existence of an eigen-pair (Ψ, λ∗,c +m ) (see (4.3)) satisfying +(4.4). Since S is finite, applying Dynkin’s formula it can be easily seen that λ∗,c +m = λ∗,c. +For S infinite, the first result concerning the ERSC problem appeared in [106] where the authors +proved the existence of an optimal stationary Markov control. One should note that the ERSC +control problem in [106] was over the set Usm. +Theorem 4.2 ([106]). Suppose that Assumptions 4.1 and 4.3 hold. In addition, we also assume +the following. +(i) supi∈S q(i) < ∞; +(ii) CTCMC X is irreducible under every stationary Markov control; +(iii) For some state z ∈ S, there exist functions w1 : S → R+, w2 : S → [1, ∞), w2 is norm-like, +and positive constants θ, κ satisfying +e−w1(i) � +j∈S +ew1(j)q(j|i, u) ≤ −θw2(i) + κ1{z}(i) +for all i ∈ S, u ∈ U(i). +(4.5) +(iv) supi∈S,ζ∈Usm Eζ +i [˘τz] < ∞, where ˘τz denotes the return time to the state z, that is, +˘τz = inf{t > 0 : Xt = z}. +Then, if we choose γ small enough so that γ supK c(i, u) < θ for θ as in (4.5), the map +Usm ∋ ζ → Ei(c, ζ) +(see(4.2)) +has a minimizer in Usm. +A variant of Theorem 4.2 was obtained by [156] where the authors replace the stability condition +(4.5) by a simultaneous Doeblin condition and also allow the transition rates to be unbounded. +Theorem 4.3 ([156]). Let Assumption 4.1- 4.3 to hold. In addition, we also assume the following. +(i) For some state z ∈ S and a control ˜ζ ∈ Usm we have Ez(c, ζ) < ∞; +(ii) CTCMC X is irreducible under every stationary Markov control; +(iii) (Simultaneous Doeblin condition) There exists t0 ∈ R+ and α ∈ (0, 1) so that Pζ +i (˘τz ≥ t0) ≤ +α for all i ∈ S and ζ ∈ Usm. +Then there exists a positive function Ψ : S → (0, ∞) satisfying +γλ∗,c +m Ψ(i) ≥ min +u∈U(i) + +� +j∈S +Ψ(j)q(j|i, u) + γc(i, u)Ψ(i) + + +for i ∈ S , +(4.6) +and every minimizing selector is an optimal control in Usm. +Proof of Theorem 4.3 in [156] is based on an approximation procedure. The ERSC problem is +first solved for a cost function c having compact support and bounded transition rate functions. +In particular, the optimality equation (4.6) is obtained for this approximate model. Then, passing +to the limit, one obtains (4.6) for the original system. In a recent work [113], the authors also +study the ERSC problem for CTCMC and establish the existence of Ψ solving (4.6) with equality. +Instead of simultaneous Doeblin condition, [113] imposes a Lyapunov type stability condition of the +form (4.5). But it also assumes infS Ψ > 0 (see [113, Assumption 5.1]) which is a bit restrictive in +nature. A thorough study of ERSC problem appears in the recent work [50] under the assumption +of blanket stability. +Assumption 4.4. Assume that the CTCMC X is irreducible under every stationary Markov +control in Usm. In (i) and (ii) below, the function V on S takes values in [1, ∞) and �C is a positive +constant. Assume also that one of the following hold. + +26 +ANUP BISWAS AND VIVEK S. BORKAR +(i) For some positive constant θ and a finite set C it holds that +sup +u∈U(i) +� +j∈S +V(j)q(j|i, u) ≤ �C1C(i) − θV(i) +∀ +i ∈ S . +Also assume that γ∥c∥∞ := γ supK c(i, u) < θ. +(ii) For a finite set C and a norm-like function ℓ : S → R+ it holds that +sup +u∈U(i) +� +j∈S +V(j)q(j|i, u) ≤ �C1K(i) − ℓ(i)V(i) +∀ +i ∈ S . +Moreover, the function ℓ(·) − maxu∈U(·) c(·, u) is norm-like. +The above conditions should be compared with Assumption 3.1. +Theorem 4.4 ([50]). Let Assumptions 4.1–4.4 hold. Also assume that there exists i0 ∈ S such that +q(j|i0, u) > 0 +for all j ̸= i0, and u ∈ U(i0). +(4.7) +Then the following hold. +(i) There exists a unique positive function Ψ, Ψ(i0) = 1, satisfying +γλ∗,c Ψ(i) = min +u∈U(i) + +� +j∈S +Ψ(j)q(j|i, u) + γc(i, u)Ψ(i) + + +for i ∈ S . +(ii) A stationary Markov control v ∈ Usm is optimal if and only if it satisfies +min +u∈U(i) + +� +j∈S +Ψ(j)q(j|i, u) + γc(i, u)Ψ(i) + + = + +� +j∈S +Ψ(j)q(j|i, v(i)) + c(i, v(i))Ψ(i) + + +for all i ∈ S. +Conditioned (4.7) can be relaxed to include different classes of CTCMC (see [50, Remark 3.2]). +As discussed before for DTCMP and controlled diffusion processes, the ERSC problem has also +been studied for near-monotone cost functions (see Definition 2.2). A work in this direction appears +in [151] where the authors prove the following +Theorem 4.5. Let Assumptions 4.1, 4.3 hold and the transition rates are bounded, that is, +sup +S +q(i) < ∞. +We also assume that following to hold. +(i) −q(i|i, u) > 0 for all (i, u) ∈ K; +(ii) One of the following hold. +(a) q(j|i, u) > 0 for all i ̸= j and u ∈ U(i). +(b) For each i ∈ S, there exists a finite set Ci such that minu∈U(i) q(j|i, u) > 0 for all +j ∈ Ci and supu∈U(i) q(j|i, u) = 0 for all j /∈ Ci. +(iii) CTCMC X is recurrent under every stationary Markov control. +(iv) For some ζ ∈ Usm, we have Ei(c, ζ) is finite for all i ∈ S. In addition, c is near-monotone +with respect to λ∗,c +m . +Then there exists a positive function Ψ, satisfying +γλ∗,c Ψ(i) ≥ min +u∈U(i) + +� +j∈S +Ψ(j)q(j|i, u) + γc(i, u)Ψ(i) + + +for i ∈ S , +and every minimizing selector is an optimal stationary Markov control. + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +27 +The conditions of Theorem 4.5 has been relaxed substantially in [50, Theorem 3.2]. +Theorem 4.6. Let Assumptions 4.1–4.3 hold and +inf +ζ∈Usm Ei(c, ζ) < ∞ +∀ i ∈ S. +We also assume the following +(i) There exists i0 ∈ S such that +q(j|i0, u) > 0 +for all j ̸= i0, and u ∈ U(i0). +(ii) c is near-monotone with respect to λ∗,c +m . +(iii) CTCMC X is recurrent under every stationary Markov control. +Then there exists a positive function Ψ, satisfying +γλ∗,c Ψ(i) ≥ min +u∈U(i) + +� +j∈S +Ψ(j)q(j|i, u) + γc(i, u)Ψ(i) + + +for i ∈ S , +and every minimizing selector is an optimal stationary Markov control. +Condition (iii) above can be relaxed for a class of CTCMC, allowing the possibility that X could +be transient for some ζ ∈ Usm. In fact, the following is proved in [50, Theorem 3.3]. +Theorem 4.7. Let S = {1, 2, . . .} and Assumption 4.1–4.3 hold. We also let +inf +ζ∈Usm Ei(c, ζ) < ∞ +∀ i ∈ S, +and +(i) c is near-monotone with respect to λ∗,c +m . CTCMC X is irreducible under every stationary +Markov control. +(ii) There exists a function W : S → [1, ∞) satisfying W(i) ≥ i for all large i and +sup +u∈U(i) +� +j∈S +W(i)q(j|i, u) ≤ g(i) +for i ∈ S, +for some function g : S → R satisfying limi→∞ g(i) = 0. Furthermore, for some η > 0 we +have1 +min +u∈U(i) +q(i − 1|i, u) +−q(i|i, u) +≥ η +for all i ∈ S. +(iii) q(·|1, u) supported on a finite set C, independent of u ∈ U(1). For Dn := {1, . . . , n}, v ∈ Usm +and any j ∈ Dn \ {1} there exists distinct i1, i2, . . . , ik ∈ Dn we have +q(i1|1, v(1))q(i2|i1, v(i1)) · · · q(j|ik, v(ik)) > 0 . +Then there exists a positive function Ψ satisfying +γλ∗,c Ψ(i) ≥ min +u∈U(i) + +� +j∈S +Ψ(j)q(j|i, u) + γc(i, u)Ψ(i) + + +for i ∈ S . +(4.8) +Furthermore, any measurable selector of (4.8) is an optimal stationary Markov control. +Theorems 4.4, 4.6 and 4.7 are proved using an approach similar to Theorem 3.8. More precisely, +the ERSC problems are first studied in bounded domains and then it is shown that the optimality +equation has a limit as we increase the domains to S. +1The following appears incorrectly in [50] + +28 +ANUP BISWAS AND VIVEK S. BORKAR +5. Risk-sensitive maximization problems and beyond +In this section, we briefly review some other types of optimization problems involving risk- +sensitive cost criterion. We mainly discuss the maximization problem and the game associated to +the risk-sensitive cost. +5.1. Risk-reward problems. The readers must have noticed that the risk-sensitive minimization +problems are not equivalent to the maximization problems. So it naturally becomes interesting to +study them separately. Surprisingly, work on risk-sensitive reward/maximization has been relatively +uncommon. +For DTCMP with a finite state space, the maximization problems are covered by +Theorem 2.2. In fact, this corresponds to the case γ < 0. Another recent work to deal the risk- +reward problem is [5] where the authors consider the maximization problem for DTCMP with a +compact state space S. More precisely, given a continuous one-stage reward function r : S×U×S → +R, the following maximization problem is considered in [5]. +β∗,m := sup +x∈S +sup +ζ∈U +Ex(r, ζ) +where +Ex(r, ζ) = lim sup +T→∞ +1 +T log Eζ +x +� +e +�T −1 +t=0 r(Xt,ζt,Xt+1)� +. +(5.1) +The result in [5, Theorem 2.2 and 2.3] is stated below. +Theorem 5.1. Let S be a compact metric space. Suppose that P(·|x, u) has full support for every +(x, u) ∈ S × U. +Then eβ∗,m is the Perron-Frobenius eigenvalue and there exists some positive +Ψ ∈ C(S) satisfying +eβ∗,mΨ(x) = +sup +v∈P(U) +� +S +� +U +P(dy|x, u)v(du)Ψ(y)er(x,u,y) := T Ψ(x) +x ∈ S. +Moreover, there exists an optimal ζ ∈ Usm for the maximization problem (5.1), and the following +Collatz-Wielandt representation hold. +eβ∗,m = +inf +0<ψ∈C(S) +sup +µ∈P(S) +� +T ψdµ +� +ψdµ += +sup +0<ψ∈C(S) +inf +µ∈P(S) +� +T ψdµ +� +ψdµ . +Recall from Theorem 3.12 ( [5, Theorem 3.2]) that a variational representation of β∗ similar to +Theorem 3.10 (and (3.29)) is obtained as a consequence of the above result. It is then extended +to the case where P(·|x, u) need not have full support, by using an approximation argument. For +controlled reflected diffusions in a bounded C3 domain (see (3.34)), risk-reward problems were +studied in [10, Theorem 2.1]. In case of Rd, as it turns out, the analysis of risk-reward problems +does not differ much under the blanket stability hypothesis (see Assumption 4.4). Furthermore, +[10] studied the maximization problem under a near-monotone condition which we describe now. +Consider controlled diffusion as in (3.12) and assume (B1)-(B3) to hold. Given a continuous reward +function c : Rd × U → R (not necessarily non-negative), Lipschitz in the first argument uniformly +with respect to the second, we define the risk-sensitive maximization problem as follows. +β∗,d := sup +x∈S +sup +ζ∈U +Ex(c, ζ) +where +Ex(c, ζ) = lim sup +T→∞ +1 +γT log Eζ +x +� +e +� T +0 γc(Xt,ζt)dt� +, γ > 0. +(5.2) +Let us define +Hf = max +u∈U{Luf + γc(x, u)f(x)}. + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +29 +For each n, it known from [148] that there exists a unique pair (wn, ̺n) ∈ C( ¯Bn) ∩ C2(Bn) × R, +Bn being the ball of radius n centered at the origin, satisfying +Hwn = γ̺nwn +in Bn, +wn > 0 +in Bn +wn = 0 +on ∂Bn, +wn(0) = 1. +(5.3) +(wn, ̺n) is called the Dirichlet generalized principal eigen-pair of H in Bn. Moreover, we have +̺n < ̺n+1 for all n ∈ N. It turns out that limn→∞ ̺n = λ1(H) where λ1(H) denotes the generalized +principal eigenvalue of H in Rd defined as follows. +λ1(H) = inf{λ : ∃ positive ψ ∈ C2(Rd) satisfying Hψ ≤ λψ in Rd}. +Definition 5.1. A continuous reward function c : Rd × U → R, which is bounded from above, is +said to be near-monotone for the maximization problem if +β∗,d > lim +r→∞ sup +Bcr×U +c(x, u). +The above definition should be compared with the near-monotonicity condition used for the +minimization problem (see Theorem 3.7). We must point out that near-monotonicity criterion in +[10] is defined using the limits of Neumann eigenvalues whereas [9] uses λ1(H). Also, note that +β∗,d is bounded from above by supRd×U c, and therefore finite. +We have the following result from [12, Theorem 4.1] +Theorem 5.2. Let (B1)-(B3) hold, c is bounded from above and |b| has at most linear growth. +Suppose that c is near-monotone for the maximization problem in the sense of Definition 5.1. Then +the following hold. +(i) λ1(H) = γβ∗,d. +(ii) There exists a unique, bounded, positive Φ ∈ C2(Rd) satisfying +HΦ(x) = γβ∗,d Φ(x), +x ∈ Rd, +and +Φ(0) = 1. +(iii) A stationary Markov control v is optimal if and only if +max +u∈U{b(x, u) · ∇Φ(x) + γc(x, u)Φ(x)} = b(x, v(x)) · ∇Φ(x) + γc(x, v(x))Φ(x) +almost surely in Rd. +5.2. Risk-sensitive games. Let us also mention a few interesting works treating the game prob- +lems with ergodic risk-sensitive criterion. For finite state DTCMC, zero-sum games are studied +in [70] whereas [25,101] consider zero-sum games with a countable state space. Some other works +dealing with zero-sum games include: [52] for controlled diffusion, [107] for controlled diffusion re- +stricted to an orthant, [104] for controlled reflected diffusion in a bounded domain, [29] for DTCMC +with a general state space. +Non zero-sum games with risk-sensitive criterion are considered in +[24,100,102,103,105,157]. +6. Algorithms +In this section we review the results on the policy iteration and value iteration for the ERSC +problem. Towards this end, we also discuss a few recent results on equivalent linear programs and +reinforcement learning. +6.1. Policy iteration. Since we already known that under suitable hypotheses, the ERSC problem +can have an optimal stationary Markov policy, it is natural to investigate if it can be determined +through policy or value iteration techniques. The first policy iteration for the ERSC problem was +considered in [120] where the authors used policy iteration algorithm to establish existence of an +eigen-pair satisfying (2.6). To begin with, suppose that the DTCMC X takes values in a finite + +30 +ANUP BISWAS AND VIVEK S. BORKAR +state space S and is irreducible under every stationary Markov control. Then the policy iteration +algorithm (PIA) can be described as follows. +Algorithm 6.1. Policy iteration. +1. Initialization: Set k = 0 and choose a ζ0 ∈ Usm. +2. Value determination: Let ψk be the unique positive eigenfunction satisfying ψk(i0) = 1 for +some prescribed i0 ∈ S, for the eigenvalue problem +eγλkψk = +� +u +ζk(u) + +eγc(x,u) � +y∈S +ψk(y)P(y|x, u) + + , +x ∈ S, +where +λk = Ex(c, ζk). +3. Policy improvement: Choose ζk+1 ∈ Usm satisfying +ζk+1(x) ∈ Arg min +u∈U(x) + +eγc(x,u) � +y∈S +ψk(y)P(y|x, u) + + . +The following result can be found in [88, Theorem 4.7] +Theorem 6.1. Suppose that S, U are finite, γ > 0 and X is irreducible under every stationary +Markov control. Then Algorithm 6.1 converges in finite number of steps, that is, there exists m ∈ N +such that {λk : 0 ≤ k ≤ m} forms a strictly deceasing sequence until it reaches λ∗. +When the state space S is countably infinite, PIA algorithms are studied under two frameworks: +(a) under an assumption of Lyapunov stability, (b) the running cost is near-monotone. Under a +Lyapunov stability condition, PIA was established in [50], which was then further improved in [74]. +Theorem 6.2 ([50]). Suppose that Assumption 2.1 and 2.3 hold with a norm-like V and in case +of Assumption 2.3(ii), there exists an η ∈ (0, 1) so that γ maxu∈U(·) c(·, u) ≤ ηℓ in S. Also suppose +that there exist states i0, z0 ∈ S satisfying +inf +u∈U(i0) P(j|i0, u) > 0 +for all j ̸= i0, +and +inf +u∈U(j) P(z0|j, u) > 0 +for all j. +(6.1) +Then Algorithm 6.1 converges in the sense that {λk} forms a decreasing sequence and limk→∞ λk = +λ∗,m. Furthermore, ψk converges to the unique solution Ψ in (2.10). +The rough idea of the proof of Theorem 6.2 goes as follows. Define ck(i) = c(i, ζk+1(i)) and +θk(i) = 1 − +1 +ψk(i) + +eγck+1(i)−γλk � +y∈S +ψk(y)P(y|i, ζk+1(i)) + + . +From Algorithm 6.1 it follows that 0 ≤ θk ≤ 1. Once we have a point-wise bound for the sequence +{ψk}, the proof of Theorem 6.2 would follow if we could establish that θk → 0 pointwise, as k → ∞. +To attain this goal, it is shown in [50, Theorem 4.1] that the Markov process Y(k) associated to +the twisted kernel ˜P (k), defined as, +˜P (k)(j|i) = +ψk(j)P(j|i, ζk(i)) +� +j∈S ψk(j)P(j|i, ζk(i)) +has a unique stationary probability measure πk. Furthermore, {πk} is tight and every sub-sequential +limit has full support. It is then show that +lim +k→∞ +� +j∈D +θ(j)πk(j) = 0 +for all finite sets D. + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +31 +This in turn proves that θk(i) → 0 for all i, as k → ∞. Note that (6.1) is a bit restrictive. This +was used to find a small set for the Markov chain Y(k). Later [74] followed the same approach but +without the condition (6.1) and established the convergence of Algorithm 6.1. +Next we come to the second setting where we do not impose any blanket stability assumption like +Assumption 2.3, but work with the near-monotone structure imposed on the running cost. This is +done under the setting of Theorem 2.5. To describe the result we need a few more notations. For +ζ ∈ Um, recall the quantity Λ(ζ) from (2.9). We say ζ ∈ Um is stabilizing if ζ ∈ Usm and Λ(ζ) < ∞. +Fix z ∈ S and define the first hitting time of z as σz = inf{n ≥ 0 : Xn = z}. Given a stabilizing +policy ζn we define the relative value function hn as follows. +hn(i) = e−γ(c(z,ζn(z))−Λ(ζn)) Eζn +x +� +e +�σz +t=0 γ(c(Xt,ζn(Xt))−Λ(ζn)� +i ∈ S. +Note that hn(z) = 1. A new policy ζn+1 is then defined through the minimization +ζn+1(i) ∈ Arg min +u∈U(i) +ec(x,u) � +j∈S +hn(j)P(j|i, u)). +This generates a sequence of stabilizing policies and relative value functions. We need two more +additional assumptions. +(H1) There exists a positive Ψ∗ satisfying +eγΛ∗Ψ∗(x) = +inf +u∈U(x) + +eγc(x,u) � +y∈S +Ψ∗(y)P(y|x, u) + + +for all x ∈ S, +(6.2) +where Λ∗ is given by (2.9). +(H2) There exists a minimizing selector w∗ of (6.2) such that the transformed kernel +˘P∗(j|i) := eγ(c(i,w∗(i))−Λ∗)Ψ∗(y)P(y|i, w∗(i)) +Ψ∗(i) +(6.3) +is positive recurrent with a unique invariant probability ˘π∗. +We say the DTCMC is skip free if for each x ∈ S there exists a finite set Nx so that P(Nx|x, u) = 1 +for all u ∈ U. Now we are ready to state the result from [57, Theorem 5.4]. +Theorem 6.3. Grant the setting of Theorem 2.5. Also, assume that (H1)-(H2) hold, DTCMC is +skip free and the relative value functions satisfy the multiplicative Poisson equation +eγΛ(ζn)hn(x) = eγc(x,ζn(x)) � +y∈S +hn(y)P(y|x, ζn(x)) +∀ x ∈ S. +Suppose moreover that +(i) ˘π∗(¯h/Ψ∗) < ∞ where ¯h(x) = lim supn→∞ hn(x); +(ii) For any limit point (h∞, w∞, c∞) of the sequence {(hn, wn, c(·, ζn)) : n ≥ 0}, the mul- +tiplicative Poisson equation has a solution ˜h∞ with transition kernel P(j|i, w∞(i)), and +the associated transformed kernel ˘P∞ (defined in the same manner as in (6.3)) is positive +recurrent with an invariant probability measure ˘π∞ and ˘π∞(h∞/˜h∞) < ∞. +Then hn → Ψ∗ and Λ(ζn) ց Λ∗, as n → ∞. +Let us now define a PIA for the controlled diffusion. For a stationary Markov control v ∈ Usm +we define the operator +Lvf(x) = trace(a(x)∇2f(x)) + b(x, v(x)) · ∇f(x) + γc(x, v(x))f(x), +and by λ1(Lv) we denote the principal eigenvalue of Lv in Rd (see (3.5)). +Algorithm 6.2. Policy iteration. + +32 +ANUP BISWAS AND VIVEK S. BORKAR +1. Initialization: Set k = 0 and choose a v0 ∈ Usm; +2. Value determination: Let Ψk be a principal eigenfunction in W2,p +loc(Rd), p > d, satisfying +Ψk(0) = 1, and +LvkΨk = γλkΨk +in Rd, +where λk = γ−1λ1(Lvk); +3. Policy improvement: Choose vk+1 ∈ Usm satisfying +vk+1(x) ∈ Arg min +u∈U(x) +[b(x, u) · ∇Ψk + γc(x, u)Ψk] . +The following convergence result can be found in [12, Theorem 3.2]. +Theorem 6.4. Assume the setting of Theorem 3.8 and let b have at most linear growth. Also, +assume in case of Assumption 3.1(ii) that supRd×U(γc) < θ. Then λk = Ex(c, vk) for all x and +the Algorithm 6.2 converges, that is, λk ց λ∗,d (given by (3.13)) and Ψk converges weakly in +W2,p +loc(Rd), p > d, to the unique solution Ψ in (3.22). +An analogous algorithm for the maximization problem has also been proved in [12, Theorem 4.2]. +Theorem 6.5. Assume the setting of Theorem 5.2 and let v0 ∈ Usm and λ0 = γ−1λ1(Lv0) be such +that +λ0 > lim +r→∞ sup +Bcr×U +c(x, u). +Generate a sequence of λk and vk as follows. Let �Ψk be the unique principal eigenfunction satisfying +Lvk �Ψk = γλk �Ψk +in Rd, +�Ψk(0) = 1, +where λ1(Lvk) = γλk. Define +vk+1(x) ∈ Arg max +u∈U(x) +� +b(x, u) · ∇�Ψk + γc(x, u)�Ψk +� +. +Then λk ր β∗,d, defined by (5.2), and �Ψk converges weakly to Φ in Theorem 5.2. +For CTCMC, PIA is studied in [106, 113] under the assumption that both S and U are finite +sets. We consider the setting of Theorem 4.1. Also, assume that supi∈S,ζ∈Usm Eζ +i [˘τz] < ∞, where +˘τz denotes the return time to a prescribed state z, that is, +˘τz = inf{t > 0 : Xt = z}. +For every ζk ∈ Usm, we let λk = Ex(c, ζk) (which would be independent of x) and define +hk(x) = Eζk +x +� +e +� ˘τz +0 +γ(c(Xt,ζk(Xt))−λk)dt� +x ∈ S. +From [106] we know that +� +j∈S +hk(j)q(j|i, ζk(i)) + γc(i, ζk(i))hk(i) = γλkhk(i) +i ∈ S. +(6.4) +As before, the improved policy ζk+1 is defined through minimization, that is, +ζk+1(i) ∈ Arg min +u∈U(i) +{ +� +y∈S +hk(j)q(j|i, u) + γc(i, u)hk(i)}. +(6.5) +Assuming S and U to be finite, it is shown in [106, Theorem 5.1] (see also [113, Lemma 6.1]) that +the above iteration converges in finite number of steps and ζk converges to an optimal stationary +Markov control. The same PIA above can be extended to countably infinite state space under the +setting of Theorem 4.4. In fact, the following is proved in [50, Theorem 4.3]. + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +33 +Theorem 6.6. Assume the setting of Theorem 4.4 with a norm-like Lyapunov function V. In case +of Assumption 4.4(ii), let there exist an η ∈ (0, 1) so that γ maxu∈U(·) c(·, u) ≤ ηℓ in S. Then the +PIA (generated by (6.4)-(6.5)) starting from any ζ0 ∈ Usm converges. +6.2. Relative value iteration. In this section we review some of the important contributions +on value iteration for the ERSC problems. Value iteration (VI) or relative value iteration (RVI) +basically provide a recursive method to generate a sequence of value functions that converge to the +solution of the optimality equation. As a by product of this method we can generate nearly optimal +controls. One of the early works dealing with VI appeared in [40]. The authors of [40] studied RVI +for finite state DTCMC. For simplicity of notation, we restrict ourselves to γ = 1 in this section. +Fix a positive function V0 : S → R and define a sequence {Vn} recursively as follows +Vn(x) = min +u∈U(x) + +ec(x,u) � +y∈S +Vn−1(y)P(y|x, u) + + +x ∈ S. +(6.6) +Fix a point z ∈ S. The relative value functions ˜Vn is defined as ˜Vn(x) = Vn(x) +Vn(z). Also, define the +n-th differential cost function as λn(x) = log Vn(x) − log Vn−1(x). Then (6.6) can be written as +˜Vn(x) = min +u∈U(x) + +ec(x,u)−λn(z) � +y∈S +˜Vn−1(y)P(y|x, u) + + +x ∈ S. +(6.7) +Then the following result is proved in [40]. +Theorem 6.7. Let S be finite and the DTCMC is irreducible under every stationary Markov +control. In addition to Assumption 2.1, also suppose that +P(x|x, u) > 0 +for all (x, u) ∈ K. +(6.8) +Then ˜Vn(x) → Ψ(x) and λn(x) → λ∗,m for all x ∈ S where (Ψ, λ∗,m) are given by Theorem 2.2. +Though (6.8) is restrictive, it plays a key role in the analysis of [40]. In particular, this condition +is used to establish a contraction phenomenon for a span semi-norm which is then used to obtain +the convergence result for the RVI sequence. Later in [71] the condition (6.8) is removed. The RVI +method in [71] works under a very general set-up. As shown in [71], one could transform the given +DTCMC model suitably so that (6.8) holds. The key hypotheses used by [71] is as follows. +(H3) S is finite and Assumption 2.1 holds. There exists an eigen-pair (ψ, λ), ψ > 0, satisfying +eλψ(x) = min +u∈U(x) + +ec(x,u) � +y∈S +ψ(x)P(y|x, u) + + +for x ∈ S. +(6.9) +It is easy to see that λ has to be λ∗. Let us now introduce the transformed model from [71]. Fix +α ∈ (0, 1) and define ˘c : K → R as follows. +˘c(x, u) = log((1 − α)ec(x,u) + α). +A transformed transition kernel ˘P is also defined as follows +˘P(y|x, u) = (1 − α)ec(x,u)P(y|x, u) + αδxy +(1 − α)ec(x,u) + α +, +where δxy denotes the Kronecker symbol on S, that is , δxy = 0 for x ̸= y and δxx = 1. Note that +˘P(x|x, u) > 0 +for all (x, u) ∈ K, +and hence (6.8) holds for the transformed system. Interestingly, if we let +˘λ = log((1 − α)eλ + α), + +34 +ANUP BISWAS AND VIVEK S. BORKAR +then it can be easily checked from (6.9) that +e +˘λψ(x) = min +u∈U(x) + +e˘c(x,u) � +y∈S +ψ(x) ˘P(y|x, u) + + +for x ∈ S. +(6.10) +Conversely, if have an eigen-pair (ψ, ˘λ) satisfying (6.10), then setting +λ = log +� +e˘λ − α +1 − α +� +, +one recovers an eigen-pair (ψ, λ) satisfying (6.9). Therefore, it is natural to investigate RVI for +(6.10). As before, given positive ˘V0, we define the sequence {( ˘Vn, ˘λn)} as follows. Set W0 = ˘V0 and +let +Wn(x) = min +u∈U(x) + +e˘c(x,u) � +y∈S +Wn−1(x) ˘P(y|x, u) + + . +For a fixed point z ∈ S, we define +˘V (x) = Wn(x) +Wn(z) +˘λn(x) = Wn(x) − Wn−1(x). +Then the following is proved in [71, Theorem 4.1] +Theorem 6.8. Assume (H3). Then ˘λn(x) → ˘λ as n → ∞, for all x ∈ S where ˘λ is given by +(6.10). +A non-stationary version of the above RVI can be found in [72]. Very recently, RVI is studied in +[15] for DTCMC with a compact state space. In particular, [15] assumes the following +(H4) S is a compact Polish space, and for some reference positive measure ν on S, with full +support, we have +S × U ∋ (x, u) �→ P(dy|x, y) = φ(y|x, y)ν(dy) ∈ P(S). +Moreover, φ(·|·, ·) is continuous. +From the argument of [5, Theorem 2.2] and (H4), we can find a positive Ψ ∈ C(S), unique up to a +multiplicative positive constant, satisfying +eλ∗,mΨ(x) = min +u∈U +� +ec(x,u) +� +S +Ψ(y)P(dy|x, u) +� +x ∈ S. +(6.11) +Following the same philosophy as before, we can define the RVI as follows: Let V0 ∈ C(S) be +positive. Fix z ∈ S and define +Vn+1(x) = +1 +Vn(z) min +u∈U +� +ec(x,u) +� +S +Vn(y)P(dy|x, u) +� +. +Theorem 6.9 ([15]). Let Assumption 2.1 and (H4) hold. Then Vn(x) → ¯V (x) ∈ C(S), ν almost +surly, as n → ∞ and ¯V (z) = eλ∗,m, where ¯V satisfies (6.11). +The above result extends to controlled diffusion in Rd. We discuss the RVI under Assumption 3.2. +[15] also consider value iteration under near-monotone setting, but it also requires some less verifi- +able conditions. Interested readers may consult [15, Theorem 3.2]. Recall from Theorem 3.8 that +under (B1)-(B3) and Assumption 3.2, there exists a positive Ψ ∈ C2(Rd) satisfying +min +u∈U{LuΨ(x) + c(x, u)Ψ(x)} = λ∗,d Ψ(x) +in Rd. +(6.12) + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +35 +Furthermore, Ψ is unique up to a positive multiplicative constant and every minimizing selector of +(6.12) is an optimal stationary Markov control. Now we define +C2 +V,+(Rd) := {g ∈ C2(Rd) : g > 0 +sup +Rd +g +V < ∞}. +It is also known that Ψ ∈ C2 +V,+(Rd). Let us now consider the parabolic equation +∂t ¯Φ(t, x) = min +u∈U{Lu ¯Φ(t, x) + c(x, u)¯Φ(t, x)} − λ∗,d ¯Φ(t, x), +t > 0, +(6.13) +and ¯Φ(0, x) = Φ0 ∈ C2 +V,+(Rd). If we choose Φ0 such that for some κ > 0 +κ−1Ψ ≤ Φ0 ≤ κΨ, +then from the proof of [15, Theorem 3.2], it can be shown that ¯Φ(t, x) → κΨ, for some κ > 0, +as t → ∞. But equation (6.13) contains λ∗,d which is unknown. To replace λ∗,d, we consider a +modified equation as follows. +∂tΦ(t, x) = min +u∈U{LuΦ(t, x) + c(x, u)Φ(t, x)} − Φ(t, 0)Φ(t, x) +t > 0, +(6.14) +with Φ(0, x) = Φ0. As can be easily checked, (6.13) and (6.14) are related by following relation +¯Φ(t, x) = Φ(t, x)e +� t +0 (Φ(s,0)−λ∗,d)ds. +(6.15) +From (6.15) one can use Φ for RVI. Note that +¯Φ(t, x) +Φ(t, x) = +¯Φ(t, 0) +Φ(t, 0). +Using (6.15) and the above relation, we obtain +d +dt +Φ(t, x) +¯Φ(t, x) = λ∗,d − Φ(t, 0) = λ∗,d − ¯Φ(t, 0)Φ(t, x) +¯Φ(t, x). +Hence +Φ(t, x) +¯Φ(t, x) = e− +� t +0 ¯Φ(s,0)ds + λ∗,d +� t +0 +e− +� t +τ ¯Φ(s,0)dsdτ. +Thus, if ¯Φ(t, x) converges to a positive function, then Φ(t,x) +¯Φ(t,x) converges to a positive constant, which +in turn proves the convergence of Φ(t, x). From (6.15), we also get that Φ(t, 0) → λ∗,d as t → ∞. +Therefore, to establish RVI, it is enough to study convergence of ¯Φ. In fact, we have the following +result from [15, Theorem 3.4]. +Theorem 6.10. Let (B1)-(B3) and Assumption 3.2 hold. Suppose that 0 < Φ0 ∈ C2(Rd) and +supRd (Φ0/V) < ∞. Then there exists a constant κ0 = κ0(Φ0) such that the value iteration ¯Φ in +(6.13) converges to κ0Ψ as t → ∞, uniformly on compact sets. +6.3. Linear programming. In this section we revisit Theorem 3.10 and formulate an equivalent +linear program using its equivalence to a single controller game. This is done in [16], building upon +the ideas from [55]. (Here we present suitably corrected statements of the results therein.) Consider +the setting of Theorem 3.10, that is, both S and U are finite. Also, fix γ = 1 for simplicity. We +do not assume irreducibility, whence Ei(c, ζ) can depend on the initial state i. Under a stationary +policy ζn = v(Xn) for all n ≥ 0 for some v : S �→ U, Ei(c, ζ) exists as a well defined limit. Let +¯λ∗,m := max +i∈S +min +ζ∈Usm Ei(c, ζ). +Let +Q := {the set of stochastic matrices Q = [q(j|i)]i,j∈S on S}. + +36 +ANUP BISWAS AND VIVEK S. BORKAR +Let Q(i) := {q(·|i)}, which is a copy of the simplex of probability vectors indexed by S, for each +i ∈ S. Also define +˜c(i, q, u) := c(i, u) − DKL(q(·|i)∥P(·|i, u)). +Consider a controlled Markov chain { ˜Xn} on S as follows. Its action space at i ∈ S is Q(i) × U. +The controlled transition probabilities are +˜P(j|i, q, u) = q(j|i), i, j ∈ S. +The running payoff is ˜c(i, u, q) as above. We shall consider only stationary policies v : S �→ U. Let +Mq denote the set of stationary distributions for q ∈ Q. Then the risk-sensitive control problem +above is equivalent to a zero sum stochastic game with payoff +min +v∈Usm max +q∈Q +�Φ(q, v) +where +�Φ(q, v) = sup +π∈Mq +� +i∈S +π(i)˜c(i, q, v). +This is a single controller game [87] in the sense that the transition probabilities are controlled +by only one of the controllers, the other controller controls only the payoff. It can be shown that +this game has a value and it equals ¯λ∗,m [16]. Then the linear program associated with the ERSC +problem over all stationary policies can be derived as in [153] and is given by: +(LP-P) Minimize � +i∈S βi subject to: +βi ≥ +� +j∈S +q(j|i)βj, (i, q) ∈ S × Q, +Vi ≥ +� +u∈U +˜c(i, q, u)yi(u) − βi + +� +j∈S +q(j|i)Vj, (i, u) ∈ S × Q, +yi(u) ≥ 0, +� +j∈S +yj(u) = 1, i ∈ S. +This is the ‘primal’ linear program. In what follows, we denote µ ∈ M(� +W) := the space of finite +non-negative measures on +� +W := +� +i∈S +({i} × Q(i)) +as µ(i, dq) instead of µ({i}, dq) for notational simplicity. The notation ‘ +� +· · · dµ(i, q)’ will indicate +integration w.r.t. the full measure, whereas ‘ +� +· · · µ(i, dq)’ will indicate integration over the second +variable with the first variable fixed at i. The dual linear program then is: +(LP-D) Maximize � +i∈S wi subject to: +� +� +W +(δij − q(j|i))dµ(i, q) = 0, j ∈ S, +� +� +W +(δij − q(j|i))dν(i, q) + µ(j, Q(j)) = 1, j ∈ S, +� +Q(j) +˜c(j, u, q)µ(j, dq) ≥ wj, j ∈ S, u ∈ U, +(6.16) +j ∈ S, µ, ν ∈ M(� +W), +where δij is the Kronecker delta. +One caveat is that unlike [153], the action spaces here are not both finite - one of them being +a probability simplex, is not. Thus one has to go via finitary approximations, using the fact that + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +37 +non-negative measures supported on a dense subset of a Polish space are dense in the space of +non-negative measures on that space, with respect to the weak∗ topology. See [16] for details. +These linear programs are precisely the counterparts of the primal and dual linear programs +for multi-chain average cost Markov decision problems [78], [132] for this specific single controller +game. +One can show that these linear programs are feasible and have bounded solutions, and +the optimal solution is precisely the value of the two person zero sum game. Furthermore, the +optimal stationary policy is optimal among all admissible policies and can be recovered from the +dual program. (See [16] for details.) +One can reverse engineer the dynamic programming equations from this. These are as follows. +Theorem 6.11. The dynamic programming equation +Ψi = max +q∈Q(i) +� +j +q(j|i)Ψj, +Ψi + Vi = min +u∈U max +q∈Bi + +˜c(i, q, u) + +� +j∈S +q(j|i)Vj + + , +where +Bi := + + +q ∈ Q(i) : +� +j +q(j|i)Ψj = Ψi + + + , +has a solution {(Ψi, Vi)} with Ψi = βi ∀ i ∈ S where {βi} is the solution to (LP-P) and βi = the +value of the above zero sum game with initial condition i for all i. Furthermore, ¯λ∗,m = maxi Ψi. +The state space S can be partitioned into disjoint subsets as S = ∪k +ℓ=1Iℓ where Ij := {j : Ψi = +βj, ∀ i ∈ Ij}, 1 ≤ j ≤ k. Performing the maximization with respect to q ∈ Q(i) exactly, one can +rewrite the dynamic programming equation as +¯λ∗,m = max +i +λi := max +i +min +ζ∈Usm Ei(c, ζ), +λiψi = min +u + +� +j∈Ii +P(j|i, u)ec(i,u)ψj + + , i ∈ Iℓ, 1 ≤ ℓ ≤ k, +λi = min +B∗ +i +� +j∈Ii +� +P(j|i, u)ec(i,u)ψj +� +j′ P(j′|i, u)ec(i,u)ψj′ +� +λj′, (i, j) ∈ Iℓ, 1 ≤ ℓ ≤ k, +where B∗ +i is the set of minimizers in the second equation above. This is the counterpart for risk- +sensitive control of the classical result of Howard [119] for average cost dynamic programming +equation for multi-chain problems. Note the appearance of the so called ‘twisted kernel’ in the last +equation. Analogous results are possible for risk-sensitive reward problems [55] ([16] restates them +correcting the order of maximization in the dynamic programming equation to as it should be.). +As a spin-off, this allows us to handle risk-sensitive control with risk-sensitive constraints2. Con- +sider minimization of Ei(c, ζ) subject to an additional constraint +lim sup +T→∞ +1 +T log E +� +e +�T −1 +t=0 k(Xt,ζt)� +≤ C, +(6.17) +for a prescribed k : S × U �→ R and a constant C > 0. Define +˜k(i, q, u) := k(i, u) − DKL(q(·|i)∥p(·|i, u)). +2This material is new, the details will appear elsewhere. + +38 +ANUP BISWAS AND VIVEK S. BORKAR +Denote the �Φ(q, v) above as �Φ(q, v, c) in order to render explicit its dependence on the per stage +cost function c. Thus we can also define �Φ(q, v, k) analogously. This leads to the convex program: +Minimize maxq∈Q �Φ(q, v, c) subject to maxq′∈Q �Φ(q′, v, k) ≤ C. +Then by standard Lagrange multiplier theory [135], one can consider the unconstrained mini- +mization of +max +q∈Q +�Φ(q, v, c) + Γ(max +q′∈Q +�Φ(q′, v, k) − C), +where Γ ≥ 0 is the Lagrange multiplier. The primal program is: +Minimize � +i∈S(βi + Γ(β′ +i − C)) subject to: +βi ≥ +� +j∈S +q(j|i)βj, (i, q) ∈ S × Q, +Vi ≥ +� +u∈U +˜c(i, q, u)yi(u) − βi + +� +j∈S +q(j|i)Vj, (i, q) ∈ S × Q, +β′ +i ≥ +� +j∈S +q′(j|i)β′ +j, (i, q′) ∈ S × Q, +V ′ +i ≥ +� +u∈U +˜k(i, q′, u)yi(u) − β′ +i + +� +j∈S +q′(j|i)V ′ +j , (i, q′) ∈ S × Q, +yi(u) ≥ 0, +� +j +yj(u) = 1. +The dual linear program is: +Maximize � +i,j∈S wi subject to: +� +� +W +(δij − q(j|i))dµ(i, q) = 0, j ∈ S, +� +� +W +(δij − q(j|i))dν(i, q) + µ(j, Q(j)) = 1, j ∈ S, +� +� +W +(δij − q′(j|i))dµ′(i, q′) = 0, j ∈ S, +� +� +W +(δij − q′(j|i))dν′(i, q′) + µ′(j, Q(j)) = 1, j ∈ S, +� +Q(i) +˜c(i, q, u)µ(i, dq) + Γ +�� +Q(i) +˜k(i, q′, u)µ′(i, dq′) − C +� +≥ wi, (j, u) ∈ S × U, +µ, µ′, ν, ν′ ∈ M(� +W). +Here the Lagrange multiplier Γ is unknown a priori. So one can use the following ‘primal-dual’ +scheme. Start with an initial guess for Γ, say Γ0 ≥ 0, and update it as follows. At step n ≥ 0, solve +the above linear programs for Γ = Γn. Let µn(·, ·) be the optimal µ′(·, ·) from the dual program +and yn +· (·) the optimal y·(·) for the primal linear program, under Γ = Γn. Perform the iterate +Γn+1 = Γn + a(n) +�� +� +W +� +u +˜k(i, q′, u)yn +i (u)dµn(i, dq′) − C +� +, + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +39 +where a(n) > 0 is a stepsize sequence satisfying a(n) → 0, � +n a(n) = ∞. +6.4. Reinforcement learning. Reinforcement learning deals with data-driven algorithms for con- +trol. They are popular for situations when the system model is not known or is too messy, but +adequate data, either real or simulated, online or offline, is available. In most cases this is based +on approximate dynamic programming. Thus these algorithms usually mimic classical iterative +schemes for solution of dynamic programming equations. In fact they are usually stochastic ap- +proximation versions thereof. The development of reinforcement learning for ergodic risk-sensitive +control, however, has been rather limited. We summarize it here. Much of the literature that talks +of risk-sensitive reinforcement learning refers to other notions of risk arising from economics and +finance. We do not consider them here, nor do we consider anything other than the ‘ergodic’ or +time-asymptotic case that we have been considering so far. That is, we do not consider the finite +horizon problem that has received some attention in literature [85]. +The risk-sensitive Q-learning algorithm is inspired by the original Q-learning scheme for dis- +counted cost [154]. Consider DTCMC with finite state S and finite action set U. Assume that +the chain is irreducible and aperiodic under any stationary Markov policy. For notational ease, we +replace λ∗,m above by λ∗. Recall the risk-sensitive dynamic programming equation +V (i) = e−λ∗ min +u∈U + +ec(i,u) � +j∈S +P(j|i, u)V (j) + + , i ∈ S. +(6.18) +Setting Q(i, u) := the term in parenthesis on the right, we have a similar equation for Q(·, ·): +Q(i, u) = + +ec(i,u)−λ∗ � +j∈S +P(j|i, u) min +u′ Q(j, u′) + + , (i, u) ∈ S × U, +(6.19) +where we have used the fact +V (i) = e−λ∗ min +u∈U Q(i, u) ∀ i ∈ S. +(6.20) +Replacing (6.18) by (6.19) has increased the dimensionality from |S| to |S|×|U|, but the advantage +is that the nonlinearity, that is, the ‘min’ operator, is now inside the conditional expectation. Since +stochastic approximation at its core is an averaging technique, this make the problem amenable +to a model-agnostic, data-driven stochastic approximation algorithm. Another advantage is that +once you know Q(·, ·) exactly or approximately, dynamic programming tells us that the best control +choice in state i is Arg min(Q(i, ·)). This does not require any knowledge of the model. +Note that both V (·) and λ∗ are unknowns. Taking cue from the average cost Q-learning [1], we +have the algorithm inspired by relative value iteration for risk-sensitive control, based on a real or +simulated run of a controlled Markov chain (Xn, ζn). This is given by: +Qn+1(i, u) = Qn(i, u) + a(ν(i, u, n))1{Xn=i,ζn=u} × +�minu′ Q(Xn+1, u′) +Qn(i0, u0) +− Qn(i, u) +� +, n ≥ 0, +(6.21) +where +ν(i, u, n) := +n +� +m=0 +1{Xm=i,ζm=u} +is assumed to satisfy +lim inf +n→∞ +ν(i, u, n) +n +> 0 a.s. ∀ i, u. + +40 +ANUP BISWAS AND VIVEK S. BORKAR +That is, all state-control pairs are sampled ‘comparably often’, a.s. (This is a standard assumption +for reinforcement learning algorithms.) The step-size sequence {a(n)} satisfies +a(n) > 0, +� +n +a(n) = ∞, +� +n +a(n)2 < ∞, +(6.22) +plus some additional technical conditions in case of this specific (that is, fully asynchronous) variant. +Ignoring the technicalities due to asynchrony, the passage from what could have been a relative +value iteration for Q(·, ·) and the foregoing is that one first replaces in the right hand side of +(6.19) the conditional expectation by an actual evaluation at a sample generated according to the +conditional probability in question (Xn+1 in this case), replaces the unknown eλ∗ by Q(i0, u0) for +some fixed choice of (i0, u0)3, and then takes a convex combination of this with the previous iterate +with weights a(n), 1 − a(n), resp. Because we are considering a scheme based on a single run of +the chain, we observe only a single transition at each time and therefore can update at time n +only the (i, u)-th component for which Xn = i, ζn = u. Hence we multiply a(n) by the indicator +1{Xn=i,ζn=u} in the above, thus leaving the rest of the components unchanged. +Stochastic approximation theory then tells us that the iterates a.s. track the differential equation +˙qt(i, u) = +� +j∈S P(j|i, u) minu′ qt(j, u′) +qt(i0, u0) +− qt(i, u), +which can be shown to converge to the solution Q of (6.19) for which the (i0, u0)th component +is eλ∗. Then so does {Qn}, a.s. This can be viewed as a stochastic approximation version of the +well known ‘power iteration’ method of computational linear algebra, albeit for a nonlinear map. +The analysis involves mapping the trajectories of this differential equation to those of a related +differential equation given by +˙q′ +t(i, u) = e−λ∗ � +j∈S +P(j|i, u) min +u′ q′ +t(j, u′) − q′ +t(i, u), +which is easier to analyze. See [54] for details. +One problem with the above ‘exact’ Q-learning scheme, known as ‘tabular form’ in machine learn- +ing literature, is that its dimensionality can be prohibitive. This prompts the use of a parametrized +family of approximate Q(·, ·; θ), where θ is a parameter vector of moderate dimensions, and then +write a recursion for {θn} to learn the ‘best’ θ in a suitable sense. One popular choice has been linear +parametrization, i.e., a linear combination of a suitable choice of basis functions, both because of +its ease and analytic tractability. One such scheme was studied in [23], albeit for policy evaluation, +that is, for learning an approximate value function for a fixed randomized Markov policy, for which +a rigorous theory is possible. (That linear function approximation may not work with the nonlin- +earity - the ‘min’ operation - in place is a known fact even for simpler cost criteria.) Interesting +approximation error bounds for the eigenvalue eλ∗ have been derived in [133]. More recently, deep +neural networks have been the favoured approximation architecture in other contexts, but they do +not seem to have been explored in the risk-sensitive scenario. +Just as Q-learning is related to value iteration or relative value iteration as the case may be, +another leading algorithm called the Actor-Critic algorithm is related to the policy iteration. Here +we replace (6.21) by +Vn+1(i) = Vn(i) + a(ν(i, n))1{Xn=i} +�V (Xn+1) +Vn(i0) +− Vn(i, u) +� +, n ≥ 0, +(6.23) +with the randomized stationary Markov control policy πn(i, u) := P(Un = u|Xn = i) given recur- +sively by a stochastic gradient scheme for the risk-sensitive cost. This recursion is performed with a +different step-size sequence {b(n)} which, in addition to satisfying the usual conditions (6.22), also +3Other choices for this ‘normalization’ are possible. + +A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL +41 +satisfies b(n) = o(a(n)), so that this iteration moves on a slower time scale. The net effect is that +(6.23) sees the latter as quasi-static, hence it can be analyzed by treating πn ≈ constant, leading to +the conclusion that it is ‘essentially’ a policy evaluation scheme that tracks the value function for +the constant policy π ≈ πn. That is, denoting by Vπ the value function for a fixed stationary policy +π, we have Vn − Vπn → 0 a.s. This emulates the policy evaluation component of policy iteration. +The {πn}-iterate, performing gradient descent as though Vn is a legitimate surrogate for Vπn (which +it is, as argued above) emulates the optimization step of policy iteration. We omit the details of the +latter, suffice to say that it is based on a sensitivity formula for risk-sensitive cost with respect to +a parameter. See [53] for details. One limitation of this work is that the optimization component +works only if you update πn directly, not its parametrized approximation, because such variants +require model knowledge for their implementation. Recently a policy gradient scheme based on +updates only at successive visits to a privileged state has been proposed as a workaround [139]. For +(6.23), however, its ‘essentially linear’ nature allows for justifiable use of linear parametrization, +that is, as in [23]. It was recently observed in [39], albeit for a different cost, that interchange of the +fast and slow time scales in Actor-Critic algorithm leads to a new algorithm that emulates value +iteration, dubbed ‘Critic-Actor algorithm’ in ibid. This also applies to risk-sensitive problem. 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MR0336813 + diff --git a/qtAyT4oBgHgl3EQfZfdb/content/tmp_files/load_file.txt b/qtAyT4oBgHgl3EQfZfdb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b52a06c9064e69841aa1cf9243a0461bdca3237 --- /dev/null +++ b/qtAyT4oBgHgl3EQfZfdb/content/tmp_files/load_file.txt @@ -0,0 +1,2639 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf,len=2638 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='00224v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='OC] 31 Dec 2022 ERGODIC RISK-SENSITIVE CONTROL—A SURVEY ANUP BISWAS† AND VIVEK S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' BORKAR‡ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Risk-sensitive control has received considerable interest since the seminal work of Howard and Matheson [120] because of its ability to account for fluctuations about the mean, its connection with H∞ control, and its application to financial mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In this article we at- tempt to put together a comprehensive survey on the research done on ergodic risk-sensitive control over the last four decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Introduction 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Risk-sensitive control of Discrete time Markov chains 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Finite state space 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Countable state space 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' General state space 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Risk-sensitive control of diffusions 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Generalized principal eigenvalue 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' ERSC for controlled diffusions 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Generalized Collatz-Wielandt formula 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Risk-sensitive control of continuous time Markov chains 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Risk-sensitive maximization problems and beyond 28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Risk-reward problems 28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Risk-sensitive games 29 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Algorithms 29 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Policy iteration 29 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Relative value iteration 33 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Linear programming 35 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Reinforcement learning 39 Acknowledgments 41 References 41 † Department of Mathematics, Indian Institute of Science Education and Research Pune, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Homi Bhabha Road, Pune 411008, India Department of Electrical Engineering, Indian Institute of Technology, Powai, Mumbai 400076, India E-mail addresses: anup@iiserpune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='in, borkar@ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='iitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Primary: 90C40, 91B06, 93E03 Secondary: 93B36, 93B52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Risk-sensitive control, Bellman equation, generalized principal eigenvalue, multiplicative dynamic programming, verification theorem, Markov decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 1 2 ANUP BISWAS AND VIVEK S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' BORKAR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Introduction Given a controlled stochastic process X = {Xt} on a state space S, controlled by the process ζ, the ergodic risk-sensitive cost is defined as Ex(c, ζ) := lim sup T→∞ 1 γT log Ex � e � T 0 γc(Xt,ζt)dt� , x ∈ S, c being the running cost and γ ̸= 0 being the risk-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The ergodic risk-sensitive control (ERSC) problem is about studying the minimization problem λ∗ = inf x∈S inf ζ Ex(c, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Replacing ‘inf’ with ‘sup’ leads to the corresponding reward maximization problem which we discuss briefly later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Observe , however, that unlike the classical cost functionals, the reward maximization problem is not equivalent to the cost minimization problem obtained by flipping the sign of the instantaneous reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Suppose c is non-negative (more generally, bounded from below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The decision maker is supposed to be risk-averse or risk-sensitive for γ > 0, risk-neutral for γ = 0 and risk-seeking for γ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The risk-neutral case, in a suitable limiting sense, corresponds to the the classical ergodic control problem which has already been studied extensively (see [6, 17] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The goal of this article is to review the development of ERSC problems when γ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The study of ERSC can be traced back to the seminal work of Howard and Matheson [120] where the problem was studied for controlled Markov chains with finite state and action sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Since then this area has been developed intensively in the past forty years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' A major fillip came from a series of works by Peter Whittle in the eighties, culminating in [159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' One major motivation was the strongly felt need for criteria going beyond those based purely on mean rewards, that did not put any weight whatsoever on fluctuations around the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The obvious extensions such as considering a weighted sum of mean and variance in some form (the Markowitz model in finance being the prime example) faced problems such as non-availability of the ‘principle of time-consistency’ or the dynamic programming principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' As the exponential function can be viewed as the weighted sum of all powers, its expectation is a weighted sum of all moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Thus its expctation does account for higher moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In addition, by facilitating a multiplicative form of dynamic programming (as opposed to the additive form for classical criteria), it does obey the principle of time consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' This made risk-sensitive control an attractive proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Two classical applications of ergodic risk-sensitive control problem motivated by such consider- ations come from robust control theory and portfolio optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' – (Robust control theory) Since it is often almost impossible to find a true model of a system, robust control theory seeks criteria that could deal with model uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The connection between risk-sensitive control and robust control started with the work of Glover and Doyle [108] (see also Whittle [158,159]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Risk-sensitive minimization problems naturally give rise to two person zero-sum differential games (first found in the work of Jacobson [123]) which are of interest in the robust control theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In the differential game formulation there are two players, one representing the disturbance entering the system which will attempt to degrade system performance, and the other representing the actual control for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Readers may also consult [84] for more on power gain inequality and its connection to the ergodic risk-sensitive value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We shall briefly discuss this and its connection to H∞ control in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Another way to deal with the model uncertainty is to consider partially observed or Hid- den Markov chain models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' ERSC has been studied in these frameworks as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' It should be note that under fairly general assumptions and suitable change of measures, partially A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL 3 observed models can be changed into a fully observed control problems (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' [124]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Inter- ested readers may consult [22,94,117,124,136] for more details in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' – (Portfolio optimization) The risk-sensitive formulation of the portfolio optimization problem was introduced in the seminal works of Bielecki and Pliska [41], Fleming and Sheu [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Since then this area has grown substantially (see [43,91,93,141,143] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In addition to the considerations already discussed, multiplicative / exponential models arise naturally in finance due to ‘compounding’ effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Suppose that there are N risky assets and the investor allocates fraction ui t of its wealth to the i-th risky asset, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The total wealth Vt, at time t, of the investor is then given by dVt = Vt � rt � 1 − N � i=1 ui t � dt + N � i=1 ui t dSi t Si t � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1) where r denotes the risk-free interest rate, Si the share price of the i-th risky asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let U ⊂ RN be a constrain set and ζt = (u1 t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' , uN t ) ∈ U for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In portfolio optimization, one wishes to maximize the long term value of γ−1 E[V γ T ], for some γ ∈ (−∞, 1) \\ {0}, over all possible investment allocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Now suppose that there are d economic factors given by the vector ˜Xt ∈ Rd that governs the market performance and evolves according to the stochastic differential equation d ˜Xt = ˜b( ˜Xt)dt + dWt, where W is a d-dimensional standard Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The share price dynamics is given by dSi t Si t = µi( ˜Xt)dt + σi D · dWt + σi I · d ˜W t, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' , N, where ˜W is an N -dimensional standard Brownian motion independent of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Assume that σi D, σi I, r are constant vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Applying Itˆo’s formula one can easily find from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1) the differential equation satisfied by log Vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then defining b(x, u) := ˜b(x) + γ N � i=1 uiσi D, ¯µi(x) := µi(x) − r, σi = [σi D, σi I] ∈ Rd+N, ℓ(x, u) := −1 2(1 − γ)| N � i=1 uiσi|2 + N � i=1 ui¯µi(x) + r, one can check that above maximization problem is equivalent to maximizing (see [93]) log E � eγ � T 0 ℓ(Xt,ζt)dt� , where ζt ∈ U and dXt = b(Xt, ζt)dt + dWt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Thus the long-term asymptotics (that is, as T → ∞) corresponds to the ergodic risk- sensitive control problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' It is worth noting that some of the early work in this direction came from information theorists, notably Thomas Cover and his associates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' See [76], Chapter 16, and its biblio- graphical note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We mention in passing another application, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' to minimizing or maximizing the asymptotic rate of exit of a controlled Markov process from a prescribed subset of its state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' This can be reduced to a risk-sensitive control problem [49], [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 4 ANUP BISWAS AND VIVEK S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' BORKAR The main focus of this article is on ERSC problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' There is also an enormous amount of work done on the finite horizon version of risk-sensitive control problems which we do not discuss in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Interested readers may look at [2,26–28,31,35, 36,127, 158].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Broadly speaking, the ERSC problems are treated in three different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The first one corresponds to the variational repre- sentation of the moment generating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' This helps us to transform the above minimization problem to an ergodic zero-sum game problem (see [70,88,96,97]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The second approach for solving ERSC problem is an approximation method based on the discounted risk-sensitive problem (see [69, 80, 138]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Discounted risk-sensitive control is not amenable to dynamic programming, but by treating the risk-snsitivity parameter as a variable, one manages to make the problem analytically tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The dynamic programming equation of the risk-sensitive control problem is a nonlinear eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The third approach is more direct where the nonlinear eigenvalue problem is analyzed using Krein-Rutman theorem (see [5,14,44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We divide the review of ERSC problems in three major parts, namely, discrete time set up, controlled diffusions and continuous time Markov chains, wherein we touch upon all three approaches above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The following is a list of the abbreviations used in this paper DTCMC discrete time controlled Markov chain ERSC ergodic risk-sensitive control CTCMC continuous time controlled Markov chain PIA policy iteration algorithm RVI relative value iteration We also summarize key notations used in this article B(X) Borel σ algebra on the topological space X Cb(X) set of all real-valued bounded,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' continuous functions on X Ck(X) set of all k-times continuous differentiable functions on X ⊂ Rd Ck +(X) subset of functions of Ck(X) that are positive on X λ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='m optimal ergodic risk-sensitive value for DTCMC λ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='d optimal ergodic risk-sensitive value for controlled diffusion λ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='c optimal ergodic risk-sensitive value for CTCMC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Risk-sensitive control of Discrete time Markov chains We begin by introducing the general setting of a controlled discrete time Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Consider a controlled Markov process X := {X0, X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' } on a Borel space S controlled by a control process ζ := {ζ0, ζ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' } taking values in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Here U is a Borel space endowed with the Borel σ algebra B(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' For every x ∈ S, U(x) ∈ B(U) stands for the nonempty compact set of all admissible actions when the system is at the state x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The space of all admissible state action pairs is given by K := {(x, u) : x ∈ S, u ∈ U(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' For each A ∈ B(S) the controlled stochastic kernel P(A|·) : K → [0, 1] is Borel measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We denote by c : K → R+ the one-stage cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' For each t ∈ N, the space Ht denotes the admissible histories up to time t, where H0 := S, Ht = K × Ht−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' A generic element ht of Ht is a vector of the form ht = (x0, u0, x1, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' , xt−1, ut−1, xt), with (xs, us) ∈ K, 0 ≤ s ≤ t − 1, x0 ∈ S, denotes the observable history of the process up to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let us also denote by Fn = B(Hn) := the Borel σ-field of Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' An admissible control is a sequence ζ = {ζ0, ζ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' } where for each t ∈ N , ζt : Ht → U is a measurable map satisfying ζt(ht) ∈ U(xt), for all ht ∈ Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The set of all admissible policies is denoted by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' It is well known that for a given initial state x ∈ S and policy ζ ∈ U there exists a unique probability measure Pζ x on (Ω, B(Ω)), where Ω = (S × U)∞, (see [118, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='4], A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL 5 [6]) satisfying the following Pζ x(X0 = x) = 1, and Pζ x(Xt+1 ∈ A|Ht, ζt) = P(A|Xt, ζt) ∀ A ∈ B(S) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1) The corresponding expectation operator is denoted by Eζ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' A policy ζ ∈ U is said to be a Markov policy if ζt(ht) = vt(xt) for all ht ∈ Ht, for some measurable map vt : S → U such that vt(x) ∈ U(x) for all x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The set of all Markov policies is denoted by Um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' If the map vt does not have any explicit time dependence, that is, ζt(ht) = v(xt) for all ht ∈ Ht, then ζ is called a stationary Markov strategy and we denote the set of all stationary Markov strategies by Usm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (We use the words ‘strategy’ and ‘policy’ interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=') From [118, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='6] (also see [6]), it is easy to see that under any Markov policy ζ ∈ Um, the corresponding stochastic process X is strong Markov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' For each ζ ∈ U, the ergodic risk-sensitive cost is given by Ex(c, ζ) := lim sup T→∞ 1 γT log Eζ x � e �T −1 t=0 γc(Xt,ζt)� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2) where γ ̸= 0 and X is the discrete time controlled Markov chain (DTCMC) corresponding to the control ζ ∈ U, with initial state x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Our aim is to minimize (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2) over all admissible policies U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In other words, we are interested in the quantity λ∗,m = inf x∈S inf ζ∈U Ex(c, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3) We refer to this as an ergodic risk-sensitive control (ERSC) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' A policy ζ∗ ∈ U is said to be optimal if for all x ∈ S Ex(c, ζ∗) = inf x∈S inf ζ∈U Ex(c, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Note that in general, Ex(c, ζ) is not independent of x for ζ ∈ Usm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let us also mention the optimality equation which will be important for characterizing the optimal stationary Markov controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' A positive function ψ : S → (0, ∞) and a real number λ are said to form an eigen-pair (ψ, λ) if sign(γ)eγλψ(x) = min u∈U(x) � sign(γ)eγc(x,u) � S ψ(y)P(dy|x, u) � for x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='4) We call ψ an eigenfunction corresponding to the eigenvalue γλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We impose the following standard assumption on our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (i) The transition kernel P(·|x, u) is weakly continuous in (x, a), that is, for every f ∈ Cb(S) we have � S f(y)P(dy|x, u) continuous in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (ii) u �→ c(x, u) is continuous in U(x) for all x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Finite state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Suppose that S is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The very first ERSC control problem appeared in the work of Howard and Matheson [120] where the authors studied an ergodic risk- reward problem under the assumption that X is irreducible and aperiodic under every stationary Markov policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Since then the finite state situation has been studied in several works [4, 58, 60, 63, 66, 67, 67–69, 94, 117, 149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' For instance, for a (uncontrolled) Markov chain X with transition matrix P, it is well-known that x �→ lim sup T→∞ 1 γT log Ex � e �T −1 t=0 γc(Xt)� is constant on each communicating class (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' [58, Lemma 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Moreover, if X is irreducible, then lim sup T→∞ 1 γT log Ex � e �T −1 t=0 γc(Xt)� = 1 γ log ρ( ˜P) 6 ANUP BISWAS AND VIVEK S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' BORKAR where ˜Pij := Pijeγc(i) and ρ( ˜P) denotes the spectral radius of ˜P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Furthermore, the existence of such an eigen-pair can be characterized by the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1 ([68]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let Pxy = P(y|x) for x, y ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (i) For each c : S → R there exists an eigen-pair (ψ, λ) satisfying eγλψ(x) = eγc(x) Ex[ψ(X1)] = eγc(x) � y∈S ψ(y)Pxy, for all x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (ii) For every cost function c the mapping x �→ lim sup T→∞ 1 γT log Ex � e �T −1 t=0 γc(Xt)� is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (iii) The transition matrix P has a unique recurrent class C ⊂ S and there exists a constant m such that Px(˘τC ≤ m) = 1 for all x ∈ S where ˘τC denotes the return time to the set C, that is, ˘τC := inf{n ≥ 1 : Xn ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Moreover, if one of the above conditions holds, then the eigenfunction ψ can be represented as follows [66,68] ψ(x) = Ex � eγ �˘τz−1 i=0 (c(Xi)−λ)� ∀ x ∈ S \\ {z}, ψ(z) = 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='5) where ˘τz = ˘τ{z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' λ is the value of the average risk-sensitive cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' This representation of the eigenfunction will be crucial in our study and will appear in several places below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The following result on the ERSC problems can be found in [66, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1] (see also [64,89]) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Suppose that under every stationary policy ζ ∈ Usm, each pair of states in S communicates under X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then the following hold for every γ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (i) There exists an eigen-pair (Ψ, λ), Ψ > 0, satisfying sign(γ)eγλΨ(x) = min u∈U(x) \uf8ee \uf8f0sign(γ)eγc(x,u) � y∈S Ψ(x)P(y|x, u) \uf8f9 \uf8fb for x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='6) (ii) infζ∈U Ex(c, ζ) = λ∗,m = λ for each x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (iii) Every minimizing selector of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='6) is an optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (iv) (Ψ, λ) satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='6) is unique provided we set Ψ(z) = 1 for a prescribed state z ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Note that the above result requires the DTCMC to be communicating under every stationary policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2 also appears in [42] where it is proved under an additional assumption that P(x|x, u) > 0 for all (x, u) ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In [59] the author shows that given any two states x, y ∈ S, if we can find a stationary policy under which y is accessible from x, then there exists Λ0 > 0 such that an eigen-pair satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='6) exists for γ satisfying γ∥c∥sp < Λ0 (∥·∥sp denotes the span semi-norm defined as ∥c∥sp = supx,u c − infx,u c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Also, note that the hypothesis of a single communicating class for every stationary control is important to ensure that infζ∈U Ex(c, ζ) is independent of x (a specific example can be found in [64, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Consider the assumption: Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2 (Simultaneous Doeblin Condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' There exists a state z ∈ S and a positive integer K such that Eζ x[˘τz] ≤ K for all x ∈ S, and ζ ∈ Usm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' A general characterization of the optimal value is then obtained in [67, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL 7 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Suppose that U is a finite set and Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then for every x ∈ S we have inf ζ∈Usm Ex(c, ζ) = inf g∈G g(x), where G denotes the collection all functions g : S → R satisfying (i) For each x ∈ S g(x) = min u∈U(x) max{g(y) : P(y|x, u) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (ii) There exists a positive function h such that eγg(x)h(x) ≥ min Bg(x) \uf8ee \uf8f0eγc(x,u) � y∈S h(y)P(y|x, u) \uf8f9 \uf8fb x ∈ S, where Bg(x) := � u ∈ U(x) : g(x) = max{g(y) : P(y|x, u) > 0} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' A generalization of the above result for DTCMC with a general state space can be found in [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Countable state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Now suppose that S is countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Without any loss of generality, assume that S = {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The analysis of ERSC problem becomes more involved due to non- compactness of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' If the running cost c is bounded, then a result analogous to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2 is possible, provided γ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='4 ([64]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Define µ = log(K + 1) − log K K + 1 , ∥c∥ = sup K |c(x, u)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then for each 0 ̸= γ ∈ (− µ 2∥c∥, µ 2∥c∥) there exists an eigen-pair (Ψ, λ) with bounded Ψ that satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Furthermore, the conclusions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2 (ii)-(iv) hold in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2 in the above theorem can be relaxed provided the state space S is communicating under every stationary policy and the cost function c is supported on a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' For more details, see [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Some other works that also study ERSC problem with bounded cost functions are [65,116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Since the simultaneous Doeblin condition in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2 is quite restrictive, we are going to impose some structural condition on the cost function, known as near-monotonicity, which also allows unbounded cost functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We say that the one-step cost function c is near-monotone with respect to ρ if lim inf x→∞ min u∈U(x) c(x, u) > ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Suppose that for some stationary Markov control ˜ζ, we have Ex(c, ˜ζ) independent of x ∈ S and c is near-monotone with respect to Ex(c, ˜ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' It is then shown in [115] that, for γ > 0, there exists a positive ψ : S → (0, ∞] satisfying eγλ∗,m m Ψ(x) ≥ inf u∈U(x) \uf8ee \uf8f0ec(x,u) � y∈S Ψ(y)P(y|x, u) \uf8f9 \uf8fb for all x ∈ S, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='7) where λ∗,m m is given by λ∗,m m = inf x∈S inf ζ∈Um Ex(c, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='8) Furthermore, if ζ∗ is a minimizing selector of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='7), then λ∗,m m = Ex(c, ζ∗) for all x ∈ {Ψ < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The main idea in [115] (motivated from [89]) is to transform the risk-sensitive minimization problem to a risk-neutral game problem using a change of variables (a ‘logarithmic transformation’ that we see 8 ANUP BISWAS AND VIVEK S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' BORKAR later) and then use the approach of discounted-control problems for the ergodic risk-neutral game to construct a solution for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We say a function F : S → R is norm-like if for each integer n the set {F ≤ n} is either empty or finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Around the same time multiplicative ergodic theorems with norm-like potential functions F are studied in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The ideas of [21] are extended to study ERSC problems for norm-like cost function c in [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' To explain the result of [57] we introduce some additional notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Fix a state z ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' For a Markov policy ζ ∈ Um, define Λ(ζ) = inf � Λ : Eζ z � e �˘τz−1 t=0 γ(c(Xt,ζt)−Λ) ≤ 1 �� , and Λ∗ = inf ζ∈Um Λ(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='9) The first entrance time to the state z is defined as σz = inf{n ≥ 0 : Xn = z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let us also define, for x ∈ S, Ψ∗(x) := inf ζ∈Um Eζ x � e �σz t=0 γ(c(Xt,ζt)−Λ∗)� , w∗(x) := Arg min u∈U(x) \uf8eb \uf8edec(x,u) � y∈S Ψ∗(y)P(y|x, u) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The following result is proved in [57, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='6] Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Suppose that U(x) is finite for all x and c(·, u) is norm-like for all u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Also assume that the chain X is communicating under every Markov policy and aperiodic under any stationary Markov policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then, provided Λ∗ is finite, the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (i) Λ∗ = λ∗,m m = Ex(c, w∗) for all x where λ∗,m m is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (i) Ψ∗ is finite on S and eγλ∗,m m Ψ∗(x) ≥ inf u∈U(x) \uf8ee \uf8f0eγc(x,u) � y∈S Ψ∗(y)P(y|x, u) \uf8f9 \uf8fb for all x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The above result requires Λ∗ to be finite and the chain to be aperiodic under each stationary Markov control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Another result of similar flavor is recently obtained in [50], which we state below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In addition to Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1 let us also assume the following to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (i) There exists a state i0 ∈ S such that min u∈U(i0) P(j|i0, u) > 0 for all j ̸= i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (ii) X is recurrent under each stationary Markov control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (iii) infζ∈Usm Ex(c, ζ) < ∞ for all x ∈ S and c is near-monotone with respect to λ∗,m m in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then there exists a positive Ψ satisfying eγλ∗,m Ψ(x) ≥ inf u∈U(x) \uf8ee \uf8f0eγc(x,u) � y∈S Ψ(y)P(y|x, u) \uf8f9 \uf8fb for all x ∈ S, and every minimizing selector is an optimal stationary Markov control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Moreover, λ∗,m m = λ∗,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' [50] also considers the ERSC problem under a blanket stability hypothesis but without the near-monotone condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL 9 Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let X be irreducible under any stationary Markov control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In (i) and (ii) below the function V on S takes values in [1, ∞) and �C is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Assume that one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (i) For some positive constant β ∈ (0, 1) and a finite set C it holds that sup u∈U(x) � y∈S V(y)P(y|x, u) ≤ (1 − β)V(x) + �C1C(x) x ∈ S, and γ supK c < θ where θ = log( 1 1−β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (ii) For a finite set C and a norm-like function ℓ : S → R+ it holds that sup u∈U(x) � y∈S V(y)P(y|x, u) ≤ (1 − β(x))V(x) + �C1C(x) x ∈ S, where 1 − e−ℓ(x) = β(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Moreover, the function ℓ − γ maxu∈U(·) c(·, u) is norm-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Condition (ii) above is useful for treating ERSC problems with an unbounded cost function c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The following result is obtained in [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Suppose that Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Also assume the condition (i) of The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then we have the following (i) There exists a unique, positive Ψ, with Ψ(i0) = 1, satisfying eγλ∗,m Ψ(x) = inf u∈U(x) \uf8ee \uf8f0eγc(x,u) � y∈S Ψ(y)P(y|x, u) \uf8f9 \uf8fb for all x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='10) (ii) A stationary Markov control is optimal if and only if it is a minimizing selector of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='7 are proved using a different approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The authors first solve a nonlinear eigenvalue problem on finite sets containing i0 and then increase the sets to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The condition (i) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='6 ensures that the limiting eigenfunction Ψ is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' This condition is recently removed in [74] where the authors used the approach of [57] (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='5) to define the eigenfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' General state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Next we describe the results known for the general state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Some of the important works in this direction are [79–81, 125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' It is natural that one needs to impose additional conditions to ensure existence of an eigen-pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We begin by recalling the following result from [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let S be a complete separable metric space, U(x) = U for all x and γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We assume the following to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (A1) There exists δ < 1 such that for all x, x′ ∈ S, B ∈ B(S) and u, u′ ∈ U we have P(B|x, u) − P(B|x′, u′) ≤ δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (A2) δeγ∥c∥sp < 1 where ∥c∥sp denotes the span semi-norm of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then there exists a bounded, positive continuous function Ψ satisfying eγλ∗,m Ψ(x) = min u∈U � eγc(x,u) � S Ψ(y)P(dy|x, u) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='11) Furthermore, any minimizing selector of the above equation is an optimal stationary Markov control for the ERSC problem, and Ψ is unique, up to a positive multiplicative constant, in the class Cb(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Note that condition (A2) above requires γ to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Writing ϕ = log Ψ we see from above that γλ∗,m + ϕ(x) = min u∈U � γc(x, u) + log � S eϕ(y)P(dy|x, u) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 10 ANUP BISWAS AND VIVEK S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' BORKAR Letting Tg(x) = min u∈U � γc(x, u) + log � S eg(y)P(dy|x, u) � , it is shown in [79] that {Tn0, n ≥ 0}, where 0 := the function identically equal to zero, converges in the space Cb(S) with respect to the span semi-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The limit of this sequence gives a fixed point (up to a positive scalar multiplier) which solves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' It can be easily checked that by (A1) , T is a local contraction [80, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2] and therefore, uniqueness is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Condition (A2) above was replaced by a more technical condition in [80] to obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Conditions (A1)- (A2) were replaced by a minorization condition and certain exponential moment bounds on the hitting time to a certain compact set in [81] in order to study the optimality equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' These results are further extended in [73,125,126] to Borel state spaces and for unbounded cost functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' These works study the ERSC control problem using discounted approximation approach which was initiated in [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' For β ∈ (0, 1), let Vβ be a positive solution to the dynamic programming equation Vβ(x) = min u∈U(x) � eγc(x,u) � S (Vβ(y))βP(dy|x, u) � x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='12) Vβ is basically the discounted value function associated with a certain dynamic game [125, Lemma 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1, there exists a unique, bounded solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='12) whenever c is bounded (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' [80, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Under some additional assumptions on the transition kernels (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' [80, The- orem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2]), it can be shown that Vβ Vβ(z), z ∈ S is a fixed point, that converges as β ↑ 1 to some Ψ satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='11) and γ−1 lim β→1(1 − β) log Vβ(z) = λ∗,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The above analysis served as the starting point for [73,125] where the authors allow the cost to be unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Suppose that x �→ U(x) is upper-semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='13) Consider c ≥ 0 and possibly unbounded and γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then one can solve (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='12) for cN = min{N, c} to obtain a sequence of Vβ,N for each β ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Letting N → ∞, it is then shown that limN→∞ Vβ,N = Vβ, and (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' [73, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1], [125, Lemma 2]) Vβ(x) = min u∈U(x) � eγc(x,u) � S (Vβ(y))βP(dy|x, u) � x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='14) Define mβ := infS Vβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Letting ˜Vβ := 1 mβ Vβ, provided mβ > 0, in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='14) gives e(1−β) log mβ ˜Vβ(x) = min u∈U(x) � eγc(x,u) � S ( ˜Vβ(y))βP(dy|x, u) � x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In [125], under the assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='13) and supβ∈(0,1) ˜Vβ < ∞, it is shown that, for any sequence βn → 1, λ∗,m = 1 γ lim n→∞(1 − βn) log mβn, and Ψ := lim inf n→∞ ˜Vβ, satisfy eγλ∗,mΨ(x) ≥ inf u∈U(x) � ec(x,u) � S Ψ(y)P(dy|x, u) � for all x ∈ S, A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL 11 and every minimizing selector is an optimal stationary Markov control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' A similar result is also obtained in [73, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2] under a milder hypothesis that requires mβ to be finite for all β ∈ (0, 1), but it also assumes (compare it with Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2) that {x ∈ S : min u∈U(x) c(x, u) ≤ λ∗,m + δ} to be compact for some δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Risk-sensitive control of diffusions In this section we review some recent progress on the ERSC problem for controlled diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' To begin with, consider the problem for uncontrolled diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Generalized principal eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let X = {Xt} be a diffusion process in Rd given by dXt = b(Xt)dt + σ(Xt)dWt, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1) where b : Rd → Rd is the drift vector, σ : Rd → Rd×d is the diffusion matrix and W is a d- dimensional standard Wiener process on a complete filtered probability space (Ω, F, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' There exists a unique strong solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1) (see [114,152,161]) for every initial data X0 = x ∈ Rd, b is Borel measurable and σ is locally Lipschitz and locally non-degenerate , provided b, σ have at-most linear growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let a(x) = 1 2σσT(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Given a continuous function c : Rd → R let us define Ex(c) = lim sup T→∞ 1 T Ex � e � T 0 c(Xs)ds� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' As in Section 2, the above quantity is related to an eigen-equation which we describe below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We define the extended generator of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1) as Lf(x) = trace(a(x)∇2f) + b(x) · ∇f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2) Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We say a pair (ψ, λ) ∈ C2(Rd) × R is an eigen-pair of L + c if ψ > 0 in Rd and Lψ(x) + c(x)ψ(x) = λψ(x) in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' To understand the relation between Ex and an eigen-pair, let us consider the problem in a smooth bounded domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' More precisely, let τ(D) be the first exit time from D, that is, τ(D) = inf{t > 0 : Xt /∈ D}, and we define Ex(c, D) := lim sup T→∞ 1 T Ex � e � T 0 c(Xs)ds1{T<τ(D)} � x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' It is then well-known that there exists a λD ∈ R such that Ex(c, D) = λD for all x ∈ D [82] and for some ψD ∈ C2(D) ∩ C( ¯D) we have LψD + c(x)ψD(x) = λDψD(x) in D ψ > 0 in D, ψ = 0 on ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3) Thus (ψD, λD) forms a Dirichlet eigen-pair for L + c in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Furthermore, λD is the generalized principal eigenvalue in the sense of [37,144,147], that is, λD = inf{λ : ∃ ψ ∈ C2 +(D) ∩ C( ¯D) satisfying Lψ + c(x)ψ ≤ λψ in D}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='4) where C2 +(D) denotes the subset of C2(D) containing functions that are positive in the interior of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (We define λD likewise for unbounded D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=') In addition, it can be easily shown that the principal eigenfunction ψD in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3) is unique up to a multiplicative constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' So we may want to ask whether Ex(c) = λRd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The answer to this question is negative in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In fact, [14, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1] shows that λRd < infx Ex(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Thus the risk-sensitive problem in the whole space becomes quite delicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 12 ANUP BISWAS AND VIVEK S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' BORKAR Let us now recall the definition of principal eigenvalue in Rd from [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The generalized principal eigenvalue of L + c in Rd is defined as follows λRd = inf{λ : ∃ positive ψ ∈ C2(Rd) satisfying Lψ + c(x)ψ ≤ λψ in Rd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='5) To illustrate explicit dependence on the potential c we would also use the notation λRd(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let us also recall the following result from [38, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' For every λ ∈ [λRd, ∞) there exists a positive ψ ∈ C2(Rd) satisfying Lψ(x) + c(x)ψ(x) = λψ(x) in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In particular, there are infinitely many eigen-pairs for L + c in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' To equate Ex(c) with the generalized principal eigenvalue λRd, we must impose additional conditions on the diffusion coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' More discussion in this direction can be found in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Another important concern is the simplicity of the principal eigenvalue λRd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We need following definition for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2 (Minimal growth at infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' An eigen-pair (ψ, λ) is said to have a minimal growth at infinity if for any compact set K and any positive v ∈ C2(Kc) ∩ C(Rd) satisfying Lv + (c − λ)v ≤ 0 in Kc, we have v ≥ κψ for some κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The above criterion was introduced by Agmon in [3] and is very useful to establish simplicity of eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let C+ (Rd) denote the collection of all non-trivial, non-negative, continuous functions which vanish at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The following notions of monotonicity are introduced in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We say the generalized principal eigenvalue λRd is strictly monotone at c if for some h ∈ C+ (Rd) we have λRd(c − h) < λRd(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We say λRd is monotone on the right at c if for all h ∈ C+ (Rd) we have λRd(c) < λRd(c + h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' It is shown in [14] that strict monotonicity at c implies λRd(c − h) < λRd(c) for all h ∈ C+ (Rd) and therefore, by convexity, it also implies monotonicity on the right at c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The following equivalence criterion is proved in [11,14] (see also [13] for its generalization to weakly-coupled systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Suppose that λRd(c) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (i) Eigen-pair (ψ, λRd(c)) has a minimal growth at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (ii) λRd is monotone on the right at c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (iii) For some compact ball B, we have ψ(x) = Ex � e � ˘τ 0 (c(Xt)−λRd)dtψ(X˘τ) � x ∈ Bc, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='6) where ˘τ = τ(Bc), the first hitting time to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Furthermore, if one of the above holds, then λRd(c) is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The analogy between (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='6) should be noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' To characterize the notion of strict monotonicity we need to introduce the twisted diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Given an eigen-pair (ψ, λ) of L + c, the twisted diffusion is given by dYt = b(Yt)dt + 2a(Yt)∇ log ψ(Yt)dt + σ(Yt)dWt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The twisted process corresponding to a principal eigen-pair is said to be a ground state process due to its interpretation in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The following result can be found in [14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1] (see also [121,122,130]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Suppose that λRd(c) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then (i) For every λ > λRd(c), the twisted process corresponding to any eigen-pair (ψ, λ) is transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (ii) The following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL 13 (a) λRd is strictly monotone at c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (b) The ground state process is exponentially ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let us remark that [14] requires c to non-negative for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3 to hold, but this restriction on c is removed in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' ERSC for controlled diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In this section we review ERSC problem for controlled diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We begin with the exponential linear-quadratic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Exponential Linear-quadratic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Exponential linear quadratic model is a risk-sensitive generalization of the classical linear-quadratic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Such problems are quite central to the optimal investment models, appearing in mathematical finance (see [92, 141–143] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' More precisely, the controlled diffusion is given by (we consider a slightly more general form) dXt = b(Xt)dt + g(Xt, ζt)dt + σ(Xt)dWt (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='7) where ζt is a progressively measurable process that is non-anticipative in the sense that for s < t, Wt −Ws is independent of the completion of the sigma-field generated by {X0, ζr, Wr : r ≤ s} with respect to (F, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The control process ζ is generally assumed take values in some Euclidean space Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' As before, we denote the set of all admissible controls by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Implicitly, we assume that under every admissible control there exists a unique strong solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='7) in the sense that, given ζ, W as above on a probability space, there exists an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' unique X satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let V : Rd → [0, ∞) and φ : Rd × Rm → [0, ∞) be two given functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We define λ∗,d = inf x∈Rd lim sup T→∞ 1 γT log J(x, T) where J(x, T) = inf ζ∈U Ex � eγ � T 0 (V (Xt)+φ(Xt,ζt))dt� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='8) It should be noted that for a given γ > 0, J(x, T) might not be finite for all T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' This is known as the breakdown phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In fact, [34, Example 1] shows that breakdown can actually happen for some large values of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Thus we need to impose conditions on the coefficients to ensure no breakdown [32,140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' As mentioned before, the above ERSC problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='8) is related to the nonlinear eigenvalue problem given by γλ∗,dΨ(x) = trace(a(x)∇2Ψ(x)) + b(x) · ∇Ψ(x) + min u∈Rm{g(x, u) · ∇Ψ + γφ(x, z)Ψ(x)} + γV (x)Ψ(x) for x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Assume γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Letting w(x) = 1 γ log Ψ(x) in the above, we obtain λ∗,d = trace(a(x)∇2w(x)) + b(x) · ∇Ψ(x) + Q0(x, ∇w) + V (x), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='9) where Q0(x, ξ) = γξa(x) · ξ + min u∈Rm{g(x, u) · ξ + φ(x, z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' If we choose a, g, φ in such a way that − κ1|ξ|2 ≤ Q0(x, ξ) ≤ −κ2|ξ|2 x, ξ ∈ Rd, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='10) for some positive constants κ1, κ2, and ���� ∂Q0(x, ξ) ∂ξ ���� ≤ κ3|ξ| + κ4, ���� ∂Q0(x, ξ) ∂x ���� ≤ κ3|x|2 + κ4, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='11) for some κ3, κ4 > 0, we are in the framework of the Hamilton-Jacobi-Isaacs equation of the ergodic type [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' More precisely, if V is coercive, the existence and uniqueness of solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='9) can be obtained from [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The following result is proved in [140, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We impose the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (i) σ, b, g, V, φ are smooth and σ, b are Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Also, all the derivatives of σ, b, V are bounded by M(1 + |x|k) for some M, k > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (ii) |g(x, z)| ≤ κ˜g(z) for some locally bounded ˜g and a constant κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 14 ANUP BISWAS AND VIVEK S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' BORKAR (iii) For some constant κ◦ > 0 we have ξa(x) · ξ ≥ κ◦|ξ|2 ξ, x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (iii) V is coercive and lim |z|→∞ φ(x, z) = ∞, lim |z|→∞ |g(x, z)| φ(x, z) = 0 uniformly in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Also, assume that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='10)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='11) hold and Q0(x, βξ) ≥ β2Q0(x, ξ) − κβ(1 − β)ξa(x) · ξ − β(1 − β)L(x), β ∈ (0, 1), for κ < κ2 and some locally bounded function L satisfying V (x) − L(x) → ∞ as |x| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then there exists a unique eigen-pair (w, λ) ∈ C2(Rd) × R, w coercive in nature, satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Furthermore, when Q0(x, ξ) = −κ ξa(x)ξT for some κ > 0, we have λ = λ∗,d, given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Similar result can also be found in [129, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3] where the authors imposed some structural assumptions on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' ERSC with a compact action set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In this section we review the result on ERSC problem when the action set U is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let X = {Xt} be a controlled diffusion in Rd governed by the Itˆo equation dXt = b(Xt, ζt)dt + σ(Xt)dWt (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='12) where ζt is an admissible control in the sense of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1, taking values in a compact metric space U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We impose the following conditions on the coefficients to guarantee the existence and uniqueness of solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (B1) Local Lipschitz continuity: The functions b : Rd × U → Rd and σ : Rd → Rd×d are continuous and satisfy |b(x, u) − b(y, u)| + ∥σ(x) − σ(y)∥ ≤ CR|x − y| ∀ x, y ∈ BR, ∀ u ∈ U, for some constant CR, depending on R > 0, where BR denotes the ball of radius R centered at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (B2) Affine growth condition: There exists a constant C0 such that max u∈U [b(x, u) · x]+ + ∥σ(x)∥2 ≤ C0(1 + |x|2) x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (B3) Local non-degeneracy: For each R > 0, there exists CR satisfying ξa(x) · ξ ≥ C−1 R |ξ|2 ∀ ξ ∈ Rd, x ∈ BR, where a(x) = 1 2σσT(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' It is well known that under (B1)-(B3), for any admissible control ζ ∈ U there exists a unique solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='12) [17, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' As before, a stationary Markov control would correspond to a Borel measurable map from Rd to U and the class of all stationary Markov controls is denoted by Usm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' It is also well known that for every stationary Markov control in Usm there exists a unique strong solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='12) which is also a strong Markov process [114, 152, 161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Now consider a continuous function c : Rd × U → [0, ∞) which is locally Lipschitz in x uniformly with respect to u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' As before, we define the ERSC problem as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' λ∗,d = inf x∈Rd inf ζ∈U Ex(c, ζ) where Ex(c, ζ) := lim sup T→∞ 1 γT log Eζ x � e � T 0 γc(Xt,ζt)dt� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='13) For the remaining part of this section, we discuss the risk-averse problem and therefore, we shall consider γ to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' As discussed before, the above ERSC problem corresponds to a nonlinear A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL 15 eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' For this purpose we introduce a family of operators Lu, parametrized by u ∈ U, defined as follows Luf(x) = trace(a(x)∇2f(x)) + b(x, u) · ∇f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We shall be interested in an eigenfunction Ψ ∈ C2(Rd), Ψ > 0, satisfying min u∈U{LuΨ(x) + γc(x, u)Ψ(x)} = γλ∗,d Ψ(x) in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='14) The first major contribution for the ERSC of diffusion came from Fleming and McEneaney [97] (see also [96]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' They prove the following in [97, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Suppose that b(·, u), c(·, u) are C1 for each u ∈ U, σ is constant, and the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (i) c, ∇xc are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (ii) ∇xb is bounded in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (iii) For some κ > 0 we have (x − y) · (b(x, u) − b(y, u)) ≤ −κ|x − y|2 ∀ x, y ∈ Rd, u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' There there exists a Ψ ∈ C2(Rd), Ψ > 0, satisfying min u∈U{LuΨ(x) + γc(x, u)Ψ(x)} = γλ∗,d Ψ(x) in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Furthermore, any measurable selector of the above equation is an optimal stationary Markov control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Apart from the condition (iii) above, the constant diffusion matrix σ also plays a key role in the above result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' These two conditions together render Lipschitz regularity to log Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' More precisely, the authors use a logarithmic transformation to change the risk-sensitive minimization problem to an ergodic game problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then using the standard method of vanishing discount, they establish the existence of solution to the Hamilton-Jacobi-Isaacs equation for the ergodic game problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In order to extend the result to a more general class of b and σ, [138] considers ERSC problem under a periodic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' This is the content of our next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Suppose that σ = √ 2I and b, c are periodic in x variable with period 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Also, assume that b, c are Lipschitz in the x variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then there exists a unique, periodic Ψ ∈ C2(Rd), Ψ > 0, satisfying ∆Ψ(x) + min u∈U{b(x, u) · ∇Ψ(x) + γc(x, u)Ψ(x)} = γλ∗,d Ψ(x) in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='15) The main idea of the proof goes as follows: one starts with an exponential of the discounted cost defined as (this is actually the continuous version of the approach that appeared in [65,80]) wα(γ, x) = inf ζ∈U Ex � exp � γ � ∞ 0 e−αtc(Xt, ζt)dt �� , α ∈ (0, 1), and shows that − αγ ∂wα ∂γ + ∆wα + min u∈U{b(x, u) · ∇wα + γc(x, u)wα} = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='16) and wα(0, x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Note that this is a parabolic equation when we treat γ as a variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Defining uα = γ−1 log wα, it is then shown that αuα, ∇xuα are globally bounded, uniformly in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' This helps us to pass the limit in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='16) to obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' This idea of [138] was then pushed in [46–48] to solve (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='15) beyond the periodic setting and under near-monotone hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We say that c is near-monotone with respect to ρ ∈ R if it satisfies lim inf |x|→∞ min u∈U c(x, u) > ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 16 ANUP BISWAS AND VIVEK S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' BORKAR In particular, it was proved in [48] that if c is near-monotone with respect to λ∗,d and the diffusion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='12) is recurrent under each stationary Markov control, then there exists a positive Ψ satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='15) and every measurable selector is an optimal stationary Markov control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' But the uniqueness of Ψ remains an issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The approach of [48, 138] establishes γλ∗,d as an eigenvalue of nonlinear operator in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='15), but in view of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1 (for nonlinear operators, see [51, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1]), it is hard to identify γλ∗,d as the principal eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Thus it is important to establish uniqueness of Ψ (up to a positive multiplicative constant) for the verification result of optimal stationary Markov controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Define the nonlinear operator G as Gf(x) = min u∈U(Luf(x) + γc(x, u)f(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='17) The generalized principal eigenvalue λ1(G) of G is defined as before (along the lines on [38,51]) : λ1(G) = inf{λ : ∃ positive ψ ∈ C2(Rd) satisfying Gψ ≤ λψ in Rd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' A natural question is: under what condition can we show that λ1(G) = γλ∗,d?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' If we start with the Dirichlet generalized eigenvalue problem for G on a sequence of increasing, smooth bounded domains and let the domains increase to Rd, then applying Harnack’s inequality and monotonicity of generalized principal eigenvalues, it can be shown that the Dirichlet principal eigenvalues con- verges to λ1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In [44], the author applies this idea to show that λ1(G) ≤ γλ∗,d, in general, and furthermore, if c is near-monotone with respect to λ∗,d and the diffusion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='12) is recurrent under each stationary Markov control, then there exists a positive Ψ satisfying GΨ = γλ∗,d Ψ in Rd, and every minimizing selector is an optimal stationary Markov control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Note that the near- monotone criterion penalizes instability of the process X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Thus it is expected that an optimal stationary Markov control would stabilize the process, that is, keep it positive recurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Using this fact, the blanket stability hypothesis was removed in [7], proving the the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Assume (B1)-(B3) and also suppose that c is bounded and is near-monotone with respect to λ∗,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In addition, suppose that b, σ are bounded, σ is Lipschitz, a is uniformly elliptic and max u∈U [b(x, u) · x]+ |x| → 0 as |x| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='18) Then there exists a Ψ ∈ C2(Rd) satisfying infRd Ψ > 0 and min u∈U{LuΨ(x) + γc(x, u)Ψ(x)} = γλ∗,d Ψ(x) in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='19) Moreover, the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (i) λ1(G) = λ∗,d = infζ∈U Ex(c, ζ) for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (ii) If v ∈ Usm is a minimizing selector of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='19), then v is stable and is an optimal stationary Markov control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (iii) In addition, if we have λ∗,d m (c) < λ∗,d m (c + h) for all h ∈ C+ (Rd) where λ∗,d m (c) := inf x∈Rd inf ζ∈Usm Ex(c, ζ), then Ψ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='19) is unique up to a positive scalar multiple and any optimal stationary Markov control is given by a measurable selector of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Note that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='7 does not impose any stability assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='18) is used to show that any minimizing selector is in fact stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The boundedness assumption on b and c was relaxed in [10, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2] where the authors allowed polynomial growth of b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The condition of monotonicity on the right in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='7(iii) is not easy to verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' To tackle this difficulty, an alternative set of conditions has also been used for the ERSC problems as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL 17 Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' There exists a positive V ∈ C2(Rd) with infRd V > 0 such that one of the following holds: (i) There exists an inf-compact, positive ℓ ∈ C(Rd) and a compact set K satisfying sup u∈U LuV ≤ ¯κ1K − ℓV in Rd, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='20) for some constant ¯κ, and ℓ − maxu∈U γc(·, u) is inf-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (ii) For some positive constants ¯κ, θ and a compact set K we have sup u∈U LuV ≤ ¯κ1K − θV in Rd, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='21) and lim sup |x|→∞ max u∈U γc(x, u) < θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We remark here that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='20) is not possible when a, b are bounded [38, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' This is the reason for introducing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Also, note that Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1 does not require c to be near-monotone, but imposes a blanket stability hypothesis on the stationary Markov controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1(ii) was also used in [45] to prove the existence of a solution Ψ of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='19) and the existence of an optimal stationary Markov control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Uniqueness and verification results are settled in [14] where the authors prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Assume that (B1)-(B3) and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then there exists a Ψ ∈ C2(Rd) satisfying Ψ > 0 and min u∈U{LuΨ(x) + γc(x, u)Ψ(x)} = γλ∗,d Ψ(x) in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='22) Moreover, the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (i) λ1(G) = λ∗,d = infζ∈U Ex(c, ζ) for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (ii) If v ∈ Usm is a minimizing selector of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='22), then it is an optimal stationary Markov control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (iii) Ψ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='19) is unique up to a positive scalar multiple and any optimal stationary Markov control is given by a measurable selector of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Incidentally, the approach of [14] does not extend to jump diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The eigenvalue approach in [7,14] crucially uses the Harnack inequality to establish the existence of principal eigenfunction of G and the Harnack inequality does not hold for the nonlocal equation with rough kernels (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' [19, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' To tackle this problem, [9] used the Lyapunov function in Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1 as a barrier function to bound the Dirichlet principal eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' More precisely, under a stability assumption analogous to Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1, [9] studies the ERSC problem for a class for jump diffusions with jump-kernels having finite measure and establishes a result analogous to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Connection to H∞ control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In this section, we briefly touch upon the connection between H∞ control and the small noise asymptotics of ERSC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Readers are encouraged to consult the book [20] to find out more on H∞ control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let us start with a (deterministic) nonlinear, controlled dynamical system dyt = g(yt, ζt, ξt)dt where ζt and ξt are two control process, taking values in some subsets U ⊂ Rm and V ⊂ Rn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Define A := L2 loc(R+, U), B := L2 loc(R+, V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We choose ζ as a causal feedback to ξ, that is, ζ = α(ξ) for some α : B → A satisfying if for some t > 0 we have ξ = ¯ξ in [0, t], then α(ξ) = α(¯ξ) on [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 18 ANUP BISWAS AND VIVEK S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' BORKAR The class of such causal feedback controls is denoted by Ucausal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The H∞ control problem can be described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Assume that the dynamical system is stable under the control ζ ≡ 0 and given a response function h : Rd × U × V → R+, we have a γ > 0 and a strategy α ∈ Ucausal satisfying, for some starting point y0 ∈ Rd, � T 0 h(yt, α(ξ)(t), ξt)dt ≤ γ2 � T 0 |ξt|2dt for all T > 0, ξ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='23) The least γ satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='23) is called the H∞ norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' When existence of α is possible, we say that the H∞ suboptimal control problem is solvable with disturbance attenuation level γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Note that the above problem can also be studied by considering the value function Vγ(x) = inf α∈Ucausal sup ξ∈B sup T≥0 � T 0 [h(yt, α(ξ)(t), ξt) − γ2|ξt|2]dt, where y0 = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Note that Vγ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The points where Vγ vanishes correspond to the points from where the H∞ problem is solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' As shown in [150], the value function Vγ is a viscosity solution to sup v∈V inf u∈U{g(x, u, v) · ∇Vγ + h(x, u, v) − γ2|v|2} = 0 in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='24) Thus the H∞ control problem is related to the study of non-negative viscosity solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In order to understand the connection of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='24) with the ERSC problem, consider the controlled diffusion dXt = b(Xt, ζt)dt + � ε 2γ2 �1/2 dWt, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='25) where ε > 0, and ζ is an admissible control process taking values in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Also, letting γ = ε−1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='13), we define Λε = inf x∈Rd inf ζ∈U Ex(c, ζ), where the controlled diffusion is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' As we have seen before, the above ERSC problem corresponds to the eigen-equation ε−1ΛεΨε(x) = ε 4γ2 ∆Ψε + min u∈U{b(x, u) · ∇Ψε + ε−1c(x, u)Ψε(x)} in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Letting Wε = ε log Ψε we obtain Λε = ε 2γ2 ∆Wε + max v∈Rd min u∈U � (b(x, u) + v) · ∇Wε + c(x, u) − γ2|v|2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='26) Thus, if we could show that the family {Wε} is locally equicontinuous and Λε → Λ0 as ε → 0 (along some subsequence), then using the stability of viscosity solutions, it can be shown from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='26) that max v∈Rd min u∈U � (b(x, u) + v) · ∇W0 + c(x, u) − γ2|v|2� = Λ0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='27) where W0 is a limit of Wε in the viscosity sense as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' If we set g(x, u, v) = b(x, u) + v and h(x, u, v) = c(x, u), then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='27) is same as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='24) when Λ0 = 0 and V = Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In fact, the following result was proved in [90, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='10] (the control process ξ does not play any role in this result) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Suppose that g(x, u, v) = b(x) + v where b satisfies the conditions in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='5 and b(0) = 0, h(x, u, v) = |h1(x)|2 for some C1 function h1 : Rd → Rm with h1(0) = 0 and h1, ∂xih1 are bounded for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then the H∞ suboptimal control problem is solvable, starting at the point 0, at the level γ, if and only if limε→0 Λε = Λ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The existence of solution to the more general equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='27) and a discussion of H∞ control can be found in [97,98], whereas uniqueness is discussed in [137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In the linear-quadratic setting, similar problems are also studied in [33,128,129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let us also mention two interesting works [95,131] where (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='27) is studied in the framework of max-plus calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Generalized Collatz-Wielandt formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Consider a non-negative, irreducible matrix A ∈ Rd×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then the celebrated Collatz-Wielandt [75,160] formula suggests λ(A) = max 0≤x=(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=',xd) min i:xi>0 (Ax)i xi = min 0≤x=(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=',xd) max i:xi>0 (Ax)i xi , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='28) where λ(A) denotes the Perron-Frobenius eigenvalue of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' An alternate characterization of λ(A) can also be given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Write A = (aij) = DR where D = diag[κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' , κd], κi := � j aij R := (p(j|i)), p(j|i) := aij κi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let G = {(π, ˜P) : π is the stationary probability of the stochastic matrix ˜P = (˜p(j|i))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then the following representation can be found in [77] log λ(A) = sup (π, ˜P )∈G �� i π(i) � κi − DKL(˜p(·|i)||p(·|i)) � � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='29) where DKL(·||·) denotes the Kullback-Leibler divergence defined as DKL(˜p(·|i)∥p(·|i, u)) = �� j∈S ˜p(j|i) log � ˜p(j|i) p(j|i,u) � if ˜p(·|i) ≪ p(·|i, u), ∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Given the connection between Perron-Frobenius eigenvalue and the risk-sensitive limits, it is natural to expect a similar representation for λ∗,m or λ∗,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let us first consider a DTCMC taking values in a finite set S and the action set U is also finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' By P(U) we denote the set of all probability vectors on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The cost function c and transition probability can be extended to P(U) in an obvious fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In particular, for v ∈ P(U), we define c(x, v) = � u∈U c(x, u)v(u), P(·|i, v) = � u∈U P(·|i, u)v(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Also, extend the set Usm by allowing the controls to take values in P(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In [16], the authors consider the ERSC problem (fix γ = 1, for simplicity) ¯λ∗,m = max i∈S inf ζ∈Usm Ei(c, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then a generalization of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='29) is obtained in [16] for the controlled problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In order to state this result, we denote by Q the set of all stochastic matrices q = (qij) satisfying qij = 0 if max u∈U P(j|i, u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let Mq denote the set of all stationary probability vectors of q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='10 ([16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Define ˜c(i, q, u) = c(i, u) − DKL(q(·|i)||P(·|i, u)) where q(j|i) := qij, ˜c(i, q, v) = � u∈U ˜c(i, q, u)v(u), v ∈ P(U), �Φ(q, v) = sup π∈Mq � i∈S π(i)˜c(i, q, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='30) Then we have ¯λ∗,m = min v∈Usm max q∈Q �Φ(q, v) = max q∈Q min v∈Usm �Φ(q, v), 20 ANUP BISWAS AND VIVEK S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' BORKAR and there exists a saddle point equilibrium point (q∗, v∗) for the above zero-sum game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The readers must have noticed the analogy between (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='29) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In fact, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='30) can be seen as an ergodic average of ˜c with respect to a suitable Markov chain dictated by the transition probability matrix q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Thus the control v has no effect on the dynamics but only on the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' This forms a single controller zero-sum ergodic game [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We revisit this theme later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' For a general state space, Donsker and Varadhan [83] proved the following min-max formula for diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let X be a compact metric space and {Tt} be a strongly continuous, positive semigroup on C(X) satisfying Tt1 = 1 for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let L be the generator of T and c be any continuous function on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then λ(c) = inf ψ∈D+ sup x∈X Lψ(x) + c(x)ψ(x) ψ(x) = sup µ∈P(X) inf ψ∈D+ � X Lψ(x) + c(x)ψ(x) ψ(x) dµ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='31) where D+ is the subset of the domain of L containing all positive functions, P(X) denotes the set of all Borel probability measures on X and λ(c) = lim t→∞ 1 T log∥T c t ∥, where {T c t } denotes the semigroup generated by L + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' If we associate the semigroup {Tt} with a Markov process {Xt} taking values in X, that is , Ttf(x) = Ex[f(Xt)], then λ(c) is nothing but λ(c) = lim T→∞ 1 T log sup x∈X Ex � e � T 0 c(Xt)dt� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Thus Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='11 gives a Collatz-Wielandt representation to the risk-sensitive value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In the context of discrete time Markov chains, the following representation is proved in [5, The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let G := {η(dx, du, dy) = η0(dx)η1(du|x)η2(dy|x, u) such that η0(dx) is invariant under the transition kernel � η2(dy|x, u)η1(du|x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let S be a compact metric space and X be a controlled Markov process with on S with a compact metric action space U and a continuous transition kernel (x, u) ∈ S × U �→ P(dy|x, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Suppose that Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1 holds and the support of P(·|x) is S for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Also, consider a continuous function c on S × U × S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then there exists a unique λ1 > 0 (the Perron- Frobenius eigenvalue) and a positive Ψ ∈ C(S) such that TΨ = λ1Ψ where T is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Tf(x) = sup ϕ ∈ P(U) � S ec(x,u,y)f(y)ϕ(du)P(dy|x) for f ∈ C(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Furthermore, the following representations hold for λ1 λ1 = inf 0<ψ∈C(S) sup µ∈P(S) � S Tψ(x)dµ � S ψdµ = sup 0<ψ∈C(S) inf µ∈P(S) � S Tψ(x)dµ � S ψdµ log λ1 = sup η∈G � � � � η(dx, du, dy)c(x, u, y)− � � η0(dx)η1(du|x)D(η2(dy|x, u)∥p(dy|x, u)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL 21 The results of [5] go far beyond the above setting where the representation is proved for the optimal value corresponding to a risk-reward problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' A similar representation is also possible for λ∗,m, the optimal value of the ERSC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In fact, the following min-max formula is established in [62, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1] for DTCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let S be a denumerable state space and we consider the setting of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let c ≥ 0 and γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Suppose that Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2 hold, and every state x is accessible from z, under every stationary Markov policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then, if λ∗,m is finite, we have λ∗,m = inf{λ : ∃ positive vector ψ satisfying eγλψ(x) ≥ min u∈U(x) eγc(x,u) � y∈S ψ(y)P(y|x, u)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='32) Furthermore, we have eγλ∗,m = inf ψ>0 sup x∈S minu∈U(x) eγc(x,u) � y∈S ψ(y)P(y|x, u) ψ(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='33) To be precise, [62] established that for some positive vector ψ∗ one has eγλ∗,m = sup x∈S minu∈U(x) eγc(x,u) � y∈S ψ∗(y)P(y|x, u) ψ∗(x) ≥ inf ψ>0 sup x∈S minu∈U(x) eγc(x,u) � y∈S ψ(y)P(y|x, u) ψ(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' But the above inequality cannot be strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Otherwise, for some ε > 0 and ψ > 0 we would have eγ(λ∗,m−ε) > sup x∈S minu∈U(x) eγc(x,u) � y∈S ψ(y)P(y|x, u) ψ(x) , which will contradict (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' This gives us (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' For controlled diffusions, similar representation was studied in [18] with the help of the nonlinear Krein-Rutman theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' To present the result of [18], we consider a bounded domain D ⊂ Rd with a C3 boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The reflected controlled diffusion on ¯D is given by dXt = b(Xt, ζt)dt + σ(Xt)dWt − δ(Xt)dξt, dξt = 1{Xt∈∂D}dξt, ξ0 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='34) where b, σ are as before (see (B1)-(B3)), ζ ∈ U, and δ : Rd → Rd is co-normal, that is, δ(x) = 2a(x)n(x) where n(x) denote the unit outward normal on ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' As before, we define the ERSC problem as λ∗,d = inf x∈Rd inf ζ∈U Ex(c, ζ) where Ex(c, ζ) := lim sup T→∞ 1 γT log Eζ x � e � T 0 γc(Xt,ζt)dt� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='35) where (Xt, ζt) satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Define C2 δ,+(D) = {ψ ∈ C2( ¯D) : ψ ≥ 0, ∇ψ · δ = 0 on ∂D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Also, recall the operator G from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The following representation of λ∗,d can be found in [18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' There exists a unique pair (ρ, Ψ) ∈ R × C2 δ,+(D) satisfying GΨ = ρΨ and max ¯Q Ψ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Moreover, ρ = λ∗,d, given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='35), and the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' λ∗,d = inf 0<ψ∈C2 δ,+(D) sup µ∈P( ¯D) � ¯D Gψ ψ dµ = sup 0<ψ∈C2 δ,+(D) inf µ∈P( ¯D) � ¯D Gψ ψ dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 22 ANUP BISWAS AND VIVEK S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' BORKAR Note that the ERSC problem in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='35) is related to the Nisio semigroup given by Stf(x) := inf ζ∈U Ex � e � t 0 c(Xs,ζs)dsf(Xt) � f ∈ C( ¯D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Thus, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='14 can be seen as a generalization of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='11 to nonlinear semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' One can also have a similar Collatz-Wielandt formula for the generalized Dirichlet principal eigenvalue which is defined replacing L + c by G in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='15 ([8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let D be a bounded, smooth domain and λD(G) denote the generalized Dirich- let principal eigenvalue of G in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then we have λD(G) = inf ψ∈C2 +(D) sup µ∈P(D) � D Gψ ψ dµ = sup ψ∈C2 +(D)∩C0(D) inf µ∈P(D) � D Gψ ψ dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' As pointed out in [8, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2], the set C2 +(D) ∩ C0(D) in the second equality cannot be extended to C2 +(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='7 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='8 we know that λ∗,d = λ1(G), the generalized principal eigenvalue of G in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' So one might expect an analog of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='15 for λ∗,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' It turns out that for a linear operator L of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2), one has (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' [8]) λRd(c) = inf 0<ψ∈C2(Rd) sup µ∈P(Rd) � Rd �Lψ ψ + c � dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' But by considering Lf = f ′′ − f ′ in R and c = 0, it is shown in [8, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='3] that λRd(c) < sup 0<ψ∈C2 b (Rd) inf µ∈P(Rd) � Rd Lψ ψ dµ ≤ sup 0<ψ∈C2(Rd) inf µ∈P(Rd) � Rd Lψ ψ dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Thus we need an additional condition on the operator in order to obtain a full Collatz-Wielandt type formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In [8], the following condition, which is slightly stronger than Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1, is used in order to obtain a Collatz-Wielandt type formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' There exists a positive V ∈ C2(Rd) with infRd V > 0 such that one of the following hold: (i) There exists an inf-compact, positive ℓ ∈ C(Rd) and a compact set K satisfying sup u∈U LuV ≤ ¯κ1K − ℓV in Rd, for some constant ¯κ, and βℓ − maxu∈U γc(·, u) is inf-compact, for some β ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (ii) For some positive constants ¯κ, θ and a compact set K, we have sup u∈U LuV ≤ ¯κ1K − θV in Rd, and lim sup |x|→∞ max u∈U γc(x, u) < θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' By o(V) we denote the class of functions growing slower than V, that is, f ∈ o(V) if and only if lim sup |x|→∞ |f(x)| V(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='16 ([8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Suppose that (B1)-(B3) and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Then we have λ∗,d = λ1(G) = inf 0<ψ∈C2(Rd) sup µ∈P(Rd) � Rd Gψ ψ dµ A SURVEY ON ERGODIC RISK-SENSITIVE CONTROL 23 = sup 0<ψ∈C2(Rd)∩o(V) inf µ∈P(Rd) � Rd Gψ ψ dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Risk-sensitive control of continuous time Markov chains In this section we review the recent developments on ERSC for continuous time controlled Markov chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' We consider a continuous time controlled Markov chain (CTCMC) X = {Xt ,t ≥ 0}, on a denumerable state space S, controlled by the control process ζt , t ≥ 0 , taking values in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' As before, U is the action space of the controller, which is assumed to be a Borel space with Borel σ-algebra B(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' For each i ∈ S, let U(i) be the space of all admissible actions of the controller when the system is at state i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let K := {(i, u) : i ∈ S, u ∈ U(i)} be the set of all feasible state action pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' As before, we denote by c : K → R+ the running cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' The transition rates q(j|i, u), u ∈ U(i) , i, j ∈ S, satisfy the condition q(j|i, u) ≥ 0 for all u ∈ U(i), i, j ∈ S and j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' In addition, we also impose the following: Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (a) For each i ∈ S, the admissible action space U(i) is a nonempty compact subset of U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (b) The model is conservative: � j∈S q(j|i, u) = 0 ∀ u ∈ U(i), i ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' (c) The model is stable: q(i) := sup u∈U(i) (−q(i|i, u)) = sup u∈U(i) � j̸=i q(j|i, u) < ∞ ∀ i ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' For each i, j ∈ S, q(j|i, u) is a measurable map on U(i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let c : S × U → R+ be the running cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Following [134] (see also [110, 112, 146]) we briefly describe the evolution of the CTCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Let S∞ := S ∪ {i∞} for an isolated point i∞ /∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Define the canonical sample space Ω := (S × (0, ∞))∞ ∪ {(i0, θ1, i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' , θm, im, ∞, i∞, ∞, i∞, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' ) | θk ̸= ∞, ik ̸= i∞ for all 0 ≤ k ≤ m, m ≥ 1} , with Borel σ-algebra B(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' For each sample point ω = (i0, θ1, i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' , θm, im, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=') ∈ Ω, we set T0(ω) = 0, Tk(ω) = θ1 + θ2 + · · · + θk, and define T∞(ω) = limk→∞ Tk(ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfZfdb/content/2301.00224v1.pdf'} +page_content=' Now we define a controlled process {Xt}t≥0 on (Ω, B(Ω)) by Xt = � k≥0 1{Tk≤t S/N. Now, S0 > S and R is a Hall σi-subgroup of S0, which contradicts +the fact that S is a σi-sylowizer of R in G. Thus S/N is a σi-sylowizer of RN/N in G/N. +Conversely, if S/N is a σi-sylowizer of RN/N in G/N, then R is a Hall σi-subgroup of S. If S is +not a σi-sylowizer of R in G, then there is a σi-sylowizer S0 of R in G such that S0 > S. Therefore +RN/N is a Hall σi-subgroup of S0/N, which contradicts the fact that S/N is a σi-sylowizer of +RN/N in G/N. Thus S is a σi-sylowizer of R in G. +� +Lemma 2.3. Let R be a σi-subgroup of a σ-full group G for some σi ∈ σ(G) and S a σi-sylowizer +of R in G. If S is σ-permutable in G, then Oσi(G) ≤ S. In particular, S = ROσi(G) is the unique +σi-sylowizer of R in G. +Proof +Let Q be a Hall σj-subgroup of G with σj ∈ σ(G) and σi∩σj = ∅. Since S is σ-permutable, +we have SQ ≤ G. Note that since R is a Hall σi-subgroup of SQ, we have QS = S by the maximality +of S. Hence Q ≤ S. It shows that Oσi(G) ≤ S. +� +Lemma 2.4. Let R be a σi-subgroup of a σ-full group of Sylow type G for some σi ∈ σ(G) and S a +σi-sylowizer of R in G. Then S is c-permutable with every Hall σj-subgroup of G for all σj ∈ σ(G) +if and only if |G : S| is a σi-number. +Proof +The sufficiency is evident, we only need to prove the necessity. +Let Q be a Hall σj-subgroup of G with σj ∈ σ(G) and σi ∩ σj = ∅. Since S is c-permutable with +Q, we have SQx = QxS for some element x ∈ G. Note that since R is a Hall σi-subgroup of SQx, +we have QxS = S by the maximality of S. Hence Qx ≤ S. It implies that |G : S| is a σi-number. � +Theorem 2.5. Let G be a σ-full group of Sylow type and H = {H1, · · ·, Ht} be a complete Hall σ-set +of G such that Hi is a nilpotent σi-subgroup for all i = 1, · · · , t. Suppose that for any σi ∈ σ(G), +every maximal subgroup of any non-cyclic Hi has a σi-sylowizer that is c-permutable with every +member of H, then G is supersoluble. +Proof +Assume that this is false and let G be a counterexample of minimal order. Then: +(1) Let N be a minimal normal subgroup of G. Then G is supersoluble. +We consider the quotient group G/N. It is clear that G/N is a σ-full group of Sylow type and +HN/N is a complete Hall σ-set of G/N such that HiN/N is nilpotent. Let H/N be a maximal +2 + +subgroup of HiN/N and Hσi be a Hall σi-subgroup of H contained in Hi. Then H = HσiN. Since +Hσi ∩ N = Nσi = Hi ∩ N, where Nσi denotes a Hall σi-subgroup of N, we have that +|Hi : Hσi| = |Hi||N| +|Hi ∩ N| · |Hσi ∩ N| +|Hσi||N| = |HiN : H| = q +for some q ∈ σi. This shows that Hσi is a maximal subgroup of Hi. If HiN/N is non-cyclic, then +so is Hi. Thus if S/N is a σi-sylowizer of H/N in G/N, then S is a σi-sylowizer of Hσi in G by +Lemma 2.2. Moreover, if S is c-permutable with every member of H, then S/N is c-permutable +with every member of HN/N by Lemma 2.4. It shows that G/N satisfies the hypotheses. Thus +G/N is supersoluble by the choice of G. +(2) N is the unique proper minimal normal subgroup of G and Φ(G) = 1. +Let p be the smallest prime divisor of G and p ∈ σi. If Hi is cyclic, then G is p-nilpotent. This +shows that G has a proper minimal normal subgroup. Thus we may assume that Hi is non-cyclic. +Let M be a maximal subgroup of Hi of index p and S a σi-sylowizer of M in G that is c-permutable +with every member of H. Then |G : S| = p by Lemma 2.4 and so S � G. Therefore we may choose +a proper minimal normal subgroup of G contained in S, say N. By Claim (1), G/N is supersoluble. +Moreover, N is the unique minimal normal subgroup of G. Since the class of all supersoluble groups +is a saturated formation, we may assume further that |Φ(G)| = 1. +(3) N is soluble. +Assume that N is not soluble. Then p = 2 and 2||N|. Let P be a Sylow 2-subgroup of Hi. +Then N2 = P ∩ N is a Sylow 2-subgroup of N. +If N2 ≤ Φ(Hi), then N2 ≤ Φ(P), and so N +is 2-nilpotent by Tate’s theorem, a contradiction. Hence N2 ≰ Φ(Hi). Thus there is a maximal +subgroup K of Hi such that Hi = KN2. Let S0 be a σi-sylowizer of K in G that is c-permutable +with every member of H. Then |G : S0| = 2 by Lemma 2.4. Thus G = S0Hi = S0N2 = S0N. Now, +|N : N ∩S0| = |G : S0| = 2, which implies that N ∩S0 �N. Since N ∩S0 �S0, we have N ∩S0 �G. +Note that N is a minimal normal subgroup of G, we have N ∩ S0 = 1. Thus |N| = |G : S0| = 2, a +contradiction. +(4) Final contradiction. +By Claim (3), we may assume that N is a q-subgroup for some prime q ∈ σj. Since Φ(G) = 1, +there is a maximal subgroup T of G such that G = TN. Let Tσj be a Hall σj-subgroup of T contained +in Hj. Then Hj = TσjN is a Hall σj-subgroup of G. If Hj is cyclic, then G is supersoluble by the +supersolublity of G/N. Thus we may assume that Hj is non-cyclic. Let Q ≥ Tσj be a maximal +subgroup of Hj and Y a σj-sylowizer of Q in G that is c-permutable with every member of H. Then +|G : Y | = q by Lemma 2.4 and N ≰ Y . Otherwise Hj = QN ≤ Y , which contradicts the fact +that Q is a Hall σj-subgroup of Y . Thus G = Y N and so |N| = |G : Y | = q. It implies that G is +supersoluble, a contradiction. This contradiction completes the proof. +� +Theorem 2.6. Let F be a soluble saturated formation containing all supersoluble groups and let +3 + +E be a normal subgroup of G with G/E ∈ F. +Suppose that G is a σ-full group of Sylow type +and H = {H1, · · ·, Ht} is a complete Hall σ-set of G such that Hi is a nilpotent σi-subgroup for +all i = 1, · · ·, t. If for any σi ∈ σ(E), every maximal subgroup of any non-cyclic Hi ∩ E has a +σi-sylowizer that is c-permutable with every member of H, then G ∈ F. +Proof +The conclusion holds when E = G by Theorem 2.5, thus we may assume that E < G. Let +N be a minimal normal subgroup of G contained in E. +(1) E is supersoluble. +Let Q be a maximal subgroup of a non-cyclic Hall σi-subgroup Hi ∩E of E and S a σi-sylowizer +of Q in G that is c-permutable with member of H. By Lemma 2.4, |G : S| is a σi-number. Let +Y = S ∩ E. Since |E : Y | = |E : S ∩ E| = |SE : S| divides |G : S|, |E : Y | is a σi-number. Hence +Y is a σi-sylowizer of Q in E and Y is c-permutable with every member of H ∩ E by Lemma 2.4. +Thus E is supersoluble by Theorem 2.5. +(2) N is the unique minimal normal subgroup of G contained in E and N ∩ Φ(G) = 1. +Consider the quotient group G/N, evidently (G/N)/(E/N) ∈ F. Since E is supersoluble by +Claim (1), we have that N is a p-group for some prime p. Without loss of generality, we may write +Ei = Hi∩E for all i ∈ {1, · · ·, t} and assume that p ∈ σi for some i. Let J/N be a maximal subgroup +of Ei/N, then J is a maximal subgroup of Ei. If S/N is a σi-sylowizer of J/N in G/N, then S is +a σi-sylowizer of J in G by Lemma 2.2. Moreover, if S is c-permutable with every member of H, +then S/N is c-permutable with every member of HN/N by Lemma 2.4. Let J/N be a maximal +subgroup of EjN/N and Jσj a Hall σj-subgroup of J contained in Ej, where i ̸= j. Then Jσj is +a maximal subgroup of Ej. If S/N is a σj-sylowizer of JσjN/N in G/N, then S is a σj-sylowizer +of Jσj in G by Lemma 2.2. Moreover, if S is c-permutable with every member of H, then S/N is +c-permutable with every member of HN/N by Lemma 2.4. This shows that (G/N, E/N) satisfies +the hypotheses. Thus we may have that G/N ∈ F by induction. Moreover, N is the unique minimal +normal subgroup of G contained in E and N ∩ Φ(G) = 1. +(3) N is an elementary abelian p-subgroup, where p is the largest prime divisor of |E|. +Since E is supersoluble by Claim (1), the Sylow p-subgroup EP of E is normal in G. Note that +N is the unique minimal normal subgroup of G contained in E, N ≤ EP is an elementary abelian +p-subgroup. +(4) G ∈ F. +Without loss of generality, we may assume that p ∈ σi. If Ei is cyclic, then |N| = p and so G ∈ F. +Assume that Ei is non-cyclic. Since N ≰ Φ(G), there is a maximal subgroup M of G such that +G = MN and M ∩ N = 1. Thus Ei = N(M ∩ Ei) and Hi = NM ∩ Hi = N(M ∩ Hi) = NMi. Since +Mi < Hi, we may choose P ⋖ Hi such that Mi ≤ P. Since M ∩ Ei ≤ P, P ∩ Ei = P ∩ N(M ∩ Ei) = +(P ∩ N)(M ∩ Ei). Note that M ∩ N = 1, we have +|Ei : Ei ∩ P| = |N(M ∩ Ei) : (P ∩ N)(M ∩ Ei)| = |N : P ∩ N| = p. +4 + +Hence R = Ei ∩ P is a maximal subgroup of Ei. Let S be a σi-sylowizer of R in G that is c- +permutable with every member of H. Then |G : S| is a σi-number by Lemma 2.4. Since G is +soluble, we may write S = RSσ′ +i and M = MiMσ′ +i. +Note also that |G : S| and |G : M| are +σi-number in G, Sσ′ +i and Mσ′ +i are also Hall σ′ +i-subgroups of G. Thus there is an element g of G +such that Sg +σ′ +i = Mσ′ +i. Since G = HiSg, we may write g = xy, where x ∈ Hi and y ∈ Sg. Note +that since R = Ei ∩ P � Hi, we have Ry = Rxy ≤ Sg and so R ≤ Sg. Thus Sg = RMσ′ +i. Since +RMi = (P ∩ Ei)Mi = P ∩ EiMi = P ∩ NMi = P ≤ G, we have RM ≤ G. Since M is a maximal +subgroup, either RM = M or RM = G. +If RM = G, then RMi = P is a Hall σi-subgroup of G, which is impossible. Thus RM = M +and so R ≤ M ∩ Ei. Since G = MN = MEi, we have Ei ≰ M. Note that since R ⋖ Ei, we have +R = M ∩ Ei. Thus |N| = |G : M| = |Ei : Ei ∩ M| = |Ei : R| = p. By [7, Theorem 2], G ∈ F, as +required. +� +References +[1] W. Gaschiitz, Sylowisatoren, Math. Z., 122 (1971), 319-320. +[2] A. N. Skiba, On σ-subnormal and σ-permutable subgroups of finite groups, J. Algebra, 436 +(2015), 1-16. +[3] W. Guo, A. N. Skiba, On Π-permutable subgroups of finite groups, Monatsh Math, 185(3) +(2018), 443-453. +[4] A. N. Skiba, On some results in the theory of finite partially soluble groups, Comm. Math. +Stat, 4 (2016), 281-309. +[5] W. Guo, K.P. Shum, A. N. Skiba, Conditionally permutable subgroups and supersolubility of +finite groups, Southeast Asian Bull. Math., 29(3)(2005), 493-510. +[6] D. Lei, X, Li, The permutability of p-sylowizers of some p-subgroups in finite groups, Arch. +Math., 114 (2020), 367-376. +[7] A. Ballester-Bolinches, M.C. Pedraza-Aguilera, On minimal subgroups of finite groups. Acta +Math. Hung., 73(4), (1996), 335-342. +5 + diff --git a/rdE0T4oBgHgl3EQfagDQ/content/tmp_files/load_file.txt b/rdE0T4oBgHgl3EQfagDQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3d23c677aac41e800bce754feeb9ffc61b6adf2f --- /dev/null +++ b/rdE0T4oBgHgl3EQfagDQ/content/tmp_files/load_file.txt @@ -0,0 +1,225 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf,len=224 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='02337v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='GR] 5 Jan 2023 The permutability of σi-sylowizers of some σi-subgroups in finite groups Zhenya Liu AND Wenbin Guo Abstract Let σ = {σi|i ∈ I} be a partition of the set of all primes P, G a finite group and σ(G) = {σi|σi ∩ π(|G|) ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' A subgroup S of a group G is called a σi-sylowizer of a σi-subgroup R in G if S is maximal in G with respect to having R as its Hall σi-subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' The main aim of this paper is to investigate the influence of σi-sylowizers on the structure of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' We obtained some new characterizations of supersoluble groups by the permutability of the σi-sylowizers of some σi-subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' 1 Introduction Let π denotes a set of primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' The concept of π-Sylowizers has been introduced by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Gaschutz [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' If R is a π-subgroup of the group G, then a π-Sylowizer of R in G is a subgroup S of G maximal with respect to containing R as a Hall π-subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' P is the set of all primes and n is a natural number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let σ = {σi|i ∈ I} is some partition of all primes P, that is, P = � i∈I σi and σi ∩ σj = ∅ for all i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' We write σ(G) = {σi|σi ∩ π(G) ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Following [5], two subgroups H and T of a group G are conditionally permutable (or in brevity, c-permutable) in G if there exists an element x ∈ G such that HT x = T xH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' 2 Preliminaries Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let H be a σi-subgroup of G for some σi ∈ σ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Assume that K is a subgroup satisfying H ≤ K ≤ G and T is a σi-sylowizer of H in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Then there is a σi-sylowizer S of H in G such that T = S ∩ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Proof Since H is a Hall σi-subgroup of T, there is a σi-sylowizer S of H in G such that S ≥ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Then H is a Hall σi-subgroup of S ∩ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since T ≤ S ∩ K and T is a σi-sylowizer of H in K, we get T = S ∩ K by the maximality of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' � Keywords: Finite group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' σ-permutable subgroup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' supersoluble group Mathematics Subject Classification (2021): 20D10, 20D15, 20D20, 20D35 1 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let R be a σi-subgroup of G for some σi ∈ σ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Assume that N is a normal subgroup of G and R is a Hall σi-subgroup of RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Then S is a σi-sylowizer of R in G if and only if S/N is a σi-sylowizer of RN/N in G/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Proof Let S be a σi-sylowizer of R in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since R is a Hall σi-subgroup of RN, R is a Hall σi-subgroup of SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus N ≤ S by the maximality of S and so RN/N is a Hall σi-subgroup of S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' If S/N is not a σi-sylowizer of RN/N in G/N, then there is a σi-sylowizer S0/N of RN/N in G/N such that S0/N > S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Now, S0 > S and R is a Hall σi-subgroup of S0, which contradicts the fact that S is a σi-sylowizer of R in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus S/N is a σi-sylowizer of RN/N in G/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Conversely, if S/N is a σi-sylowizer of RN/N in G/N, then R is a Hall σi-subgroup of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' If S is not a σi-sylowizer of R in G, then there is a σi-sylowizer S0 of R in G such that S0 > S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Therefore RN/N is a Hall σi-subgroup of S0/N, which contradicts the fact that S/N is a σi-sylowizer of RN/N in G/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus S is a σi-sylowizer of R in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' � Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let R be a σi-subgroup of a σ-full group G for some σi ∈ σ(G) and S a σi-sylowizer of R in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' If S is σ-permutable in G, then Oσi(G) ≤ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' In particular, S = ROσi(G) is the unique σi-sylowizer of R in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Proof Let Q be a Hall σj-subgroup of G with σj ∈ σ(G) and σi∩σj = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since S is σ-permutable, we have SQ ≤ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Note that since R is a Hall σi-subgroup of SQ, we have QS = S by the maximality of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Hence Q ≤ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' It shows that Oσi(G) ≤ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' � Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let R be a σi-subgroup of a σ-full group of Sylow type G for some σi ∈ σ(G) and S a σi-sylowizer of R in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Then S is c-permutable with every Hall σj-subgroup of G for all σj ∈ σ(G) if and only if |G : S| is a σi-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Proof The sufficiency is evident, we only need to prove the necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let Q be a Hall σj-subgroup of G with σj ∈ σ(G) and σi ∩ σj = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since S is c-permutable with Q, we have SQx = QxS for some element x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Note that since R is a Hall σi-subgroup of SQx, we have QxS = S by the maximality of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Hence Qx ≤ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' It implies that |G : S| is a σi-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' � Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let G be a σ-full group of Sylow type and H = {H1, · · ·, Ht} be a complete Hall σ-set of G such that Hi is a nilpotent σi-subgroup for all i = 1, · · · , t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Suppose that for any σi ∈ σ(G), every maximal subgroup of any non-cyclic Hi has a σi-sylowizer that is c-permutable with every member of H, then G is supersoluble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Proof Assume that this is false and let G be a counterexample of minimal order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Then: (1) Let N be a minimal normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Then G is supersoluble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' We consider the quotient group G/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' It is clear that G/N is a σ-full group of Sylow type and HN/N is a complete Hall σ-set of G/N such that HiN/N is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let H/N be a maximal 2 subgroup of HiN/N and Hσi be a Hall σi-subgroup of H contained in Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Then H = HσiN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since Hσi ∩ N = Nσi = Hi ∩ N, where Nσi denotes a Hall σi-subgroup of N, we have that |Hi : Hσi| = |Hi||N| |Hi ∩ N| · |Hσi ∩ N| |Hσi||N| = |HiN : H| = q for some q ∈ σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' This shows that Hσi is a maximal subgroup of Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' If HiN/N is non-cyclic, then so is Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus if S/N is a σi-sylowizer of H/N in G/N, then S is a σi-sylowizer of Hσi in G by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Moreover, if S is c-permutable with every member of H, then S/N is c-permutable with every member of HN/N by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' It shows that G/N satisfies the hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus G/N is supersoluble by the choice of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' (2) N is the unique proper minimal normal subgroup of G and Φ(G) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let p be the smallest prime divisor of G and p ∈ σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' If Hi is cyclic, then G is p-nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' This shows that G has a proper minimal normal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus we may assume that Hi is non-cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let M be a maximal subgroup of Hi of index p and S a σi-sylowizer of M in G that is c-permutable with every member of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Then |G : S| = p by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='4 and so S � G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Therefore we may choose a proper minimal normal subgroup of G contained in S, say N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' By Claim (1), G/N is supersoluble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Moreover, N is the unique minimal normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since the class of all supersoluble groups is a saturated formation, we may assume further that |Φ(G)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' (3) N is soluble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Assume that N is not soluble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Then p = 2 and 2||N|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let P be a Sylow 2-subgroup of Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Then N2 = P ∩ N is a Sylow 2-subgroup of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' If N2 ≤ Φ(Hi), then N2 ≤ Φ(P), and so N is 2-nilpotent by Tate’s theorem, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Hence N2 ≰ Φ(Hi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus there is a maximal subgroup K of Hi such that Hi = KN2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let S0 be a σi-sylowizer of K in G that is c-permutable with every member of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Then |G : S0| = 2 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus G = S0Hi = S0N2 = S0N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Now, |N : N ∩S0| = |G : S0| = 2, which implies that N ∩S0 �N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since N ∩S0 �S0, we have N ∩S0 �G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Note that N is a minimal normal subgroup of G, we have N ∩ S0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus |N| = |G : S0| = 2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' (4) Final contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' By Claim (3), we may assume that N is a q-subgroup for some prime q ∈ σj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since Φ(G) = 1, there is a maximal subgroup T of G such that G = TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let Tσj be a Hall σj-subgroup of T contained in Hj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Then Hj = TσjN is a Hall σj-subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' If Hj is cyclic, then G is supersoluble by the supersolublity of G/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus we may assume that Hj is non-cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let Q ≥ Tσj be a maximal subgroup of Hj and Y a σj-sylowizer of Q in G that is c-permutable with every member of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Then |G : Y | = q by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='4 and N ≰ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Otherwise Hj = QN ≤ Y , which contradicts the fact that Q is a Hall σj-subgroup of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus G = Y N and so |N| = |G : Y | = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' It implies that G is supersoluble, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' This contradiction completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' � Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let F be a soluble saturated formation containing all supersoluble groups and let 3 E be a normal subgroup of G with G/E ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Suppose that G is a σ-full group of Sylow type and H = {H1, · · ·, Ht} is a complete Hall σ-set of G such that Hi is a nilpotent σi-subgroup for all i = 1, · · ·, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' If for any σi ∈ σ(E), every maximal subgroup of any non-cyclic Hi ∩ E has a σi-sylowizer that is c-permutable with every member of H, then G ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Proof The conclusion holds when E = G by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='5, thus we may assume that E < G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let N be a minimal normal subgroup of G contained in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' (1) E is supersoluble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let Q be a maximal subgroup of a non-cyclic Hall σi-subgroup Hi ∩E of E and S a σi-sylowizer of Q in G that is c-permutable with member of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='4, |G : S| is a σi-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let Y = S ∩ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since |E : Y | = |E : S ∩ E| = |SE : S| divides |G : S|, |E : Y | is a σi-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Hence Y is a σi-sylowizer of Q in E and Y is c-permutable with every member of H ∩ E by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus E is supersoluble by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' (2) N is the unique minimal normal subgroup of G contained in E and N ∩ Φ(G) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Consider the quotient group G/N, evidently (G/N)/(E/N) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since E is supersoluble by Claim (1), we have that N is a p-group for some prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Without loss of generality, we may write Ei = Hi∩E for all i ∈ {1, · · ·, t} and assume that p ∈ σi for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let J/N be a maximal subgroup of Ei/N, then J is a maximal subgroup of Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' If S/N is a σi-sylowizer of J/N in G/N, then S is a σi-sylowizer of J in G by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Moreover, if S is c-permutable with every member of H, then S/N is c-permutable with every member of HN/N by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let J/N be a maximal subgroup of EjN/N and Jσj a Hall σj-subgroup of J contained in Ej, where i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Then Jσj is a maximal subgroup of Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' If S/N is a σj-sylowizer of JσjN/N in G/N, then S is a σj-sylowizer of Jσj in G by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Moreover, if S is c-permutable with every member of H, then S/N is c-permutable with every member of HN/N by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' This shows that (G/N, E/N) satisfies the hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus we may have that G/N ∈ F by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Moreover, N is the unique minimal normal subgroup of G contained in E and N ∩ Φ(G) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' (3) N is an elementary abelian p-subgroup, where p is the largest prime divisor of |E|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since E is supersoluble by Claim (1), the Sylow p-subgroup EP of E is normal in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Note that N is the unique minimal normal subgroup of G contained in E, N ≤ EP is an elementary abelian p-subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' (4) G ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Without loss of generality, we may assume that p ∈ σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' If Ei is cyclic, then |N| = p and so G ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Assume that Ei is non-cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since N ≰ Φ(G), there is a maximal subgroup M of G such that G = MN and M ∩ N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus Ei = N(M ∩ Ei) and Hi = NM ∩ Hi = N(M ∩ Hi) = NMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since Mi < Hi, we may choose P ⋖ Hi such that Mi ≤ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since M ∩ Ei ≤ P, P ∩ Ei = P ∩ N(M ∩ Ei) = (P ∩ N)(M ∩ Ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Note that M ∩ N = 1, we have |Ei : Ei ∩ P| = |N(M ∩ Ei) : (P ∩ N)(M ∩ Ei)| = |N : P ∩ N| = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' 4 Hence R = Ei ∩ P is a maximal subgroup of Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Let S be a σi-sylowizer of R in G that is c- permutable with every member of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Then |G : S| is a σi-number by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since G is soluble, we may write S = RSσ′ i and M = MiMσ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Note also that |G : S| and |G : M| are σi-number in G, Sσ′ i and Mσ′ i are also Hall σ′ i-subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus there is an element g of G such that Sg σ′ i = Mσ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since G = HiSg, we may write g = xy, where x ∈ Hi and y ∈ Sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Note that since R = Ei ∩ P � Hi, we have Ry = Rxy ≤ Sg and so R ≤ Sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus Sg = RMσ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since RMi = (P ∩ Ei)Mi = P ∩ EiMi = P ∩ NMi = P ≤ G, we have RM ≤ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since M is a maximal subgroup, either RM = M or RM = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' If RM = G, then RMi = P is a Hall σi-subgroup of G, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus RM = M and so R ≤ M ∩ Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Since G = MN = MEi, we have Ei ≰ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Note that since R ⋖ Ei, we have R = M ∩ Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Thus |N| = |G : M| = |Ei : Ei ∩ M| = |Ei : R| = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' By [7, Theorem 2], G ∈ F, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' � References [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Gaschiitz, Sylowisatoren, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Shum, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Skiba, Conditionally permutable subgroups and supersolubility of finite groups, Southeast Asian Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=', 29(3)(2005), 493-510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' [6] D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Pedraza-Aguilera, On minimal subgroups of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' Hung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=', 73(4), (1996), 335-342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} +page_content=' 5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE0T4oBgHgl3EQfagDQ/content/2301.02337v1.pdf'} diff --git a/rdFST4oBgHgl3EQfPzg1/content/tmp_files/2301.13756v1.pdf.txt b/rdFST4oBgHgl3EQfPzg1/content/tmp_files/2301.13756v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f6b539bef17227ca413851965de8dd392a685118 --- /dev/null +++ b/rdFST4oBgHgl3EQfPzg1/content/tmp_files/2301.13756v1.pdf.txt @@ -0,0 +1,1890 @@ +arXiv:2301.13756v1 [cs.GT] 31 Jan 2023 +PAC learning and stabilizing Hedonic Games: towards a unifying approach. +Simone Fioravanti,1 Michele Flammini,1 Bojana Kodric,2, 1 Giovanna Varricchio 3 +1 Gran Sasso Science Institute (GSSI), L’Aquila, Italy +2 Ca’ Foscari University of Venice, Venice, Italy +3 Goethe-Universit¨at, Frankfurt am Main, Germany +simone.fioravanti@gssi.it, michele.flammini@gssi.it, bojana.kodric@unive.it, varricchio@em.uni-frankfurt.de +Abstract +We study PAC learnability and PAC stabilizability of He- +donic Games (HGs), i.e., efficiently inferring preferences +or core-stable partitions from samples. We first expand the +known learnability/stabilizability landscape for some of the +most prominent HGs classes, providing results for Friends +and Enemies Games, Bottom Responsive, and Anonymous +HGs. Then, having a broader view in mind, we attempt to +shed light on the structural properties leading to learnabil- +ity/stabilizability, or lack thereof, for specific HGs classes. +Along this path, we focus on the fully expressive Hedonic +Coalition Nets representation of HGs. We identify two sets +of conditions that lead to efficient learnability, and which en- +compass all of the known positive learnability results. On the +side of stability, we reveal that, while the freedom of choosing +an ad hoc adversarial distribution is the most obvious hur- +dle to achieving PAC stability, it is not the only one. First, +we show a distribution independent necessary condition for +PAC stability. Then, we focus on W-games, where players +have individual preferences over other players and evaluate +coalitions based on the least preferred member. We prove that +these games are PAC stabilizable under the class of bounded +distributions, which assign positive probability mass to all +coalitions. Finally, we discuss why such a result is not easily +extendable to other HGs classes even in this promising sce- +nario. Namely, we establish a purely computational property +necessary for achieving PAC stability. +1 +Introduction +Hedonic Games (HGs) (Dreze and Greenberg 1980) are a +formal model for describing selfish individuals gathering to- +gether in order to form coalitions. Both HGs and general +coalition formation games attracted considerable research +attention in the last years due to their applicability to multi- +agent environments. Solution concepts for HGs are usually +in the form of agent partitions with some suitable proper- +ties. The one we consider in this paper is core stability. A +partition is said to be core-stable (or in the core) if there ex- +ists no subset of players that could regroup into a so-called +core-blocking coalition, which is preferred by all of them. +The usual assumption when considering any solution con- +cept is that the preferences of the agents are fully known, +which is arguably unrealistic. Could we instead efficiently +infer the whole game structure, or even directly learn so- +lution concepts, while having only partial knowledge of the +preferences? Questions of this kind are naturally captured by +the probably approximately correct (PAC) learning frame- +work (Valiant 1984), which formalizes the problem of learn- +ing a target concept from a limited number of samples from +any possible unknown but fixed distribution. +Sliwinski and Zick (2017) were the first to leverage the +PAC framework to study the problem of learning HGs pref- +erences and core-stable partitions from samples. In partic- +ular, they define PAC stabilizability of a HGs class as the +property of being able to, upon seeing a limited number of +samples, either report that the core is empty or propose a par- +tition that is unlikely to be core-blocked by further coalitions +sampled from the same distribution. In a recent paper, Lev +et al. (2021a) apply the notion of PAC stabilizability of HGs +in the context of political coalition formation. In particular, +they use the publicly available Israeli parliament voting data +to fit a Friends Appreciation HG, and compare the actual +political parties of the voters to the PAC-stable coalitions re- +sulting from the model. This example shows how learning +concepts have the potential to create space for applications +of mainly theoretical models, as HGs. +While the work of Sliwinski and Zick (2017) and the ones +that followed considered PAC learnability and stabilizabil- +ity of many specific classes of HGs, the overall picture is +still far from being complete. Most prominently, the char- +acterization of the underlying general conditions explaining +the existing results is missing. Furthermore, PAC stabiliz- +ability seems very hard to achieve and it is natural to won- +der whether some restrictions on the PAC stability definition +can yield better results. Here, we address these questions, +attempting to provide a deeper theoretical understanding of +what makes HGs learnable and stabilizable. +1.1 +Our Contribution +We first extend the knowledge on PAC learnable and PAC +stabilizable classes of HGs. We start by focusing on Friends +and Enemies Games, examining whether the negative re- +sults on stabilizability of Additively Separable HGs transfer +to this simple subclass. By exploiting previous results and +proposing an algorithm stabilizing Friends and Enemies un- +der Enemies Aversion, we deduce that Friends and Enemies +Games belong to the very few lucky HGs classes that can +both be learned and stabilized. Next, we study Bottom Re- +sponsive HGs and show that while they are not efficiently + +learnable, they are stabilizable. Finally, we turn our atten- +tion to Anonymous HGs and show that the opposite holds +here, i.e., they are efficiently learnable but not stabilizable. +After exploring specific HGs classes, we use the gained +insights to follow a more general research direction, devoted +to a deeper understanding of the structural properties that +make HGs learnable and/or stabilizable. +We first consider the learning problem. Additively Sep- +arable, Anonymous, W and B-games are all known to be +learnable, and we investigate why this is the case. To this +aim, we consider Hedonic Coalition Nets (HCNs), a general +framework for representing HGs that is universally expres- +sive, i.e., it can represent any HGs class. We identify two +sets of conditions on the HCNs representation that imply ef- +ficient learnability, and as special cases explain the learn- +ability of all of the aforementioned HGs classes. +We then turn our attention to stability. Achieving PAC sta- +bility does not seem possible for most HGs classes, and we +try to find general reasons causing this fact. First, we show a +simple necessary condition for PAC stability, abstracting the +proof pattern of all the known negative results for specific +HGs classes. Then, we consider the problem of PAC stabil- +ity with bounded probability distributions and prove that un- +der this restriction it is possible to PAC stabilize W-games, +which is known not to be possible in general. Finally, we dis- +cuss why the same result cannot be easily extended to other +HGs. In particular, we determine a general purely computa- +tional property necessary for achieving PAC stability. +Due to space limitations, all the missing proofs are de- +ferred to the Appendix. +1.2 +Related Work +Many works have dealt with learning game-theoretic solu- +tion concepts from data. Sliwinski and Zick (2017) first in- +troduced the PAC learning framework into the study of HGs. +Their work was extended by Igarashi, Sliwinski, and Zick +(2019) to tackle HGs with underlying players’ interaction +networks. Moreover, Jha and Zick (2020) laid further foun- +dations for learning game-theoretic solution concepts from +samples. More recently, Trivedi and Hemachandra (2021) +studied the problem of learning HGs with noisy preferences. +Other +works +have +considered +learning +cooperative +games (Balcan, Procaccia, and Zick 2015), markets (Lev +et al. 2021b), auctions (Balcan, Sandholm, and Viter- +cik 2018) but also, more generally, combinatorial func- +tions (Balcan, Vitercik, and White 2016; Balcan 2015). +There is a vast body of literature on HGs. For a thor- +ough introduction to the main concepts and results, we refer +to Aziz and Savani (2016), where both all the HGs classes +studied in this paper and also HCNs are discussed. +2 +Preliminaries +Let N be a set of n players. We call any non-empty subset +S ⊆ N a coalition and denote by Ni the set of all coalitions +which contain a given player i ∈ N. We call any coalition +of size one a singleton. We denote by ≿i any binary prefer- +ence relation of player i over the coalitions in Ni, which is +reflexive, transitive, and complete. A Hedonic Game (HG) +is then a pair H = (N, ≿), where ≿= (≿i, . . . , ≿n) is a +preference profile, i.e., the collection of all players’ pref- +erences. Throughout this work we will assume that play- +ers’ preferences are expressed as real numbers by means of +valuation functions vi. In other words, given S, T ∈ Ni: +vi(S) ≥ vi(T ) if and only if S ≿i T . We will denote by +⃗v = (v1, . . . , vn) the collections of players’ valuations and +assume that vi(S) = ∅ for S /∈ Ni. Let H be a HG and π a +coalition structure, i.e., a partition of players into coalitions. +A set S is said to core-block π if vi(S) > vi(π(i)) for each +i ∈ S, where π(i) denotes the coalition containing i in π. +A coalition structure π is core-stable if there does not exist +a core-blocking coalition S ⊆ N. Among the many possi- +ble solution concepts, the one we will consider in this paper +is core stability, as it is the most prominent one in the PAC +stability model. +2.1 +Defining Classes of Hedonic Games +In this subsection, we provide the definitions of some HGs +classes already considered from the perspective of PAC +learning by (Sliwinski and Zick 2017), that will be fre- +quently mentioned in the sequel. In all of these classes, for a +player i ∈ N and a coalition S ∈ Ni, the valuation vi(S) is +completely determined by the values vi(j) for j ∈ S \ {i}. +More precisely, the valuation of i for S is equal to: +1. Additively Separable: the sum of the values of its mem- +bers, i.e., vi(S) = � +j∈S\{i} vi(j); +2. Fractional: the sum of the values of its members, but +normalized by the size of the coalition, i.e., vi(S) = +� +j∈S\{i} vi(j)/|S|; +3. W-games: the value of the worst player in the coalition; +4. B-games: the value of the best player in the coalition, but +coalitions of smaller size are preferred. +2.2 +PAC Learning +The PAC learning model, originally introduced by Valiant +(1984), mathematically formalizes the process of learning a +target concept v belonging to a hypothesis class H, by us- +ing a sample of labeled examples as input. There are many +variants, which adapt to different learning paradigms. In the +following, we will formally present only the one that we will +use in this work. Our aim is to learn an unknown valua- +tion function v : 2N → R within a class H, given as in- +put S = {(S1, v(S1)), . . . , (Sm, v(Sm))}, i.e., a collection +of coalition/valuations pairs. The distribution D, according +to which the i.i.d. input coalitions are sampled, is unknown, +while the class H is determined by the HG instance one con- +siders, e.g., if one is studying Additively Separable HGs, H +will be the class of additively separable functions over (the +other) n−1 players. Starting from a sample S, learning is the +process of producing a hypothesis v∗ ∈ H which is as close +as possible to the real v. Formally, a hypothesis v∗ ∈ H is +ε-approximately correct w.r.t. a distribution D over 2N and +a function v ∈ H, if the following holds: +Pr +S∼D [ v∗(S) ̸= v(S) ] < ε . +Given ε, δ > 0, class H is (ε, δ) probably approximately +correctly (PAC) learnable if there exists an algorithm A that, + +for every distribution D over 2N, and any v ∈ H, given a +sample drawn from D, is able to produce a hypothesis v∗ +which is ε-approximately correct with probability at least +1 − δ. A class H is said to be PAC learnable if it is (ε, δ) +PAC learnable for all ε, δ > 0. Furthermore, if the sample +size m and the running time of A are polynomial in 1 +ε, log 1 +δ +and n, H is said to be efficiently PAC learnable. +The inherent complexity of efficiently PAC learning a +concept class of real functions H is usually measured by the +so-called pseudo-dimension (see, e.g., Anthony and Bartlett +(2002)), which is the analog of the more renowned VC- +dimension (Kearns and Vazirani 1994) defined only for +classes of binary functions. In order to formally define +pseudo-dimension, we first need to introduce the concept +of pseudo-shattering. Given a collection of coalition/value +pairs S = {(S1, r1), . . . , (Sq, rq)}, we say that a class H +can pseudo-shatter S if, for every possible binary labeling +l1, . . . , lq of S, there exists a function f ∈ H such that +f(Sj) > rj +⇐⇒ +lj = 1. Intuitively, the more H is ex- +pressive, the bigger the sets that it can pseudo-shatter. The +pseudo-dimension of H, denoted as Pdim(H), is the size of +the maximal set S that can be pseudo-shattered by H. +We conclude this section by reporting the theorem which +bridges learning and pseudo-dimension. +Theorem 2.1 (Anthony and Bartlett 2002). A hypothe- +sis class H with Pdim(H) polynomial in n is (ε, δ) PAC +learnable using m samples, where m is polynomial in +Pdim(H), 1 +ε and log 1 +δ , by any algorithm A that returns a hy- +pothesis f ∗ consistent with the sample, i.e., f ∗(Si) = f(Si) +for all i. Furthermore, if Pdim(H) is superpolynomial in n, +H is not efficiently PAC learnable. +2.3 +PAC Stabilizing Hedonic Games +The concept of PAC stabilizing HGs was first introduced +in (Sliwinski and Zick 2017). A coalition structure π +is said to be ε-PAC stable under a distribution D if +PrS∼D [ S core-blocks π ] < ε. A class of HGs H is PAC +stabilizable if there exists an algorithm A that for any HG +in H, any ε, δ > 0, and any D over 2N, given a sample +S = {(S1,⃗v(S1)), . . . , (Sm,⃗v(Sm))} of coalitions drawn +according to D, produces an ε-PAC stable coalition struc- +ture π under D with probability at least 1 − δ, or reports +that the core is empty. If the sample size m and the running +time of A meet the same conditions required for efficient +PAC learnability, we say that H is efficiently PAC stabiliz- +able. Intuitively, this concept formalizes the learnability of +a solution concept for a HGs class, independently from the +learnability of the class itself. We will rely on the following +theorem in the next section. +Theorem 2.2 (Jha and Zick 2020). A class of HGs H is effi- +ciently PAC stabilizable iff there exists an algorithm that out- +puts a partition π consistent with the sample, i.e., no coali- +tion from the sample core-blocks π. +3 +Learnability and Stabilizability of New +Classes of Hedonic Games +In this section we broaden the picture of learnability and sta- +bilizability of different classes of HGs, studying the follow- +HGs class +Learnable +Stabilizable +Friends and Enemies +Friends Appreciation +✓∗ +✓∗ +Enemies Aversion +✓∗ +✓ +Bottom Responsive +✗ +✓ +Anonymous +✓ +✗ +Table 1: A summary of the learnability and stabilizability +landscape discussed in Section 3. Entries marked by an as- +terisk symbol are consequences of previous work. +ing HG classes that were not considered by previous work. +Friends and Enemies. +Friends and Enemies Games have +been traditionally investigated under two types of preference +profiles, called Friends Appreciation and Enemies Aversion, +where agents prefer coalitions with a greater number of +friends (and smaller number of enemies in case of ties) or +with a smaller number of enemies (and greater number of +friends in case of ties), respectively. +Bottom Responsive. +The bottom responsiveness property +was first defined by Suzuki and Sung (2010) as bottom re- +fuseness and then further considered by Aziz and Brandl +(2012), where it was renamed in analogy to a related prop- +erty called top responsiveness. Intuitively, it models pes- +simistic agents who rank coalitions based on sets of players +that they would like to avoid. +Definition 3.1. For each player i ∈ N and S ∈ Ni, we +define the avoid set of player i in coalition S as +Av(i, S) = {S′ ⊆ S : (i ∈ S′) ∧ (∀S′′ ⊆ S, S′ ⪯i S′′)}. +A game satisfies bottom responsiveness if for each i ∈ N +and for each pair S, T ∈ Ni the following conditions hold: +(i) if for each S′ ∈ Av(i, S) and for each T ′ ∈ Av(i, T ) it +holds that if S′ ≻i T ′, then S ≻i T ; +(ii) if Av(i, S) ∩ Av(i, T ) ̸= ∅ ∧ |S| ≥ |T |, then S ⪰i T . +In what follows, we assume a minimum a priori +knowledge of the values. Namely, we assume to know +vi({i}), ∀i ∈ N. A similar, yet significantly stronger, as- +sumption was used by (Sliwinski and Zick 2017) to prove +that Top Responsive HGs (i.e. HGs which satisfy top respon- +siveness) are efficiently PAC stabilizable. +Anonymous. +A HG is said to satisfy anonimity, as defined +in (Banerjee, Konishi, and S¨onmez 2001; Bogomolnaia and +Jackson 2002), if vi(S) = vi(T ) for any player i ∈ N and +any S, T ∈ Ni with |S| = |T |, i.e., players evaluate coali- +tions only according to their size. +We are now ready to state the following theorem, summa- +rizing our results for the just defined HGs classes. +Theorem 3.2. The results described in Table 1 hold. +Proof. We give here just a sketch of the proof. while the full +version can be found in Appendix A. +Friends and Enemies. +The efficient PAC learnability of +both Friends Appreciation and Enemies Aversion profiles + +follows directly by observing that they are both subclasses +of Additively Separable HGs (see (Dimitrov et al. 2006)), +known to be efficiently PAC learnable by the results of Sli- +winski and Zick (2017). +For what concerns stabilizability, Suzuki and Sung (2010) +showed that Friends and Enemies Games under Friends Ap- +preciation are a subclass of Top Responsive HGs. Sliwinski +and Zick (2017) proved that Top Responsive HGs are ef- +ficiently PAC stabilizable, which then implies the same for +Friends Appreciation. For Enemies Aversion, Dimitrov et al. +(2006) prove that core-stable partitions always exist, while +Dimitrov and Sung (2004) provide an algorithm returning +such a partition. Inspired by their algorithm, we provide an +algorithm PAC stabilizing this class. +Bottom Responsive. +To show that the class is not efficiently +PAC learnable, we prove that its pseudo-dimension is lower +bounded by 2 +n−1 +2 , and thus is exponential in n. The result +then follows by Theorem 2.1. The construction in our proof +bears similarities to the one of Sliwinski and Zick (2017) for +Top Responsive HGs. +Regarding stabilizability, we first observe that Suzuki and +Sung (2010) show that a core-stable coalition structure al- +ways exists for this class. Moreover, a simple necessary con- +dition for S to be part of a core-stable partition π, is that for +all i ∈ S it must hold that {i} ∈ Av(i, S). Indeed, if this +condition is not satisfied, at least one player prefers to devi- +ate to a singleton. To give a viable alternative for checking +the condition while knowing the values of the singletons, +we prove the following property: Given a Bottom Respon- +sive HG H = (N, v), for every i ∈ N and every S ∈ Ni, it +holds that {i} ∈ Av(i, S) ⇐⇒ vi({i}) ≤ vi(S). Starting +from this property, we construct Algorithm 1 which, given a +sample, returns a coalition structure that is not core-blocked +by any coalition from the sample. By Theorem 2.2 this is +sufficient for concluding the efficient PAC stabilizability. +Anonymous. To show efficient PAC learnability, we prove +that the pseudo-dimension of this class is upper bounded by +n(1 + log n), and thus is polynomial. Then, for each i ∈ N, +the following procedure computes a hypothesis v∗ +i consis- +tent with the sample in time polynomial in n and m: For +every coalition C of size k ∈ [n], if there exists Sj s.t. +i ∈ Sj and |Sj| = k, then set vi(C) = vi(Sj), otherwise +set vi(C) = −∞. +For what concerns stabilizability, we can provide a +counter-example showing that the class is not PAC stabiliz- +able, even in the case of natural single-peaked preferences, +where every player has a given preferred size, and the valua- +tion decreases as the distance from such size increases. +Notice that, according to the above theorem, the nega- +tive results on stabilizability of Additively Separable HGs +of (Sliwinski and Zick 2017) do not transfer to Friends and +Enemies Games. Furthermore, while the Bottom Reponsive +HGs class is not PAC learnable but efficiently PAC stabiliz- +able, exactly the opposite holds for Anonymous HGs. +Algorithm 1: Stabilizing Bottom Responsive HGs +Input: N, S = {(Sj,⃗v(Sj))}m +j=1 +Output: π: an ε-stable partition of N +1 π ← ∅, T ← ∅ +2 for ⟨S,⃗v(S)⟩ ∈ S do +3 +f ← 1 +4 +if ∃i ∈ S s.t. vi(S) < vi({i}) then f ← 0 break +5 +if f=1 then T ← T ∪ {S} +6 +while T ̸= ∅ do +7 +T + ← argmaxT ∈T |T \ � +P ∈π P| +8 +π ← π ∪ +� +T + \ � +P ∈π P +� +9 +T ← T \ {T +} +10 +N ← N \ T + +11 +for i ∈ N do π ← π ∪ {{i}} +12 return π +4 +A General Framework for Learnability: +Hedonic Coalition Nets +To provide a general unifying framework for learnability +of HGs, a direction worth investigating is the one of de- +termining a suitable superclass or a small number of super- +classes encompassing all the learnable HG classes. Such re- +sults would contribute to the general understanding of the +crucial properties leading to learnability, or lack thereof, and +would also provide means to easily determine whether a spe- +cific class of HGs is learnable. +A universal HGs class, maintaining the full expressive- +ness for representing any HG, is the one of the so-called +Hedonic Coalition Nets (HCNs) (Elkind and Wooldridge +2009). Before giving the definition, we note that, since there +exist classes of HGs that are not learnable, it is not possible +to get a positive result for the learnability of any fully ex- +pressive HGs representation, so not for HCNs either, without +imposing further restrictions. Thus, our goal here is to deter- +mine suitable restrictions allowing for efficient learnability. +Definition 4.1. A hedonic coalition net (HCN) is a tuple +(N, R1, . . . , Rn) where N is a set of variables (each corre- +sponding to a player) and Ri is the set of rules for player +i. A single rule in Ri is given by a pair (φ, β), where φ is a +formula of propositional logic over N and β ∈ R is a real +number. We will denote a rule in Ri by φ �→i β. Then, as- +suming the conventionalsemantic satisfaction relation “|=”, +the valuation of player i for a coalition S ∈ Ni is +vi(S) = +� +φj�→iβj∈Ri: +S|=φj +βj. +(1) +The first HCNs subclass we consider comprises HCNs in +which the formulas appearing in each set of rules Ri are +known a priori. Namely, for any rule φ �→i β ∈ Ri we only +need to learn β. We first show that, in this case, the pseudo- +dimension depends on the number of rules. +Proposition 4.2. Let H(Ri) be the class of valuation func- +tions that can be expressed with a fixed set of a priori known +distinct rules Ri. Then, Pdim (H(Ri)) = O (|Ri|). + +Proof. Let r = |Ri|. We will show that no set of size +r + 1 can be pseudo-shattered by H(Ri). As a conse- +quence Pdim (H(Ri)) ≤ r, which implies the result. Let +S = {Sj}r+1 +j=1 be any set of coalitions from Ni of size r + 1, +and (t1, . . . , tr+1) any sequence of r+1 real numbers. Given +any labeling l, the condition vi(Sj) > tj ⇔ ℓj = 1 can be +written as a system of r + 1 linear inequalities of the form +r +� +k=1 +ajkβk > tj if ℓj = 1, and +r +� +k=1 +ajkβk ≤ tj if ℓj = 0, +where ajk = 1 if Sj |= φk and 0 otherwise. This is a system +of r + 1 inequalities with r unknowns β1, . . . , βr, thus the +coefficient matrix A = (ajk) must have linearly dependent +rows. Let us w.l.o.g. assume that the last row Ar+1 can be +written as Ar+1 = �r +j=1 yjAj where the coefficients yj are +not all null. Let us define the labelings ℓ(1), ℓ(2) in this way: +ℓ(1) +j += 1 ⇔ yj < 0 for j ∈ [r], ℓ(1) +r+1 = 1 and ℓ(2) +j += 0 ⇔ +ℓ(1) +j += 1. By contradiction, assume that there exist solutions +⃗b1 and ⃗b2 that satisfy the respective systems of inequalities. +Let us consider the first system. By definition of ℓ(1) and +⃗b1, if ℓ(1) +j +< 0 then +� +Aj ·⃗b1 +� +> tj but yj < 0 implying +that yj +� +Aj ·⃗b1 +� +< yjtj. When ℓ(1) +j += 0, instead, it holds +that yj +� +Aj ·⃗b1 +� +≤ yjtj. We can then conclude that this +last inequality holds for all j ∈ [r]. Regarding ℓ(2), with the +same line of reasoning one can prove that yj +� +Aj ·⃗b2 +� +≥ +�r +j=1 yjtj for all j ∈ [r]. Writing Ar+1 as a combination of +the other rows, and including the inequalities associated to +Sr+1, we obtain the following: +tr+1 < Ar+1 ·⃗b1 = +r +� +j=1 +yj +� +Aj ·⃗b1 +� +≤ +r +� +j=1 +yjtj +tr+1 ≥ Ar+1 ·⃗b2 = +r +� +j=1 +yj +� +Aj ·⃗b2 +� +≥ +r +� +j=1 +yjtj +implying tr+1 < �r +j=1 yjtj ≤ tr+1, a contradiction. +We say that H admits a compact HCN representation if +it is possible to represent every v ∈ H with a polynomial +number of rules for each player i. Observe that so far ev- +ery class that has been shown to be learnable, also admits +a compact HCNs representation. Elkind and Wooldridge +(2009) give HCNs representations for Additively Separable, +Anonymous, W and B-games. We describe these represen- +tations and provide one for Fractional HGs in Appendix B. +The following result shows that HGs admitting a compact +HCN representation, for which we know the formulas a pri- +ori, are efficiently PAC learnable. +Theorem 4.3. Let H be a class of HGs that admits a com- +pact HCN representation. Suppose that for every set of rules +Ri, we know the corresponding set of formulas Φ. Then, H +is efficiently PAC learnable. +While the class presented above includes Additively Sep- +arable, Fractional, and Anonymous HGs, which have all +been shown to be efficiently PAC learnable, there exist other +learnable classes which do not fall within the above charac- +terization. Indeed, for W-games and B-games, knowing the +φ for each rule a priori is not possible, since the formulas +themselves depend on the ordered preferences that we need +to learn. On the other hand, the maximum number of distinct +coalition values in both cases is only n. +To capture these remaining classes of learnable HGs +through another suitable subclass of HCNs, we resort to de- +cision lists, which were introduced by Rivest (1987) as al- +ternative representations for Boolean functions. +Definition 4.4. A decision list (DL) L is defined by a set of l +rules L = {(κ1, b1), . . . , (κl, bl)} such that κi is a conjunc- +tion of literals, bi ∈ {0, 1}, ∀i ∈ [l], and κl is the constant +function true. Given L and a truth assignment x, L(x) is +equal to bj where j is the least index such that κj(x) = 1. +We use the term k-decision lists (k-DL) if all the conjunc- +tions in the DL are of size at most k. +For our purposes, for any coalition S and i ∈ [l], κi(S) = +1 if S |= κi and κi(S) = 0 otherwise. It is convenient to +think of a DL as an “if – then – else if – ... - else –” rule. +if κ1(S) = 1 then return b1 +else if κ2(S) = 1 then return b2 +. . . +else return bl +Note that all the HGs classes we mentioned, other than +Anonymous HGs (see Appendix C), can be represented as +k-DL with k constant. It is known that for constant k, k-DL +are efficiently PAC learnable: Rivest (1987) shows an effi- +cient learning procedure LEARN-k-DL(k, S) that takes the +size k and a sample S as input and returns a k-DL L (see Ap- +pendix D). Furthermore, in the same work it is proven that +k-DL are strictly more expressive than k-CNF and k-DNF +formulas, and decision trees of depth k, meaning that every +Boolean function that is representable in one of these forms +admits a representation as a k-DL, but not viceversa. +Now, if we consider HCNs which contain rules that are +represented by k-DL and additionally restrict our attention +to representations in which every coalition satisfies exactly +one rule, it turns out that we can again efficiently PAC learn +the valuations, as shown in the following. +Theorem 4.5. Let H be a class of HGs that admits a HCN +representation such that +(i) every coalition S ∈ Ni satisfies exactly one rule in Ri, +(ii) every rule is of the form L �→ β, where L is a k-DL with +k constant, and β unique, i.e., no pair of distinct rules +have the same value β. +Then, H is efficiently PAC learnable by Algorithm 2. +While the second assumption in Theorem 4.5 seems rather +strong, we argue that asking for unique values β actually +does not impose a further restriction, even though it seems +fundamental for proving the result (see Appendix D). In- +deed, if there is more than one k-DL associated with the + +Algorithm 2: Learning HCN in k-DL form +Input: k ∈ N, S = {(Sj, vi(Sj))}m +j=1 +Output: Ri consistent with S +1 Ri ← ∅ +2 for β in {vi(Sj) : Sj ∈ S} do +3 +for Sj ∈ S do +4 +if vi(S) = β then bj = 1 else bj = 0 +5 +S′ = {(Sj, bj)}m +j=1 +6 +L ← LEARN-k-DL(k, S′) +7 +Ri ← Ri ∪ {L �→ β} +8 return Ri +same value β, using the assumption that every coalition sat- +isfies exactly one rule, it is always possible to merge them +into one k-DL. +Theorem 4.5 includes as a special case all HGs that can be +represented by sets of mutually exclusive conjunctions, each +containing at most k positive literals. This is so, because we +can phrase the negative literals positively within the DL, by +associating the presence of such a variable with returning 0. +Thus, the conjunction size depends only on the number of +positive literals. The case k = 1 includes W- and B-games. +5 +Stabilizability of Hedonic Games +We start this section by identifying a property that a HGs +class needs to satisfy if it has any ambitions of being PAC +stabilizable. To this end, we first define the set of core stable +partitions w.r.t. a fixed sample S, and equivalence classes in +a HGs class H w.r.t. a fixed sample S. Then, we state a theo- +rem that abstracts on the arguments used in proofs showing +that a specific HGs class is not PAC stabilizable. Recall that +a sample S is a set {(S1,⃗v(S1)), . . . , (Sm,⃗v(Sm))}. +Definition 5.1. Let H be a class of HGs, S a sample and let +H ∈ H. We denote by: +(i) CS(H) = {π : ∀S ∈ S, S does not core block π}, the +set of partitions consistent with the sample S; +(ii) H[S] the set of all instances H = (N,⃗v ′) ∈ H such that +⃗v ′(S) = ⃗v(S), for each (S,⃗v(S)) ∈ S. +We are now able to define the following property. +Definition 5.2. HGs class H satisfies the sample resistant +core property, or has SRC in short, if for every S ⊆ 2N +• CS(H) = ∅, ∀H ∈ H[S], or +• � +H∈H[S] CS(H) ̸= ∅. +Theorem 5.3. If H is PAC stabilizable, then H has SRC. +Notice that Theorem 5.3 (whose proof can be found in +Appendix 5) formalizes the standard approach of Sliwinski +and Zick (2017) and of our work, to show that a specific HG +class is not PAC stabilizable. Furthermore, property SRC, +which is of course satisfied by Top and Bottom Responsive +HGs (as they can be PAC stabilized), does not seem to be a +common HGs property. +One could argue that aiming to PAC stabilize a specific +HG class without having any a priori knowledge on the dis- +tribution D is too ambitious. Thus, a natural question is +whether restricting the attention to special distributions in- +creases the prospect of stabilizing some classes of HGs. This +direction was left as an open question by Sliwinski and Zick +(2017) and is our focus in the remaining part of the paper. +The motivation for limiting the scope of allowed distribu- +tions is to get a more fine-grained insight into PAC stabil- +ity. The simple counterexamples from (Sliwinski and Zick +2017), while providing valuable understandings, do not re- +veal “how far away” from achieving PAC stability certain +classes of HGs are. Thus, we proceed by studying PAC sta- +bility under a class of distributions that excludes the usual +adversarial examples. In particular, we focus on distributions +having a fair amount of probability mass on all coalitions. +Definition 5.4. A distribution D is said to be bounded if +there exists λ ≥ 1 such that, for every two coalitions S1, S2, +it holds that PrD [ S1 ] ≤ λ PrD [ S2 ]. +Observe that the uniform distribution is a special case of +the above definition, obtained by setting λ = 1. +A useful property that we will use extensively in our cal- +culations is that, if D is bounded with a factor λ, then +1 +λ2n ≤ +1 +λ(2n − 1) ≤ Pr +S∼D [ S ] ≤ +λ +2n − 1 . +(2) +These simple bounds follow from the definition of a +bounded distribution and the fact that � +T ∈2N PrD [ T ] = 1, +where the sum goes over the 2n − 1 non-empty coalitions. +By Equation (2), every coalition now has a positive proba- +bility of being sampled. Since the counterexamples to PAC +stabilizability of specific HGs classes usually rely on ad hoc +distributions where most of the coalitions are never sampled, +this feature provides hope of obtaining better results. +5.1 +W-games under Bounded Distributions +As a case study we consider W-games with no ties. This +class admits a polynomial algorithm for finding a core sta- +ble partition (Cechl´arov´a and Hajdukov´a 2004), but, despite +that, it has been shown not to be PAC stabilizable (Sliwinski +and Zick 2017). Thus, it seems a natural first candidate for +being PAC stabilizable under bounded distributions. In the +rest of this subsection, we indeed show the following result. +Theorem 5.5. W-games under bounded distributions are +efficiently PAC stabilizable. +To this end, in what follows, when focusing on a fixed +player i, w.l.o.g. we assume that the other players are or- +dered such that vi(1) < vi(2) < . . . < vi(n − 1). We start +by exploiting the fact that the distribution is bounded. +Lemma 5.6. Let ε > 0 be fixed. If we denote by A(i) +j +the +event that a sampled coalition S satisfies i, j ∈ S and S \ +{i, j} ⊆ {j + 1, . . . , n − 1}, and by B(i) +j +the event that +a sampled coalition S satisfies i, j ∈ S and S \ {i, j} ⊆ + +Algorithm 3: Stabilizing W-games +Input: N players, S = {(Sj,⃗v(Sj))}m +j=1, ε > 0 +Output: A partition π +1 ⃗v∗ ← LEARN-W-GAMES(N, S) +2 π ← ∅ +3 while N ̸= ∅ do +4 +Pick i ∈ N +5 +if N \ {i} ̸= ∅ then +6 +j ← argmaxk∈N\{i} v∗ +i (k) +7 +π ← π ∪ {{i, j}}, N ← N \ {i, j} +8 +else π ← π ∪ {{i}} +9 return π +�� +log2 +1 +ε +� ++ 2, . . . , n − 1 +� +, it holds that +Pr +S∼D +� +A(i) +j +� +≥ ε +2λ +for 1 ≤ j ≤ +� +log2 +1 +ε +� +, and +Pr +S∼D +� +B(i) +j +� +≥ ε +4λ +for j > +� +log2 +1 +ε +� +. +The proof of Lemma 5.6 and all the others missing proofs +of this subsection can be found in Appendix 5). Sliwinski +and Zick (2017) presented a simple procedure, that we will +refer to as LEARN-W-GAMES, which takes in input the set +of players and a sample, and returns a consistent estimate +⃗v∗ for the players’ valuations in W-games. This procedure +sets v∗ +i (j) to be the maxS∈Sij vi(S) where Sij = {S ∈ S : +{i, j} ⊆ S}, if Sij is non-empty, −∞ otherwise. Next, we +define what we call an ε-estimate of a function, and show +that the output of LEARN-W-GAMES is actually such an es- +timate. +Definition 5.7. Function v′ +i is an ε-estimate of vi if +� +v′ +i(j) = vi(j) +for 1 ≤ j ≤ +� +log2 +1 +ε +� +, and +v′ +i(j) > vi( +� +log2 +1 +ε +� +) +for j > +� +log2 +1 +ε +� +. +(3) +Proposition 5.8. Let ε, δ > 0 and S be a sample of size m. +If m ≥ 2λ +ε log n2 +δ , LEARN-W-GAMES returns an ε-estimate +⃗v∗ of ⃗v with confidence 1 − δ. +We want to show that, by relying on the ε-estimate given +by LEARN-W-GAMES for ε “not too small”, Algorithm 3 +returns an ε-stable partition. We first state a technical lemma. +Lemma 5.9. Let π be the output of Algorithm 3 and let us +call a player i green if it is not in a coalition with one of his +� +log2 +1 +ε +� +least preferred choices according to v∗ +i . Then, +a) for i green, PrS∼D [ i ∈ S ∧ vi(S) > vi(π(i)) ] < λε, +b) for ε ≥ +3� +λ2 +2n , PrS∼D [ S does not contain green i ] < ε. +We are now finally ready to prove the main theorem, +stated in the beginning of this subsection. +Proof of Theorem 5.5. For δ > 0 and ε ≥ +3� +λ5 +2n−3 , we +call Algorithm 3 with ε′ = ε/2λ and m ≥ +2λ +ε′ log n2 +δ += +1 +ε log n2 +δ , and obtain an ε-stable partition with probability at +least 1 − δ. By point a) of Lemma 5.9, a green node has +probability < λε′ = ε/2 to get a better outcome by moving +from π(i) to S. Furthermore, ε′ satisfies the requirement of +point b) in Lemma 5.9, so the probability of sampling an S +without a green node is < ε′ ≤ ε/2. In conclusion, if we +call G the event that S contains a green player, then since +PrS∼D [ S core blocks π ] equals +Pr +S∼D [ S core blocks π | G ] + Pr +S∼D +� +S core blocks π | G +� +, +we see that PrS∼D [ S core blocks π ] ≤ ε/2 + ε/2 = ε. +For δ > 0 and ε < +3� +λ5 +2n−3 , with a sample of size +m ≥ 8λ6 +ε3 log n2 +δ , we can reveal the exact valuation functions +with probability at least 1 − δ and return the core stable par- +tition π using the algorithm of Cechl´arov´a and Hajdukov´a +(2004). Indeed, since the probability of drawing any coali- +tion is by Equation (2) at least 1/λ2n, this also holds for the +coalition containing only agents i and j, which provides both +vi(j) and vj(i). The probability of not drawing a particular +coalition of size 2 is ≤ (1 − 1/λ2n)m ≤ e−m/λ2n ≤ δ/n2. +Taking a union bound over all the +�n +2 +� +< n2 coalitions of +size 2, we see that the probability of not seeing all the exact +valuations is upper bounded by δ. +5.2 +Barriers to the Restricted Distributions +Approach +Encouraged by the positive results of the last subsection, one +could try to extend the approach of focusing on bounded dis- +tributions in the hope that other classes that are known not +to be PAC stabilizable, such as Additively Separable, Frac- +tional, and Anonymous HGs, are in fact stabilizable under +such distributions. Unfortunately, this does not seem to be +always the case, as we discuss below. +Definition 5.10. For a HGs class H , let T (H) be the time +complexity of the best algorithm solving the core for this +class, i.e., the runtime of the fastest algorithm that for every +input instance either correctly replies that the core is empty +or returns a core-stable partition. +Theorem 5.11. If T (H) = ω(poly(2n)) for a HG class H, +then H is not efficiently PAC stabilizable, even under the +uniform distribution. +Notice that the assumption T (H) ∈ ω(poly(2n)) is not +that strong. In most of the HGs classes the complexity of the +problem of deciding the existence of the core is either Σp +2- +complete or NP-hard. Although this does not imply that it +is not possible to find a O(poly(2n)) algorithm (in the case +of the total collapse of the polynomial hierarchy, this would +even be possible in polynomial time), such algorithms are +currently not known and at this point it seems that finding +them is unlikely. In particular, the brute force approach, that +searches for an element in the core by examining all the pos- +sible partitions, has a running time of Ω((n/2)n/4) (as this +is one possible lower bound on the Bell number), and thus +its running time is also in ω(poly(2n)). + +6 +Conclusions +In this work, we initiated the study of a unified approach +for determining the learnability and stabilizability of spe- +cific HGs classes. One of the obvious goals for future work +is finding a unique characterization of HCN representations +that imply learnability. Another one is exploring further con- +sequences of Theorem 5.11 and expanding the knowledge +on the exact computational complexity of solving the core +for the different classes of HGs. +Acknowledgements +Giovanna Varricchio was supported by DFG grant Ho +3831/5-1. The authors would also like to thank the anony- +mous reviewers for their comments and suggestions. +References +Anthony, M.; and Bartlett, P. 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We will show that: +(i) Friends and Enemies Games are efficiently PAC learn- +able and stabilizable; +(ii) Bottom Responsive HGs are not PAC learnable but effi- +ciently stabilizable, if we require as a baseline to know +the values v(i) for each i ∈ N; +(iii) Anonymous HGs are efficiently PAC learnable but not +stabilizable, even in the case of natural single-peaked +preferences. +A.1 +Friends and Enemies Games +Since the learnability for both Friends Appreciation and En- +emies Aversion profiles and the stabilizability for Friends +Appreciation are a consequence of previous results, in the +following we will focus on proving efficient PAC stabiliz- +ability for Enemies Aversion profiles. +For this type of preferences, Dimitrov et al. (2006) proved +that core stable partitions always exist. An algorithm for +computing a core stable partition was provided by Dimitrov +and Sung (2004), which we describe here only informally. +Initially all the agents are “unassigned”. At every step a new +coalition is created, consisting of the maximum subgroup of +still unassigned agents that all consider each other friends. +Once a new coalition is created, its members are marked as +“assigned”. This process continues as long as there are unas- +signed agents. Singleton coalitions may be created once no +larger set of agents forms a friendship clique. The just de- +scribed algorithm clearly has an exponential running time. +In fact, due to its close relation to the MAXCLIQUE problem, +computing a core stable partition in this setting is known to +be NP-hard (Dimitrov et al. 2006). +Here, we present Algorithm 4 which PAC stabilizes this +class. Our algorithm, inspired by the one of Dimitrov and +Sung (2004), stores in T the sampled coalitions that are +cliques (lines 2-11). Then, it repeatedly extracts a maximum +clique from T and creates a corresponding coalition (lines +12-17). Every time a coalition is formed, since any subset +of a clique is a clique, the algorithm refines the sets in T +by removing agents which have been assigned to a coali- +tion. This process takes place as long as T ̸= ∅. If there are +agents which have not been assigned to any coalition, they +are placed into singletons (lines 18-20). +Let H = (N, v) be a Friends and Enemies Game under +Enemy Aversion. To prove that Algorithm 4 stabilizes H, it +is enough to prove that the partition π output by the algo- +rithm cannot be core blocked by any of the coalitions inside +the sample. Then the result follows by Theorem 2.2. To this +end, observe that any coalition S from the sample ends up +in the candidate set T if and only if all the members of the +coalition consider each other to be friends. Furthermore, the +all-friends coalitions are added to π by cardinality, starting +from the largest. Now, first notice that a coalition S that is +not an all-friends coalition, cannot block π. Furthermore, for +an all-friends coalition S from the sample, it is not possible +that |S| > π(i) for all i ∈ S, concluding the statement. +Notice that the exact valuation function values are not im- +portant for Algorithm 4, since any valuation function that +respects Enemy Aversion will have a negative value as soon +as the coalition contains at least one enemy relation. +Algorithm 4: Stabilizing Enemy Aversion +1 Input: N, S = {(Sj,⃗v(Sj))}m +j=1 +2 Output: π: an ε-stable partition of N +1: π ← ∅, T ← ∅ +2: for ⟨S,⃗v(S)⟩ ∈ S do +3: +f ← 1 +4: +if ∃i ∈ S s.t. vi(S) < 0 then +5: +f ← 0 +6: +break +7: +if f=1 then +8: +T ← T ∪ {S} +9: while T ̸= ∅ do +10: +T + ← argmaxT ∈T |T \ � +P ∈π P| +11: +π ← π ∪ +� +T + \ � +P ∈π P +� +12: +T ← T \ {T +} +13: +N ← N \ T + +14: for i ∈ N do +15: +π ← π ∪ {{i}} +16: return π +A.2 +Bottom Responsive HGs +Non-Learnability. +Consider the class R of HGs such that, +for S, T ∈ Ni, |S| < |T | ⇒ vi(S) > vi(T ). First, notice +that class R satisfies bottom responsiveness. Indeed, as for +each S ∈ Ni it holds that Av(i, S) = {S}, it is easy to +check that conditions (i) and (ii) hold. +Let now X = {S ∈ Ni : |S| = +� n +2 +� +}. In particular, for +S1, S2 ∈ X, it holds that Av(i, S1) ∩ Av(i, S2) = {S1} ∩ +{S2} = ∅. The cardinality of X is +|X| = +� n +� n +2 +� +� +> +� n +n−1 +2 +� +> +�n − 1 +n−1 +2 +� +> 2 +n−1 +2 +where the last inequality follows from the fact that 2m = +�m +k=1 +�m +k +� +< �m +k=1 +�m +k +�� m +m−k +� += +�2m +m +� +. +In what follows, we show that S = {(S, r(S))}S∈X is +pseudo-shattered by R, for any sequence of real numbers +{r(S)}S∈X. Indeed, given any labeling ℓ : X → {0, 1}, +we are free to choose the vi(S) in such a way that vi(S) > +r(S) ⇔ ℓ(S) = 1 is satisfied, as long as we respect the +following: +min +T :|T |<⌈ n +2 ⌉ +vi(T ) > max +S∈X vi(S), +max +T :|T |>⌈ n +2 ⌉ +vi(T ) < min +S∈X vi(S). +As a consequence Pdim(R) > |X| > 2 +n−1 +2 , which implies +the result by Theorem 2.1. +Stabilizability. +Let H = (N, v) be a Bottom Responsive +HG. First we will prove that the following condition holds +for every i ∈ N and S ∈ Ni. +{i} ∈ Av(i, S) ⇐⇒ vi({i}) ≤ vi(S). +(4) + +One direction follows by the definition of the avoid set. Now, +assume that vi({i}) ≤ vi(S). By contradiction, let us as- +sume that {i} /∈ Av(i, S) and let T ∈ Av(i, S). By the defi- +nition of the avoid set, we know that vi(T ) < vi({i}). Since +this holds for every T ∈ Av(i, S) and the game is bottom re- +sponsive, since Av(i, {i}) = {{i}}, by (i) of Definition 3.1 +it follows that vi(S) < vi({i}), a contradiction. +Let H = (N,⃗v) a Bottom Responsive game for which we +know v(i) for each i ∈ N. In the following, we will show +that Algorithm 1 PAC stabilizes H. Again, by Theorem 2.2, +it is sufficient to prove that it returns a partition which is +consistent with the sample i.e. that is not core-blocked by +any sampled coalition. +Since we assume that the values assigned by the players +to singletons are known, Algorithm 1 uses the equivalence +from Equation (4) to sort out every element of the sam- +ple that does not satisfy the necessary condition to be in a +core-stable partition. The remaining sets after this operation +are stored in T . Observe that any coalition S core block- +ing π must satisfy both {i} ∈ Av(i, S) and |S| > |π(i)|, +∀i ∈ S. The former is, as already remarked, a necessary +condition for being in a core stable partition, while the lat- +ter is a consequence of this condition and the requirement +(ii) in Definition 3.1. Now, consider any S ∈ S. Indeed, if +{i} ∈ Av(i, S) for all i ∈ S, then the algorithm adds S to +T . Furthermore, the coalitions in π are created from T by +maximising the cardinality of the chosen set at each step, +which makes it impossible for S ∈ T to satisfy |S| > |π(i)| +for all i ∈ S. +A.3 +Anonymous HGs +Recall that a HG is said to satisfy anonimity, as defined +in (Banerjee, Konishi, and S¨onmez 2001; Bogomolnaia and +Jackson 2002), if vi(S) = vi(T ) for any player i ∈ N and +any S, T ∈ Ni with |S| = |T |. That is, players evaluate +coalitions only according to their size. For this reason, we +will write s1 ⪰i s2 to denote that player i prefers size s1 to +size s2. +Learnability. +We will first prove that the pseudo-dimension +of v is bounded by n(log n + 1). Let S1, . . . , Sm be a +list of subsets of N and r1, . . . , rm a list of real val- +ues. Since in Anonymous HGs the valuation of a coalition +depends only on its size, we will not be able to shatter +(⟨S1, r1⟩, . . . , ⟨Sm, rm⟩) if the list of sets contains at least +two coalitions of the same size that have a non-empty inter- +section. Indeed, if we w.l.o.g. assume that |S1| = |S2| and +S1 ∩ S2 ̸= ∅, we see that for any labelling in which ℓ1 ̸= ℓ2 +we arrive at a contradiction. This is so, because for an agent +i ∈ S1 ∩ S2 it has to hold that vi(S1) = vi(S2) and at +the same time l1 ̸= l2 implies either vi(S1) > vi(S2) or +vi(S1) < vi(S2). What now remains is to see that any list of +sets of length at least n(log n+1) contains at least one pair of +sets with the same cardinality and a non-empty intersection. +To this end, notice that there are n sets of cardinality 1 with +an empty intersection, ⌊n/2⌋ sets of cardinality 2 with an +empty intersection, and generally ⌊n/k⌋ sets of cardinality k +with an empty intersection. Since, n+⌊n/2⌋+· · ·+⌊n/(n− +1)⌋+⌊n/n⌋ ≤ n+n/2+· · ·+n/(n−1)+1 ≤ n(log n+1), +we see that the pseudo-dimension of the class of Anonymous +HGs is bounded by n(log n + 1). +What remains to be proven, is that we can give a con- +sistent hypothesis in polynomial time. The following proce- +dure computes a hypothesis v∗ +i consistent with the sample in +time polynomial in n and m. For every coalition C of size +k ∈ [n], if there exists Sj s.t. i ∈ Sj and |Sj| = k, then set +vi(C) = vi(Sj), otherwise set vi(C) = −∞. +Non-Stabilizability. The anonymity condition is often com- +plemented by the so called single-peakedness, where every +player has a given preferred size, and the valuation decreases +as the distance from such size increases. Formally: +Definition A.1. An instance of Anonymous HGs is single- +peaked if there exists a permutation (s1, . . . , sn) of +{1, . . ., n} in which every player i ∈ N admits a peak p(i) +such that j < k ≤ p(i) or j > k ≥ p(i) imply sk ⪰i sj. +Moreover if the permutation is the identity function, we say +that the preference is single peaked in the natural ordering. +In the following we will prove that Anonymous HGs are +not PAC stabilizable, even if we restrict our attention to +single-peaked instances in the natural ordering. +Let +us +consider +instances +with +7 +agents +N += +{a1, a2, a3, a4, b1, c1, c2} where a,b and c are the agent +types, and agents of the same type have the same prefer- +ences over coalition sizes. Consider a distribution D that se- +lects uniformly at random a coalition in {S ⊆ N : if |S| = +5, then c1, c2 ̸∈ S}. In other words, the distribution never +samples coalitions of size 5 containing c1 or c2. In partic- +ular, this means that we will not know where coalitions of +size 5 appear in the preference list of type c. +Let I1 be the instance in which the agents’ types have the +following preferences over coalition sizes: +Type a: 6 ≻a 5 ≻a 4 ≻a 3 ≻a 2 ≻a 1 ≻a 7, +Type b: 5 ≻b 4 ≻b 3 ≻b 6 ≻b 2 ≻b 1 ≻b 7, +Type c: 4 ≻c 3 ≻c 5 ≻c 6 ≻c 2 ≻c 1 ≻c 7. +The preferences in I1 are single peaked in the natural or- +dering. Moreover, Banerjee, Konishi, and S¨onmez (2001) +showed that instance I1 has an empty core. In particular, +coalitions of size 5 containing c1 or c2 are never used as +blocking coalitions. Therefore, a PAC stabilizing algorithm +applied on I1 should always report that the core is empty. +Let us now consider a slightly different instance I2, +obtained by changing the preferences of agents c1 and +c2 to type b. Preferences in I2 are still single peaked +in the natural ordering, but now the partition π += +{{a1, a2}, {a3, a4, b1, c1, c2}} is core stable. +On the one hand, since the distribution D never samples +coalitions of size 5 containing c1 or c2, no algorithm is able +to distinguish between I1 and I2. On the other hand, I1 has +an empty core and I2 does not. Note that in instance I1, for +any partition π there exists at least one blocking coalition +in the support of D. Since the probability of selecting any +coalition in the support of D is fixed (a positive constant), +the algorithm indeed has to report that the core is empty. +Thus, no algorithm can PAC stabilize this HGs class. + +B +Expressing Classes of Hedonic Games as +Hedonic Coalition Nets +In this section, we will show how it is possible to express as +HCN the classes of HGs mentioned in the paper. These rep- +resentations have all been given by Elkind and Wooldridge +(2009), except for the Fractional HGs one. For Anonymous +HGs, only the existence of the formulas used in the rules is +mentioned by Elkind and Wooldridge (2009). Writing these +formulas explicitly is a non-trivial task that is, for interested +readers, discussed thoroughly in Appendix C. In the follow- +ing, we consider a player i and describe how to write the +rules in Ri. +1. Additively Separable: xj �→i vi(j), ∀j ̸= i. +2. Anonymous: formulas φk, k ∈ [n] such that a subset of +variables ξ satisfies φk iff |ξ| = k; φk �→i vi(k), ∀k ∈ [n] +are known to exist and to have polynomial length in the +number of variables. For further details see Appendix C. +3. Fractional: assuming φk, k ∈ [n] as above; xj ∧ φk �→i +vi(j)/k for 2 ≤ k ≤ n, ∀j ̸= i. +4. W-games: denote by ij the j-th player in i’s prefer- +ence list (in descending order); xn−1 +�→i +vi(in−1), +xn−2∧¬xn−1 �→i vi(in−2), xn−3∧¬xn−1∧¬xn−2 �→i +vi(in−3) and so on. +5. B-games: denote by ij the j-th player in i’s preferences +(in descending order); x1 �→i vi(i1), x2 ∧¬x1 �→i vi(i2) +and so on, xj �→i −α, ∀j ̸= i, where α sufficiently small. +We observe that, as stated in Section 4, we can assume to +know the formulas a priori for the first three classes. This is +so, because the rules are symmetric and we can freely asso- +ciate variables to players in every Ri, and write φ indepen- +dently from β. +For W and B-games, the formulas clearly depend on the +ordered preferences. Thus, it is crucial that the variables xj +are associated to the players in the right order (which is un- +known). +C +Expressing Anonymous Preferences as +Hedonic Coalition Nets +Anonymous HGs admit a compact HCN representation, by +assuming formulas φk, k ∈ [n] such that a subset of vari- +ables ξ satisfies φk iff |ξ| = k. Here, we briefly and infor- +mally describe how we know that such formulas exist and +why we do not state them explicitly, as for the other HGs +classes. For further details, we refer the interested reader to +Paterson (1976) and Wegener (1987). +Let us denote by Bn the set of n-argument Boolean func- +tions {f : {0, 1}n → {0, 1}}. Functions in Bn are to be +computed by acyclic circuits over the basis B2, containing +all 16 Boolean functions with two arguments. +An acyclic circuit may be represented as a finite directed +acyclic graph with n input nodes and one output node, where +each input node corresponds to one of the arguments and +each intermediate node is associated to an element of B2. +Furthermore, the indegree of the input nodes is zero, while +every intermediate node has an ordered pair of incoming +arcs. An immediate complexity measure of circuits is the cir- +cuit size, c, which counts the number of intermediate nodes, +which we refer to as logical gates. Another parameter, mo- +tivated by the fact that if each logical gate in the circuit re- +quires the same execution time, the bottleneck in parallel +computation will be the circuit depth, d, which counts the +maximum number of logical gates on any path from an in- +put node to the output node. +Now, if we want to represent a Boolean function as a lin- +ear expression over the input variables with function sym- +bols corresponding to elements of B2, we need to build an +acyclic circuit where all the logical gates have outdegree +one. This transformation can be easily done by replicating +inputs multiple times, as we do not longer allow for using +a result of any logical gate computation more than once. +Such circuits are then called formulas, and the size of the +formula, ℓ, is equal to the number of its internal nodes. It +is easy to see that for any Boolean function f, it holds that +c(f) ≤ ℓ(f) ≤ 2d(f). +A Boolean function is called symmetric, if its value de- +pends only of the number of 1s in the input. It is known that +ℓ(Sn) ∈ O(n), where Sn ⊂ Bn is the class of all symmet- +ric Boolean functions. The proof of this claim easily follows +from a two-stage construction of the formula, in which the +first stage is to construct a circuit which for any input vector +(xi)i∈[n] computes the binary representation of � +i xi and +in the second stage, the required function is computed from +the ⌈log(n + 1)⌉ results of the first stage. The first stage can +be computed by using a recursive procedure in which the bi- +nary representations of the two halves of the argument set +are first computed separately and then added together. Such +a circuit has depth O(log n), and each node performs an ad- +dition of two ⌈log(n + 1)⌉ bit numbers, which can naively +be done with a circuit of depth O(log n) but is also possible +to do with a circuit with depth O(log⌈log(n + 1)⌉). Further- +more, for the second stage, since it is also known that the +circuit depth of Bm is bounded by m + 1, we arrive at the +claimed bound, as the number of arguments in the second +stage is ⌈log(n + 1)⌉ and by the fact that ℓ(f) ≤ 2d(f). +All the claims above were made for the logical gates +corresponding to elements of B2. We say that a basis Ω +is complete if any Boolean function can be computed in +an Ω-circuit. Already some smaller bases, as for instance +{∧, ∨, ¬}, are complete, and so is the base consisting of +all the conventional Boolean operators and truth constants, +that we assume in HCNs. Since the complexity and depth +of Boolean functions can increase only by a constant fac- +tor when switching from one complete basis to another, we +were free to restrict our attention to B2. +The specific symmetric function that fits the purpose +of representing Anonymous HGs is the exactly-k-function, +En +k (x1, . . . , xn) = 1 +⇐⇒ +� +i xi = k. Alternatively, as +implicitly proposed by Elkind and Wooldridge (2009), one +could use the threshold function, T n +k (x1, . . . , xn) = 1 ⇐⇒ +� +i xi ≥ k, together with a construction similar to the one +for W-games for building the HCN representation. +There are many further known results regarding the for- +mula size both for general and specific symmetric functions +that can be used to derive the existence of a compact rep- + +resentation for Anonymous HGs via HCNs (see also Dunne +(1988)), but all of them, to the best of our knowledge, in- +clude constructions that result in formulas that are much +more complicated to write down than what was necessary +for any of the other considered HGs classes. +Finally, we remark that it is not possible to represent +Anonymous HGs as k-DL. To be able to recognize a coali- +tion of size m, the DL cannot contain only conjunctions of +size strictly less than m. Otherwise, the output would depend +on less than m variables, which is a contradiction. +D +Omitted Proofs from Section 4 +Here we report the proofs of the results in Section 4, starting +with Theorem 4.3. +Theorem 4.3. Let H be a class of HGs that admits a com- +pact HCN representation. Suppose that for every set of rules +Ri, we know the corresponding set of formulas Φ. Then, H +is efficiently PAC learnable. +Proof. By Proposition 4.2, we know that the pseudo- +dimension of the class of valuation functions of every player +is polynomial in n. Now, by Theorem 2.1, if we can show +that for any sample S we can give a hypothesis ⃗v∗ consis- +tent with the sample, we obtain the result. The following +procedure infers the real values βj to associate to each rule +φj ∈ Φ. To every S ∈ S ∩ Ni, by defining φj(S) = 1 if +S |= φj and 0 otherwise, we associate a linear equation of +the form +vi(S) = +� +φj∈Φ +βjφj(S), +where βj are the unknown variables. We obtain a linear sys- +tem given by all the equations associated to S ∈ S ∩Ni. De- +pending on the rank of this system induced by the sample, +we can either solve it exactly or get multiple solutions. +We conclude with the proof regarding the learnability of +HCNs whose formulas are k-DL. For completeness, we also +report the procedure LEARN-k-DL(k, S) given by Rivest +(1987), which is employed by Algorithm 2. +Algorithm 5: LEARN-k-DL (Rivest 1987) +1 Input: k ∈ N, S = {(Sj, bj))}m +j=1 +2 Output: L, a k-DL consistent with S +1: L ← ∅ +2: while S ̸= ∅ do +3: +for κ ∈ Cn +k do +4: +if every set in K = {S ∈ S : S |= κ} has the +same label b then +5: +L ← L ∪ {(κ, b)} +6: +S ← S \ K +7: +break +8: return L +Theorem 4.5. Let H be a class of HGs that admits a HCN +representation such that +(i) every coalition S ∈ Ni satisfies exactly one rule in Ri, +(ii) every rule is of the form L �→ β, where L is a k-DL with +k constant, and β unique, i.e., no pair of distinct rules +have the same value β. +Then, H is efficiently PAC learnable by Algorithm 2. +Proof. Let us first show that the pseudo-dimension of the +class of players’ valuations is polynomial in n. Note that in +this setting we assume the exact formulas to be unknown, +thus we cannot use Proposition 4.2. By (i), every coalition +S satisfies only one rule and its value is unique by (ii). As a +consequence, if the number of rules equals r, then no sample +of size r(n) + 1 can be pseudo-shattered. Indeed, any sam- +ple of size r(n) + 1 contains two coalitions with the same +associated value, thus every labeling containing different la- +bels for these coalitions cannot be satisfied. The largest set +of rules which respects our assumption is the one in which +every DL contains all of the possible conjunctions of size at +most k, and only one of them returns 1. In this case, the num- +ber of different rules r ∈ Θ(nk), so generally it holds that +Pdim(H) = O(nk), i.e., polynomial in n since k is constant. +Now, let us show that Algorithm 2 produces a hypothe- +sis consistent with the sample. The learning problem in this +case is to associate the correct rule to each different β. By +the second assumption, every rule is represented as a k-DL +and Algorithm 5 is known to return a k-DL consistent with a +binary labeled sample (Rivest 1987). Algorithm 2, thus, pro- +ceeds by assigning binary labels for every possible β sepa- +rately, setting the label to 1 for coalitions whose value equals +β and 0 for all others. Then, it calls Algorithm 5 as a sub- +routine for determining the DL associated to this particular +value β. The correctness of Algorithm 2 therefore follows +from the correctness of Algorithm 5. +E +Omitted Proofs from Section 5 +Here we report the proofs of the results in Section 5 starting +with Theorem 5.3. +Theorem 5.3. If H is PAC stabilizable, then H has SRC. +Proof. Assume that H does not satisfy SRC. This means +that there exists a sample S∗ such that there exist H1, H2 ∈ +H[S∗] such that CS∗(H1) ̸= ∅ and CS∗(H1) ∩ CS∗(H2) = +∅. If we now define a distribution D with positive probability +mass only on the elements of S∗, we will be able to conclude +that H is not PAC stabilizable. Indeed, no algorithm can dis- +tinguish between H1 and H2 when sampling from D and, +thus, cannot return a correct answer on both instances. +The following are the missing proofs for the part on PAC +stability of W-games with no ties. +Lemma 5.6. Let ε > 0 be fixed. If we denote by A(i) +j +the +event that a sampled coalition S satisfies i, j ∈ S and S \ +{i, j} ⊆ {j + 1, . . . , n − 1}, and by B(i) +j +the event that +a sampled coalition S satisfies i, j ∈ S and S \ {i, j} ⊆ + +�� +log2 +1 +ε +� ++ 2, . . . , n − 1 +� +, it holds that +Pr +S∼D +� +A(i) +j +� +≥ ε +2λ +for 1 ≤ j ≤ +� +log2 +1 +ε +� +, and +Pr +S∼D +� +B(i) +j +� +≥ ε +4λ +for j > +� +log2 +1 +ε +� +. +Proof. Let j ≤ +� +log2 +1 +ε +� +. By Equation (2) we have that +Pr +S∼D +� +A(i) +j +� +≥ 2n−j−1 +λ2n += +1 +λ2j+1 ≥ +1 +λ2⌊log2 +1 +ε⌋+1 ≥ ε +2λ. +Similarly, if j > +� +log2 +1 +ε +� +, we see that +Pr +S∼D +� +B(i) +j +� +≥ 2n−⌊log2 +1 +ε⌋−2 +λ2n += +1 +λ2⌊log2 +1 +ε⌋+2 ≥ ε +4λ. +Proposition 5.8. Let ε, δ > 0 and S be a sample of size m. +If m ≥ 2λ +ε log n2 +δ , LEARN-W-GAMES returns an ε-estimate +⃗v∗ of ⃗v with confidence 1 − δ. +Proof. Consider a fixed player i, let A(i) +j , B(i) +j +be the events +defined as in Lemma 5.6 and Sij as defined in Algorithm ??. +First, let j ≤ +� +log2 +1 +ε +� +. Observe that, if A(i) +j +holds for at +least one S ∈ S, then v∗ +i (j) = vi(j). Indeed, in that case +v∗ +i (j) = maxS∈Sij vi(S) = vi(j). By Lemma 5.6, +Pr +S∼D +� +A(i) +j +� +≤ +� +1 − ε +2λ +�m +≤ e−mε/2λ ≤ δ +n2 . +Now, let j > +� +log2 +1 +ε +� +. Observe that, if B(i) +j +holds for at least +one S ∈ S, then v∗ +i (j) > vi( +� +log2 +1 +ε +� +). Indeed, v∗ +i (j) = +maxS∈Sij vi(S) ≥ vi(⌊log2 +1 +ε⌋ + 1). Again by Lemma 5.6, +Pr +S∼D +� +B(i) +j +� +≤ +� +1 − ε +4λ +�m +≤ e−mε/4λ ≤ +� δ +n2 +�2 +≤ δ +n2 . +Now, let A(i) += +� +1≤j≤⌊log2 +1 +ε⌋ A(i) +j +and B(i) += +� +j>⌊log2 +1 +ε⌋ B(i) +j . From the previously shown bounds, +Pr +S∼D +� +A(i) ∪ B(i) +� +≤ (n − 1) δ +n2 ≤ δ +n, +which means that the probability that v∗ +i is not a ε-estimate +of vi is less then δ/n. Considering the whole ⃗v, by using a +union bound, we obtain that ⃗v∗ is an ε-estimate of ⃗v with +confidence 1 − δ. +Lemma 5.9. Let π be the output of Algorithm 3 and let us +call a player i green if it is not in a coalition with one of his +� +log2 +1 +ε +� +least preferred choices according to v∗ +i . Then, +a) for i green, PrS∼D [ i ∈ S ∧ vi(S) > vi(π(i)) ] < λε, +b) for ε ≥ +3� +λ2 +2n , PrS∼D [ S does not contain green i ] < ε. +Proof. To prove a), it is enough to notice that for green i +Pr +S∼D [ i ∈ S ∧ vi(S) > vi(π(i)) ] +≤ Pr +S∼D +� +S \ {i} ⊆ +�� +log2 +1 +ε +� ++ 2, . . . , n − 1 +� � +≤2n−⌊log2 +1 +ε⌋−2λ +2n − 1 +< +λ +2⌊log2 +1 +ε⌋+1 ≤ λε. +To prove b), we start by observing that at least (n − +� +log2 +1 +ε +� ++ 2)/2 players are green. Indeed, notice that at the +beginning of iteration t there are exactly n − 2t + 2 players +left. A player picked by the algorithm is associated with the +remaining player ranked the highest in their preference list. +If t < (n− +� +log2 +1 +ε +� ++2)/2, the number of remaining nodes +is > +� +log2 +1 +ε +� +, so the player picked at iteration t is green. +Now, we arrive at the statement by observing that +Pr +S∼D [ S does not contain green i ] +≤ +λ +2n − 1(2n−(n−⌊log2 +1 +ε⌋+2)/2 − 1) +< λ2n/2−1+⌊log2 +1 +ε⌋/2 +2n−1 +≤ λ +� +1/ε +2n/2 +≤ ε +where the first inequality follows from the bound on the +number of green players and Equation (2), and the last one +from the assumption on ε. +We conclude with the proof of the last result of the paper. +Theorem 5.11. If T (H) = ω(poly(2n)) for a HG class H, +then H is not efficiently PAC stabilizable, even under the +uniform distribution. +Proof. Let D be the uniform distribution on 2N and let +us assume that a PAC stabilizing algorithm exists. Given +ε = +1 +2n+1 , the PAC stabiliz ing algorithm must solve the +core. Indeed, the PAC stabilizing algorithm cannot return a +partition that is not core stable, as the probability of sam- +pling a blocking coalition is at least +1 +2n−1 > ε. By defini- +tion, the running time of the PAC stabilizing algorithm is +polynomial in 1/ε and n, implying that its time complexity +is O(poly(2n)). Therefore, we reached a contradiction. + diff --git a/rdFST4oBgHgl3EQfPzg1/content/tmp_files/load_file.txt b/rdFST4oBgHgl3EQfPzg1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..896a2b2aa6922d21ddf7b10fa7296745126f3698 --- /dev/null +++ b/rdFST4oBgHgl3EQfPzg1/content/tmp_files/load_file.txt @@ -0,0 +1,989 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf,len=988 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='13756v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='GT] 31 Jan 2023 PAC learning and stabilizing Hedonic Games: towards a unifying approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Simone Fioravanti,1 Michele Flammini,1 Bojana Kodric,2, 1 Giovanna Varricchio 3 1 Gran Sasso Science Institute (GSSI), L’Aquila, Italy 2 Ca’ Foscari University of Venice, Venice, Italy 3 Goethe-Universit¨at, Frankfurt am Main, Germany simone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='fioravanti@gssi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='it, michele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='flammini@gssi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='it, bojana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='kodric@unive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='it, varricchio@em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='uni-frankfurt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='de Abstract We study PAC learnability and PAC stabilizability of He- donic Games (HGs), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', efficiently inferring preferences or core-stable partitions from samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We first expand the known learnability/stabilizability landscape for some of the most prominent HGs classes, providing results for Friends and Enemies Games, Bottom Responsive, and Anonymous HGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Then, having a broader view in mind, we attempt to shed light on the structural properties leading to learnabil- ity/stabilizability, or lack thereof, for specific HGs classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Along this path, we focus on the fully expressive Hedonic Coalition Nets representation of HGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We identify two sets of conditions that lead to efficient learnability, and which en- compass all of the known positive learnability results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' On the side of stability, we reveal that, while the freedom of choosing an ad hoc adversarial distribution is the most obvious hur- dle to achieving PAC stability, it is not the only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' First, we show a distribution independent necessary condition for PAC stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Then, we focus on W-games, where players have individual preferences over other players and evaluate coalitions based on the least preferred member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We prove that these games are PAC stabilizable under the class of bounded distributions, which assign positive probability mass to all coalitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Finally, we discuss why such a result is not easily extendable to other HGs classes even in this promising sce- nario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Namely, we establish a purely computational property necessary for achieving PAC stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 1 Introduction Hedonic Games (HGs) (Dreze and Greenberg 1980) are a formal model for describing selfish individuals gathering to- gether in order to form coalitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Both HGs and general coalition formation games attracted considerable research attention in the last years due to their applicability to multi- agent environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Solution concepts for HGs are usually in the form of agent partitions with some suitable proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The one we consider in this paper is core stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A partition is said to be core-stable (or in the core) if there ex- ists no subset of players that could regroup into a so-called core-blocking coalition, which is preferred by all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The usual assumption when considering any solution con- cept is that the preferences of the agents are fully known, which is arguably unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Could we instead efficiently infer the whole game structure, or even directly learn so- lution concepts, while having only partial knowledge of the preferences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Questions of this kind are naturally captured by the probably approximately correct (PAC) learning frame- work (Valiant 1984), which formalizes the problem of learn- ing a target concept from a limited number of samples from any possible unknown but fixed distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Sliwinski and Zick (2017) were the first to leverage the PAC framework to study the problem of learning HGs pref- erences and core-stable partitions from samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In partic- ular, they define PAC stabilizability of a HGs class as the property of being able to, upon seeing a limited number of samples, either report that the core is empty or propose a par- tition that is unlikely to be core-blocked by further coalitions sampled from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In a recent paper, Lev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' (2021a) apply the notion of PAC stabilizability of HGs in the context of political coalition formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In particular, they use the publicly available Israeli parliament voting data to fit a Friends Appreciation HG, and compare the actual political parties of the voters to the PAC-stable coalitions re- sulting from the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' This example shows how learning concepts have the potential to create space for applications of mainly theoretical models, as HGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' While the work of Sliwinski and Zick (2017) and the ones that followed considered PAC learnability and stabilizabil- ity of many specific classes of HGs, the overall picture is still far from being complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Most prominently, the char- acterization of the underlying general conditions explaining the existing results is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Furthermore, PAC stabiliz- ability seems very hard to achieve and it is natural to won- der whether some restrictions on the PAC stability definition can yield better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Here, we address these questions, attempting to provide a deeper theoretical understanding of what makes HGs learnable and stabilizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='1 Our Contribution We first extend the knowledge on PAC learnable and PAC stabilizable classes of HGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We start by focusing on Friends and Enemies Games, examining whether the negative re- sults on stabilizability of Additively Separable HGs transfer to this simple subclass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' By exploiting previous results and proposing an algorithm stabilizing Friends and Enemies un- der Enemies Aversion, we deduce that Friends and Enemies Games belong to the very few lucky HGs classes that can both be learned and stabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Next, we study Bottom Re- sponsive HGs and show that while they are not efficiently learnable, they are stabilizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Finally, we turn our atten- tion to Anonymous HGs and show that the opposite holds here, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', they are efficiently learnable but not stabilizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' After exploring specific HGs classes, we use the gained insights to follow a more general research direction, devoted to a deeper understanding of the structural properties that make HGs learnable and/or stabilizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We first consider the learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Additively Sep- arable, Anonymous, W and B-games are all known to be learnable, and we investigate why this is the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' To this aim, we consider Hedonic Coalition Nets (HCNs), a general framework for representing HGs that is universally expres- sive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', it can represent any HGs class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We identify two sets of conditions on the HCNs representation that imply ef- ficient learnability, and as special cases explain the learn- ability of all of the aforementioned HGs classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We then turn our attention to stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Achieving PAC sta- bility does not seem possible for most HGs classes, and we try to find general reasons causing this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' First, we show a simple necessary condition for PAC stability, abstracting the proof pattern of all the known negative results for specific HGs classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Then, we consider the problem of PAC stabil- ity with bounded probability distributions and prove that un- der this restriction it is possible to PAC stabilize W-games, which is known not to be possible in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Finally, we dis- cuss why the same result cannot be easily extended to other HGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In particular, we determine a general purely computa- tional property necessary for achieving PAC stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Due to space limitations, all the missing proofs are de- ferred to the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='2 Related Work Many works have dealt with learning game-theoretic solu- tion concepts from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Sliwinski and Zick (2017) first in- troduced the PAC learning framework into the study of HGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Their work was extended by Igarashi, Sliwinski, and Zick (2019) to tackle HGs with underlying players’ interaction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Moreover, Jha and Zick (2020) laid further foun- dations for learning game-theoretic solution concepts from samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' More recently, Trivedi and Hemachandra (2021) studied the problem of learning HGs with noisy preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Other works have considered learning cooperative games (Balcan, Procaccia, and Zick 2015), markets (Lev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 2021b), auctions (Balcan, Sandholm, and Viter- cik 2018) but also, more generally, combinatorial func- tions (Balcan, Vitercik, and White 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Balcan 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' There is a vast body of literature on HGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' For a thor- ough introduction to the main concepts and results, we refer to Aziz and Savani (2016), where both all the HGs classes studied in this paper and also HCNs are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 2 Preliminaries Let N be a set of n players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We call any non-empty subset S ⊆ N a coalition and denote by Ni the set of all coalitions which contain a given player i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We call any coalition of size one a singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We denote by ≿i any binary prefer- ence relation of player i over the coalitions in Ni, which is reflexive, transitive, and complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A Hedonic Game (HG) is then a pair H = (N, ≿), where ≿= (≿i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , ≿n) is a preference profile, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', the collection of all players’ pref- erences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Throughout this work we will assume that play- ers’ preferences are expressed as real numbers by means of valuation functions vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In other words, given S, T ∈ Ni: vi(S) ≥ vi(T ) if and only if S ≿i T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We will denote by ⃗v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , vn) the collections of players’ valuations and assume that vi(S) = ∅ for S /∈ Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let H be a HG and π a coalition structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', a partition of players into coalitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A set S is said to core-block π if vi(S) > vi(π(i)) for each i ∈ S, where π(i) denotes the coalition containing i in π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A coalition structure π is core-stable if there does not exist a core-blocking coalition S ⊆ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Among the many possi- ble solution concepts, the one we will consider in this paper is core stability, as it is the most prominent one in the PAC stability model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='1 Defining Classes of Hedonic Games In this subsection, we provide the definitions of some HGs classes already considered from the perspective of PAC learning by (Sliwinski and Zick 2017), that will be fre- quently mentioned in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In all of these classes, for a player i ∈ N and a coalition S ∈ Ni, the valuation vi(S) is completely determined by the values vi(j) for j ∈ S \\ {i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' More precisely, the valuation of i for S is equal to: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Additively Separable: the sum of the values of its mem- bers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', vi(S) = � j∈S\\{i} vi(j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Fractional: the sum of the values of its members, but normalized by the size of the coalition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', vi(S) = � j∈S\\{i} vi(j)/|S|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' W-games: the value of the worst player in the coalition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' B-games: the value of the best player in the coalition, but coalitions of smaller size are preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='2 PAC Learning The PAC learning model, originally introduced by Valiant (1984), mathematically formalizes the process of learning a target concept v belonging to a hypothesis class H, by us- ing a sample of labeled examples as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' There are many variants, which adapt to different learning paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In the following, we will formally present only the one that we will use in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Our aim is to learn an unknown valua- tion function v : 2N → R within a class H, given as in- put S = {(S1, v(S1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , (Sm, v(Sm))}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', a collection of coalition/valuations pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The distribution D, according to which the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' input coalitions are sampled, is unknown, while the class H is determined by the HG instance one con- siders, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', if one is studying Additively Separable HGs, H will be the class of additively separable functions over (the other) n−1 players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Starting from a sample S, learning is the process of producing a hypothesis v∗ ∈ H which is as close as possible to the real v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Formally, a hypothesis v∗ ∈ H is ε-approximately correct w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' a distribution D over 2N and a function v ∈ H, if the following holds: Pr S∼D [ v∗(S) ̸= v(S) ] < ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Given ε, δ > 0, class H is (ε, δ) probably approximately correctly (PAC) learnable if there exists an algorithm A that, for every distribution D over 2N, and any v ∈ H, given a sample drawn from D, is able to produce a hypothesis v∗ which is ε-approximately correct with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A class H is said to be PAC learnable if it is (ε, δ) PAC learnable for all ε, δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Furthermore, if the sample size m and the running time of A are polynomial in 1 ε, log 1 δ and n, H is said to be efficiently PAC learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The inherent complexity of efficiently PAC learning a concept class of real functions H is usually measured by the so-called pseudo-dimension (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', Anthony and Bartlett (2002)), which is the analog of the more renowned VC- dimension (Kearns and Vazirani 1994) defined only for classes of binary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In order to formally define pseudo-dimension, we first need to introduce the concept of pseudo-shattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Given a collection of coalition/value pairs S = {(S1, r1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , (Sq, rq)}, we say that a class H can pseudo-shatter S if, for every possible binary labeling l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , lq of S, there exists a function f ∈ H such that f(Sj) > rj ⇐⇒ lj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Intuitively, the more H is ex- pressive, the bigger the sets that it can pseudo-shatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The pseudo-dimension of H, denoted as Pdim(H), is the size of the maximal set S that can be pseudo-shattered by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We conclude this section by reporting the theorem which bridges learning and pseudo-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='1 (Anthony and Bartlett 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A hypothe- sis class H with Pdim(H) polynomial in n is (ε, δ) PAC learnable using m samples, where m is polynomial in Pdim(H), 1 ε and log 1 δ , by any algorithm A that returns a hy- pothesis f ∗ consistent with the sample, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', f ∗(Si) = f(Si) for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Furthermore, if Pdim(H) is superpolynomial in n, H is not efficiently PAC learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='3 PAC Stabilizing Hedonic Games The concept of PAC stabilizing HGs was first introduced in (Sliwinski and Zick 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A coalition structure π is said to be ε-PAC stable under a distribution D if PrS∼D [ S core-blocks π ] < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A class of HGs H is PAC stabilizable if there exists an algorithm A that for any HG in H, any ε, δ > 0, and any D over 2N, given a sample S = {(S1,⃗v(S1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , (Sm,⃗v(Sm))} of coalitions drawn according to D, produces an ε-PAC stable coalition struc- ture π under D with probability at least 1 − δ, or reports that the core is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' If the sample size m and the running time of A meet the same conditions required for efficient PAC learnability, we say that H is efficiently PAC stabiliz- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Intuitively, this concept formalizes the learnability of a solution concept for a HGs class, independently from the learnability of the class itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We will rely on the following theorem in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='2 (Jha and Zick 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A class of HGs H is effi- ciently PAC stabilizable iff there exists an algorithm that out- puts a partition π consistent with the sample, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', no coali- tion from the sample core-blocks π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 3 Learnability and Stabilizability of New Classes of Hedonic Games In this section we broaden the picture of learnability and sta- bilizability of different classes of HGs, studying the follow- HGs class Learnable Stabilizable Friends and Enemies Friends Appreciation ✓∗ ✓∗ Enemies Aversion ✓∗ ✓ Bottom Responsive ✗ ✓ Anonymous ✓ ✗ Table 1: A summary of the learnability and stabilizability landscape discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Entries marked by an as- terisk symbol are consequences of previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' ing HG classes that were not considered by previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Friends and Enemies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Friends and Enemies Games have been traditionally investigated under two types of preference profiles, called Friends Appreciation and Enemies Aversion, where agents prefer coalitions with a greater number of friends (and smaller number of enemies in case of ties) or with a smaller number of enemies (and greater number of friends in case of ties), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Bottom Responsive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The bottom responsiveness property was first defined by Suzuki and Sung (2010) as bottom re- fuseness and then further considered by Aziz and Brandl (2012), where it was renamed in analogy to a related prop- erty called top responsiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Intuitively, it models pes- simistic agents who rank coalitions based on sets of players that they would like to avoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' For each player i ∈ N and S ∈ Ni, we define the avoid set of player i in coalition S as Av(i, S) = {S′ ⊆ S : (i ∈ S′) ∧ (∀S′′ ⊆ S, S′ ⪯i S′′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A game satisfies bottom responsiveness if for each i ∈ N and for each pair S, T ∈ Ni the following conditions hold: (i) if for each S′ ∈ Av(i, S) and for each T ′ ∈ Av(i, T ) it holds that if S′ ≻i T ′, then S ≻i T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' (ii) if Av(i, S) ∩ Av(i, T ) ̸= ∅ ∧ |S| ≥ |T |, then S ⪰i T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In what follows, we assume a minimum a priori knowledge of the values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Namely, we assume to know vi({i}), ∀i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A similar, yet significantly stronger, as- sumption was used by (Sliwinski and Zick 2017) to prove that Top Responsive HGs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' HGs which satisfy top respon- siveness) are efficiently PAC stabilizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Anonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A HG is said to satisfy anonimity, as defined in (Banerjee, Konishi, and S¨onmez 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Bogomolnaia and Jackson 2002), if vi(S) = vi(T ) for any player i ∈ N and any S, T ∈ Ni with |S| = |T |, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', players evaluate coali- tions only according to their size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We are now ready to state the following theorem, summa- rizing our results for the just defined HGs classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The results described in Table 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We give here just a sketch of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' while the full version can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Friends and Enemies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The efficient PAC learnability of both Friends Appreciation and Enemies Aversion profiles follows directly by observing that they are both subclasses of Additively Separable HGs (see (Dimitrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 2006)), known to be efficiently PAC learnable by the results of Sli- winski and Zick (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' For what concerns stabilizability, Suzuki and Sung (2010) showed that Friends and Enemies Games under Friends Ap- preciation are a subclass of Top Responsive HGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Sliwinski and Zick (2017) proved that Top Responsive HGs are ef- ficiently PAC stabilizable, which then implies the same for Friends Appreciation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' For Enemies Aversion, Dimitrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' (2006) prove that core-stable partitions always exist, while Dimitrov and Sung (2004) provide an algorithm returning such a partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Inspired by their algorithm, we provide an algorithm PAC stabilizing this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Bottom Responsive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' To show that the class is not efficiently PAC learnable, we prove that its pseudo-dimension is lower bounded by 2 n−1 2 , and thus is exponential in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The result then follows by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The construction in our proof bears similarities to the one of Sliwinski and Zick (2017) for Top Responsive HGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Regarding stabilizability, we first observe that Suzuki and Sung (2010) show that a core-stable coalition structure al- ways exists for this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Moreover, a simple necessary con- dition for S to be part of a core-stable partition π, is that for all i ∈ S it must hold that {i} ∈ Av(i, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Indeed, if this condition is not satisfied, at least one player prefers to devi- ate to a singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' To give a viable alternative for checking the condition while knowing the values of the singletons, we prove the following property: Given a Bottom Respon- sive HG H = (N, v), for every i ∈ N and every S ∈ Ni, it holds that {i} ∈ Av(i, S) ⇐⇒ vi({i}) ≤ vi(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Starting from this property, we construct Algorithm 1 which, given a sample, returns a coalition structure that is not core-blocked by any coalition from the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='2 this is sufficient for concluding the efficient PAC stabilizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Anonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' To show efficient PAC learnability, we prove that the pseudo-dimension of this class is upper bounded by n(1 + log n), and thus is polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Then, for each i ∈ N, the following procedure computes a hypothesis v∗ i consis- tent with the sample in time polynomial in n and m: For every coalition C of size k ∈ [n], if there exists Sj s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' i ∈ Sj and |Sj| = k, then set vi(C) = vi(Sj), otherwise set vi(C) = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' For what concerns stabilizability, we can provide a counter-example showing that the class is not PAC stabiliz- able, even in the case of natural single-peaked preferences, where every player has a given preferred size, and the valua- tion decreases as the distance from such size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Notice that, according to the above theorem, the nega- tive results on stabilizability of Additively Separable HGs of (Sliwinski and Zick 2017) do not transfer to Friends and Enemies Games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Furthermore, while the Bottom Reponsive HGs class is not PAC learnable but efficiently PAC stabiliz- able, exactly the opposite holds for Anonymous HGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Algorithm 1: Stabilizing Bottom Responsive HGs Input: N, S = {(Sj,⃗v(Sj))}m j=1 Output: π: an ε-stable partition of N 1 π ← ∅, T ← ∅ 2 for ⟨S,⃗v(S)⟩ ∈ S do 3 f ← 1 4 if ∃i ∈ S s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' vi(S) < vi({i}) then f ← 0 break 5 if f=1 then T ← T ∪ {S} 6 while T ̸= ∅ do 7 T + ← argmaxT ∈T |T \\ � P ∈π P| 8 π ← π ∪ � T + \\ � P ∈π P � 9 T ← T \\ {T +} 10 N ← N \\ T + 11 for i ∈ N do π ← π ∪ {{i}} 12 return π 4 A General Framework for Learnability: Hedonic Coalition Nets To provide a general unifying framework for learnability of HGs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' a direction worth investigating is the one of de- termining a suitable superclass or a small number of super- classes encompassing all the learnable HG classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Such re- sults would contribute to the general understanding of the crucial properties leading to learnability, or lack thereof, and would also provide means to easily determine whether a spe- cific class of HGs is learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A universal HGs class, maintaining the full expressive- ness for representing any HG, is the one of the so-called Hedonic Coalition Nets (HCNs) (Elkind and Wooldridge 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Before giving the definition, we note that, since there exist classes of HGs that are not learnable, it is not possible to get a positive result for the learnability of any fully ex- pressive HGs representation, so not for HCNs either, without imposing further restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Thus, our goal here is to deter- mine suitable restrictions allowing for efficient learnability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A hedonic coalition net (HCN) is a tuple (N, R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , Rn) where N is a set of variables (each corre- sponding to a player) and Ri is the set of rules for player i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A single rule in Ri is given by a pair (φ, β), where φ is a formula of propositional logic over N and β ∈ R is a real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We will denote a rule in Ri by φ �→i β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Then, as- suming the conventionalsemantic satisfaction relation “|=”, the valuation of player i for a coalition S ∈ Ni is vi(S) = � φj�→iβj∈Ri: S|=φj βj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' (1) The first HCNs subclass we consider comprises HCNs in which the formulas appearing in each set of rules Ri are known a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Namely, for any rule φ �→i β ∈ Ri we only need to learn β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We first show that, in this case, the pseudo- dimension depends on the number of rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let H(Ri) be the class of valuation func- tions that can be expressed with a fixed set of a priori known distinct rules Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Then, Pdim (H(Ri)) = O (|Ri|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let r = |Ri|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We will show that no set of size r + 1 can be pseudo-shattered by H(Ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' As a conse- quence Pdim (H(Ri)) ≤ r, which implies the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let S = {Sj}r+1 j=1 be any set of coalitions from Ni of size r + 1, and (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , tr+1) any sequence of r+1 real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Given any labeling l, the condition vi(Sj) > tj ⇔ ℓj = 1 can be written as a system of r + 1 linear inequalities of the form r � k=1 ajkβk > tj if ℓj = 1, and r � k=1 ajkβk ≤ tj if ℓj = 0, where ajk = 1 if Sj |= φk and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' This is a system of r + 1 inequalities with r unknowns β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , βr, thus the coefficient matrix A = (ajk) must have linearly dependent rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let us w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' assume that the last row Ar+1 can be written as Ar+1 = �r j=1 yjAj where the coefficients yj are not all null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let us define the labelings ℓ(1), ℓ(2) in this way: ℓ(1) j = 1 ⇔ yj < 0 for j ∈ [r], ℓ(1) r+1 = 1 and ℓ(2) j = 0 ⇔ ℓ(1) j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' By contradiction, assume that there exist solutions ⃗b1 and ⃗b2 that satisfy the respective systems of inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let us consider the first system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' By definition of ℓ(1) and ⃗b1, if ℓ(1) j < 0 then � Aj ·⃗b1 � > tj but yj < 0 implying that yj � Aj ·⃗b1 � < yjtj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' When ℓ(1) j = 0, instead, it holds that yj � Aj ·⃗b1 � ≤ yjtj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We can then conclude that this last inequality holds for all j ∈ [r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Regarding ℓ(2), with the same line of reasoning one can prove that yj � Aj ·⃗b2 � ≥ �r j=1 yjtj for all j ∈ [r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Writing Ar+1 as a combination of the other rows, and including the inequalities associated to Sr+1, we obtain the following: tr+1 < Ar+1 ·⃗b1 = r � j=1 yj � Aj ·⃗b1 � ≤ r � j=1 yjtj tr+1 ≥ Ar+1 ·⃗b2 = r � j=1 yj � Aj ·⃗b2 � ≥ r � j=1 yjtj implying tr+1 < �r j=1 yjtj ≤ tr+1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We say that H admits a compact HCN representation if it is possible to represent every v ∈ H with a polynomial number of rules for each player i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Observe that so far ev- ery class that has been shown to be learnable, also admits a compact HCNs representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Elkind and Wooldridge (2009) give HCNs representations for Additively Separable, Anonymous, W and B-games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We describe these represen- tations and provide one for Fractional HGs in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The following result shows that HGs admitting a compact HCN representation, for which we know the formulas a pri- ori, are efficiently PAC learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let H be a class of HGs that admits a com- pact HCN representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Suppose that for every set of rules Ri, we know the corresponding set of formulas Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Then, H is efficiently PAC learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' While the class presented above includes Additively Sep- arable, Fractional, and Anonymous HGs, which have all been shown to be efficiently PAC learnable, there exist other learnable classes which do not fall within the above charac- terization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Indeed, for W-games and B-games, knowing the φ for each rule a priori is not possible, since the formulas themselves depend on the ordered preferences that we need to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' On the other hand, the maximum number of distinct coalition values in both cases is only n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' To capture these remaining classes of learnable HGs through another suitable subclass of HCNs, we resort to de- cision lists, which were introduced by Rivest (1987) as al- ternative representations for Boolean functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A decision list (DL) L is defined by a set of l rules L = {(κ1, b1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , (κl, bl)} such that κi is a conjunc- tion of literals, bi ∈ {0, 1}, ∀i ∈ [l], and κl is the constant function true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Given L and a truth assignment x, L(x) is equal to bj where j is the least index such that κj(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We use the term k-decision lists (k-DL) if all the conjunc- tions in the DL are of size at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' For our purposes, for any coalition S and i ∈ [l], κi(S) = 1 if S |= κi and κi(S) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' It is convenient to think of a DL as an “if – then – else if – .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' - else –” rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' if κ1(S) = 1 then return b1 else if κ2(S) = 1 then return b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' else return bl Note that all the HGs classes we mentioned, other than Anonymous HGs (see Appendix C), can be represented as k-DL with k constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' It is known that for constant k, k-DL are efficiently PAC learnable: Rivest (1987) shows an effi- cient learning procedure LEARN-k-DL(k, S) that takes the size k and a sample S as input and returns a k-DL L (see Ap- pendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Furthermore, in the same work it is proven that k-DL are strictly more expressive than k-CNF and k-DNF formulas, and decision trees of depth k, meaning that every Boolean function that is representable in one of these forms admits a representation as a k-DL, but not viceversa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Now, if we consider HCNs which contain rules that are represented by k-DL and additionally restrict our attention to representations in which every coalition satisfies exactly one rule, it turns out that we can again efficiently PAC learn the valuations, as shown in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let H be a class of HGs that admits a HCN representation such that (i) every coalition S ∈ Ni satisfies exactly one rule in Ri, (ii) every rule is of the form L �→ β, where L is a k-DL with k constant, and β unique, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', no pair of distinct rules have the same value β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Then, H is efficiently PAC learnable by Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' While the second assumption in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='5 seems rather strong, we argue that asking for unique values β actually does not impose a further restriction, even though it seems fundamental for proving the result (see Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In- deed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' if there is more than one k-DL associated with the Algorithm 2: Learning HCN in k-DL form Input: k ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' S = {(Sj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' vi(Sj))}m j=1 Output: Ri consistent with S 1 Ri ← ∅ 2 for β in {vi(Sj) : Sj ∈ S} do 3 for Sj ∈ S do 4 if vi(S) = β then bj = 1 else bj = 0 5 S′ = {(Sj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' bj)}m j=1 6 L ← LEARN-k-DL(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' S′) 7 Ri ← Ri ∪ {L �→ β} 8 return Ri same value β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' using the assumption that every coalition sat- isfies exactly one rule,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' it is always possible to merge them into one k-DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='5 includes as a special case all HGs that can be represented by sets of mutually exclusive conjunctions, each containing at most k positive literals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' This is so, because we can phrase the negative literals positively within the DL, by associating the presence of such a variable with returning 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Thus, the conjunction size depends only on the number of positive literals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The case k = 1 includes W- and B-games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 5 Stabilizability of Hedonic Games We start this section by identifying a property that a HGs class needs to satisfy if it has any ambitions of being PAC stabilizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' To this end, we first define the set of core stable partitions w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' a fixed sample S, and equivalence classes in a HGs class H w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' a fixed sample S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Then, we state a theo- rem that abstracts on the arguments used in proofs showing that a specific HGs class is not PAC stabilizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Recall that a sample S is a set {(S1,⃗v(S1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , (Sm,⃗v(Sm))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let H be a class of HGs, S a sample and let H ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We denote by: (i) CS(H) = {π : ∀S ∈ S, S does not core block π}, the set of partitions consistent with the sample S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' (ii) H[S] the set of all instances H = (N,⃗v ′) ∈ H such that ⃗v ′(S) = ⃗v(S), for each (S,⃗v(S)) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We are now able to define the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' HGs class H satisfies the sample resistant core property, or has SRC in short, if for every S ⊆ 2N CS(H) = ∅, ∀H ∈ H[S], or � H∈H[S] CS(H) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' If H is PAC stabilizable, then H has SRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Notice that Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='3 (whose proof can be found in Appendix 5) formalizes the standard approach of Sliwinski and Zick (2017) and of our work, to show that a specific HG class is not PAC stabilizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Furthermore, property SRC, which is of course satisfied by Top and Bottom Responsive HGs (as they can be PAC stabilized), does not seem to be a common HGs property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' One could argue that aiming to PAC stabilize a specific HG class without having any a priori knowledge on the dis- tribution D is too ambitious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Thus, a natural question is whether restricting the attention to special distributions in- creases the prospect of stabilizing some classes of HGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' This direction was left as an open question by Sliwinski and Zick (2017) and is our focus in the remaining part of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The motivation for limiting the scope of allowed distribu- tions is to get a more fine-grained insight into PAC stabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The simple counterexamples from (Sliwinski and Zick 2017), while providing valuable understandings, do not re- veal “how far away” from achieving PAC stability certain classes of HGs are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Thus, we proceed by studying PAC sta- bility under a class of distributions that excludes the usual adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In particular, we focus on distributions having a fair amount of probability mass on all coalitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A distribution D is said to be bounded if there exists λ ≥ 1 such that, for every two coalitions S1, S2, it holds that PrD [ S1 ] ≤ λ PrD [ S2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Observe that the uniform distribution is a special case of the above definition, obtained by setting λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A useful property that we will use extensively in our cal- culations is that, if D is bounded with a factor λ, then 1 λ2n ≤ 1 λ(2n − 1) ≤ Pr S∼D [ S ] ≤ λ 2n − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' (2) These simple bounds follow from the definition of a bounded distribution and the fact that � T ∈2N PrD [ T ] = 1, where the sum goes over the 2n − 1 non-empty coalitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' By Equation (2), every coalition now has a positive proba- bility of being sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Since the counterexamples to PAC stabilizability of specific HGs classes usually rely on ad hoc distributions where most of the coalitions are never sampled, this feature provides hope of obtaining better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='1 W-games under Bounded Distributions As a case study we consider W-games with no ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' This class admits a polynomial algorithm for finding a core sta- ble partition (Cechl´arov´a and Hajdukov´a 2004), but, despite that, it has been shown not to be PAC stabilizable (Sliwinski and Zick 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Thus, it seems a natural first candidate for being PAC stabilizable under bounded distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In the rest of this subsection, we indeed show the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' W-games under bounded distributions are efficiently PAC stabilizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' To this end, in what follows, when focusing on a fixed player i, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' we assume that the other players are or- dered such that vi(1) < vi(2) < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' < vi(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We start by exploiting the fact that the distribution is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let ε > 0 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' If we denote by A(i) j the event that a sampled coalition S satisfies i, j ∈ S and S \\ {i, j} ⊆ {j + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , n − 1}, and by B(i) j the event that a sampled coalition S satisfies i, j ∈ S and S \\ {i, j} ⊆ Algorithm 3: Stabilizing W-games Input: N players, S = {(Sj,⃗v(Sj))}m j=1, ε > 0 Output: A partition π 1 ⃗v∗ ← LEARN-W-GAMES(N, S) 2 π ← ∅ 3 while N ̸= ∅ do 4 Pick i ∈ N 5 if N \\ {i} ̸= ∅ then 6 j ← argmaxk∈N\\{i} v∗ i (k) 7 π ← π ∪ {{i, j}}, N ← N \\ {i, j} 8 else π ← π ∪ {{i}} 9 return π �� log2 1 ε � + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , n − 1 � , it holds that Pr S∼D � A(i) j � ≥ ε 2λ for 1 ≤ j ≤ � log2 1 ε � , and Pr S∼D � B(i) j � ≥ ε 4λ for j > � log2 1 ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='6 and all the others missing proofs of this subsection can be found in Appendix 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Sliwinski and Zick (2017) presented a simple procedure, that we will refer to as LEARN-W-GAMES, which takes in input the set of players and a sample, and returns a consistent estimate ⃗v∗ for the players’ valuations in W-games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' This procedure sets v∗ i (j) to be the maxS∈Sij vi(S) where Sij = {S ∈ S : {i, j} ⊆ S}, if Sij is non-empty, −∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Next, we define what we call an ε-estimate of a function, and show that the output of LEARN-W-GAMES is actually such an es- timate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Function v′ i is an ε-estimate of vi if � v′ i(j) = vi(j) for 1 ≤ j ≤ � log2 1 ε � , and v′ i(j) > vi( � log2 1 ε � ) for j > � log2 1 ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' (3) Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let ε, δ > 0 and S be a sample of size m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' If m ≥ 2λ ε log n2 δ , LEARN-W-GAMES returns an ε-estimate ⃗v∗ of ⃗v with confidence 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We want to show that, by relying on the ε-estimate given by LEARN-W-GAMES for ε “not too small”, Algorithm 3 returns an ε-stable partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We first state a technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let π be the output of Algorithm 3 and let us call a player i green if it is not in a coalition with one of his � log2 1 ε � least preferred choices according to v∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Then, a) for i green, PrS∼D [ i ∈ S ∧ vi(S) > vi(π(i)) ] < λε, b) for ε ≥ 3� λ2 2n , PrS∼D [ S does not contain green i ] < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We are now finally ready to prove the main theorem, stated in the beginning of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' For δ > 0 and ε ≥ 3� λ5 2n−3 , we call Algorithm 3 with ε′ = ε/2λ and m ≥ 2λ ε′ log n2 δ = 1 ε log n2 δ , and obtain an ε-stable partition with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' By point a) of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='9, a green node has probability < λε′ = ε/2 to get a better outcome by moving from π(i) to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Furthermore, ε′ satisfies the requirement of point b) in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='9, so the probability of sampling an S without a green node is < ε′ ≤ ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In conclusion, if we call G the event that S contains a green player, then since PrS∼D [ S core blocks π ] equals Pr S∼D [ S core blocks π | G ] + Pr S∼D � S core blocks π | G � , we see that PrS∼D [ S core blocks π ] ≤ ε/2 + ε/2 = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' For δ > 0 and ε < 3� λ5 2n−3 , with a sample of size m ≥ 8λ6 ε3 log n2 δ , we can reveal the exact valuation functions with probability at least 1 − δ and return the core stable par- tition π using the algorithm of Cechl´arov´a and Hajdukov´a (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Indeed, since the probability of drawing any coali- tion is by Equation (2) at least 1/λ2n, this also holds for the coalition containing only agents i and j, which provides both vi(j) and vj(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The probability of not drawing a particular coalition of size 2 is ≤ (1 − 1/λ2n)m ≤ e−m/λ2n ≤ δ/n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Taking a union bound over all the �n 2 � < n2 coalitions of size 2, we see that the probability of not seeing all the exact valuations is upper bounded by δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='2 Barriers to the Restricted Distributions Approach Encouraged by the positive results of the last subsection, one could try to extend the approach of focusing on bounded dis- tributions in the hope that other classes that are known not to be PAC stabilizable, such as Additively Separable, Frac- tional, and Anonymous HGs, are in fact stabilizable under such distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Unfortunately, this does not seem to be always the case, as we discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' For a HGs class H , let T (H) be the time complexity of the best algorithm solving the core for this class, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', the runtime of the fastest algorithm that for every input instance either correctly replies that the core is empty or returns a core-stable partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' If T (H) = ω(poly(2n)) for a HG class H, then H is not efficiently PAC stabilizable, even under the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Notice that the assumption T (H) ∈ ω(poly(2n)) is not that strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In most of the HGs classes the complexity of the problem of deciding the existence of the core is either Σp 2- complete or NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Although this does not imply that it is not possible to find a O(poly(2n)) algorithm (in the case of the total collapse of the polynomial hierarchy, this would even be possible in polynomial time), such algorithms are currently not known and at this point it seems that finding them is unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In particular, the brute force approach, that searches for an element in the core by examining all the pos- sible partitions, has a running time of Ω((n/2)n/4) (as this is one possible lower bound on the Bell number), and thus its running time is also in ω(poly(2n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 6 Conclusions In this work, we initiated the study of a unified approach for determining the learnability and stabilizability of spe- cific HGs classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' One of the obvious goals for future work is finding a unique characterization of HCN representations that imply learnability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Another one is exploring further con- sequences of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='11 and expanding the knowledge on the exact computational complexity of solving the core for the different classes of HGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Acknowledgements Giovanna Varricchio was supported by DFG grant Ho 3831/5-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The authors would also like to thank the anony- mous reviewers for their comments and suggestions.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Hedonic coalition forma- tion in conservative societies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Available at SSRN 1700921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Trivedi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' and Hemachandra, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Learning Noisy He- donic Games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' CoRR, abs/2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='07738.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Valiant, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A Theory of the Learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' ACM, 27(11): 1134–1142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Wegener, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The complexity of Boolean functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' John Wiley & Sons, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='2 In the following, we report the complete proof of Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='2 sketched in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We will show that: (i) Friends and Enemies Games are efficiently PAC learn- able and stabilizable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' (ii) Bottom Responsive HGs are not PAC learnable but effi- ciently stabilizable, if we require as a baseline to know the values v(i) for each i ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' (iii) Anonymous HGs are efficiently PAC learnable but not stabilizable, even in the case of natural single-peaked preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='1 Friends and Enemies Games Since the learnability for both Friends Appreciation and En- emies Aversion profiles and the stabilizability for Friends Appreciation are a consequence of previous results, in the following we will focus on proving efficient PAC stabiliz- ability for Enemies Aversion profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' For this type of preferences, Dimitrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' (2006) proved that core stable partitions always exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' An algorithm for computing a core stable partition was provided by Dimitrov and Sung (2004), which we describe here only informally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Initially all the agents are “unassigned”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' At every step a new coalition is created, consisting of the maximum subgroup of still unassigned agents that all consider each other friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Once a new coalition is created, its members are marked as “assigned”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' This process continues as long as there are unas- signed agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Singleton coalitions may be created once no larger set of agents forms a friendship clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The just de- scribed algorithm clearly has an exponential running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In fact, due to its close relation to the MAXCLIQUE problem, computing a core stable partition in this setting is known to be NP-hard (Dimitrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Here, we present Algorithm 4 which PAC stabilizes this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Our algorithm, inspired by the one of Dimitrov and Sung (2004), stores in T the sampled coalitions that are cliques (lines 2-11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Then, it repeatedly extracts a maximum clique from T and creates a corresponding coalition (lines 12-17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Every time a coalition is formed, since any subset of a clique is a clique, the algorithm refines the sets in T by removing agents which have been assigned to a coali- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' This process takes place as long as T ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' If there are agents which have not been assigned to any coalition, they are placed into singletons (lines 18-20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let H = (N, v) be a Friends and Enemies Game under Enemy Aversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' To prove that Algorithm 4 stabilizes H, it is enough to prove that the partition π output by the algo- rithm cannot be core blocked by any of the coalitions inside the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Then the result follows by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' To this end, observe that any coalition S from the sample ends up in the candidate set T if and only if all the members of the coalition consider each other to be friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Furthermore, the all-friends coalitions are added to π by cardinality, starting from the largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Now, first notice that a coalition S that is not an all-friends coalition, cannot block π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Furthermore, for an all-friends coalition S from the sample, it is not possible that |S| > π(i) for all i ∈ S, concluding the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Notice that the exact valuation function values are not im- portant for Algorithm 4, since any valuation function that respects Enemy Aversion will have a negative value as soon as the coalition contains at least one enemy relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Algorithm 4: Stabilizing Enemy Aversion 1 Input: N, S = {(Sj,⃗v(Sj))}m j=1 2 Output: π: an ε-stable partition of N 1: π ← ∅, T ← ∅ 2: for ⟨S,⃗v(S)⟩ ∈ S do 3: f ← 1 4: if ∃i ∈ S s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' vi(S) < 0 then 5: f ← 0 6: break 7: if f=1 then 8: T ← T ∪ {S} 9: while T ̸= ∅ do 10: T + ← argmaxT ∈T |T \\ � P ∈π P| 11: π ← π ∪ � T + \\ � P ∈π P � 12: T ← T \\ {T +} 13: N ← N \\ T + 14: for i ∈ N do 15: π ← π ∪ {{i}} 16: return π A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='2 Bottom Responsive HGs Non-Learnability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Consider the class R of HGs such that, for S, T ∈ Ni, |S| < |T | ⇒ vi(S) > vi(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' First, notice that class R satisfies bottom responsiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Indeed, as for each S ∈ Ni it holds that Av(i, S) = {S}, it is easy to check that conditions (i) and (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let now X = {S ∈ Ni : |S| = � n 2 � }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In particular, for S1, S2 ∈ X, it holds that Av(i, S1) ∩ Av(i, S2) = {S1} ∩ {S2} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The cardinality of X is |X| = � n � n 2 � � > � n n−1 2 � > �n − 1 n−1 2 � > 2 n−1 2 where the last inequality follows from the fact that 2m = �m k=1 �m k � < �m k=1 �m k �� m m−k � = �2m m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In what follows, we show that S = {(S, r(S))}S∈X is pseudo-shattered by R, for any sequence of real numbers {r(S)}S∈X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Indeed, given any labeling ℓ : X → {0, 1}, we are free to choose the vi(S) in such a way that vi(S) > r(S) ⇔ ℓ(S) = 1 is satisfied, as long as we respect the following: min T :|T |<⌈ n 2 ⌉ vi(T ) > max S∈X vi(S), max T :|T |>⌈ n 2 ⌉ vi(T ) < min S∈X vi(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' As a consequence Pdim(R) > |X| > 2 n−1 2 , which implies the result by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Stabilizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let H = (N, v) be a Bottom Responsive HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' First we will prove that the following condition holds for every i ∈ N and S ∈ Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' {i} ∈ Av(i, S) ⇐⇒ vi({i}) ≤ vi(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' (4) One direction follows by the definition of the avoid set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Now, assume that vi({i}) ≤ vi(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' By contradiction, let us as- sume that {i} /∈ Av(i, S) and let T ∈ Av(i, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' By the defi- nition of the avoid set, we know that vi(T ) < vi({i}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Since this holds for every T ∈ Av(i, S) and the game is bottom re- sponsive, since Av(i, {i}) = {{i}}, by (i) of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='1 it follows that vi(S) < vi({i}), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let H = (N,⃗v) a Bottom Responsive game for which we know v(i) for each i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In the following, we will show that Algorithm 1 PAC stabilizes H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Again, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='2, it is sufficient to prove that it returns a partition which is consistent with the sample i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' that is not core-blocked by any sampled coalition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Since we assume that the values assigned by the players to singletons are known, Algorithm 1 uses the equivalence from Equation (4) to sort out every element of the sam- ple that does not satisfy the necessary condition to be in a core-stable partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The remaining sets after this operation are stored in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Observe that any coalition S core block- ing π must satisfy both {i} ∈ Av(i, S) and |S| > |π(i)|, ∀i ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The former is, as already remarked, a necessary condition for being in a core stable partition, while the lat- ter is a consequence of this condition and the requirement (ii) in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Now, consider any S ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Indeed, if {i} ∈ Av(i, S) for all i ∈ S, then the algorithm adds S to T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Furthermore, the coalitions in π are created from T by maximising the cardinality of the chosen set at each step, which makes it impossible for S ∈ T to satisfy |S| > |π(i)| for all i ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='3 Anonymous HGs Recall that a HG is said to satisfy anonimity, as defined in (Banerjee, Konishi, and S¨onmez 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Bogomolnaia and Jackson 2002), if vi(S) = vi(T ) for any player i ∈ N and any S, T ∈ Ni with |S| = |T |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' That is, players evaluate coalitions only according to their size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' For this reason, we will write s1 ⪰i s2 to denote that player i prefers size s1 to size s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Learnability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We will first prove that the pseudo-dimension of v is bounded by n(log n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , Sm be a list of subsets of N and r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , rm a list of real val- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Since in Anonymous HGs the valuation of a coalition depends only on its size, we will not be able to shatter (⟨S1, r1⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , ⟨Sm, rm⟩) if the list of sets contains at least two coalitions of the same size that have a non-empty inter- section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Indeed, if we w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' assume that |S1| = |S2| and S1 ∩ S2 ̸= ∅, we see that for any labelling in which ℓ1 ̸= ℓ2 we arrive at a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' This is so, because for an agent i ∈ S1 ∩ S2 it has to hold that vi(S1) = vi(S2) and at the same time l1 ̸= l2 implies either vi(S1) > vi(S2) or vi(S1) < vi(S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' What now remains is to see that any list of sets of length at least n(log n+1) contains at least one pair of sets with the same cardinality and a non-empty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' To this end, notice that there are n sets of cardinality 1 with an empty intersection, ⌊n/2⌋ sets of cardinality 2 with an empty intersection, and generally ⌊n/k⌋ sets of cardinality k with an empty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Since, n+⌊n/2⌋+· · ·+⌊n/(n− 1)⌋+⌊n/n⌋ ≤ n+n/2+· · ·+n/(n−1)+1 ≤ n(log n+1), we see that the pseudo-dimension of the class of Anonymous HGs is bounded by n(log n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' What remains to be proven, is that we can give a con- sistent hypothesis in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The following proce- dure computes a hypothesis v∗ i consistent with the sample in time polynomial in n and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' For every coalition C of size k ∈ [n], if there exists Sj s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' i ∈ Sj and |Sj| = k, then set vi(C) = vi(Sj), otherwise set vi(C) = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Non-Stabilizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The anonymity condition is often com- plemented by the so called single-peakedness, where every player has a given preferred size, and the valuation decreases as the distance from such size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Formally: Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' An instance of Anonymous HGs is single- peaked if there exists a permutation (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , sn) of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', n} in which every player i ∈ N admits a peak p(i) such that j < k ≤ p(i) or j > k ≥ p(i) imply sk ⪰i sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Moreover if the permutation is the identity function, we say that the preference is single peaked in the natural ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In the following we will prove that Anonymous HGs are not PAC stabilizable, even if we restrict our attention to single-peaked instances in the natural ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let us consider instances with 7 agents N = {a1, a2, a3, a4, b1, c1, c2} where a,b and c are the agent types, and agents of the same type have the same prefer- ences over coalition sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Consider a distribution D that se- lects uniformly at random a coalition in {S ⊆ N : if |S| = 5, then c1, c2 ̸∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In other words, the distribution never samples coalitions of size 5 containing c1 or c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In partic- ular, this means that we will not know where coalitions of size 5 appear in the preference list of type c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let I1 be the instance in which the agents’ types have the following preferences over coalition sizes: Type a: 6 ≻a 5 ≻a 4 ≻a 3 ≻a 2 ≻a 1 ≻a 7, Type b: 5 ≻b 4 ≻b 3 ≻b 6 ≻b 2 ≻b 1 ≻b 7, Type c: 4 ≻c 3 ≻c 5 ≻c 6 ≻c 2 ≻c 1 ≻c 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The preferences in I1 are single peaked in the natural or- dering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Moreover, Banerjee, Konishi, and S¨onmez (2001) showed that instance I1 has an empty core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In particular, coalitions of size 5 containing c1 or c2 are never used as blocking coalitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Therefore, a PAC stabilizing algorithm applied on I1 should always report that the core is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let us now consider a slightly different instance I2, obtained by changing the preferences of agents c1 and c2 to type b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Preferences in I2 are still single peaked in the natural ordering, but now the partition π = {{a1, a2}, {a3, a4, b1, c1, c2}} is core stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' On the one hand, since the distribution D never samples coalitions of size 5 containing c1 or c2, no algorithm is able to distinguish between I1 and I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' On the other hand, I1 has an empty core and I2 does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Note that in instance I1, for any partition π there exists at least one blocking coalition in the support of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Since the probability of selecting any coalition in the support of D is fixed (a positive constant), the algorithm indeed has to report that the core is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Thus, no algorithm can PAC stabilize this HGs class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' B Expressing Classes of Hedonic Games as Hedonic Coalition Nets In this section, we will show how it is possible to express as HCN the classes of HGs mentioned in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' These rep- resentations have all been given by Elkind and Wooldridge (2009), except for the Fractional HGs one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' For Anonymous HGs, only the existence of the formulas used in the rules is mentioned by Elkind and Wooldridge (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Writing these formulas explicitly is a non-trivial task that is, for interested readers, discussed thoroughly in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In the follow- ing, we consider a player i and describe how to write the rules in Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Additively Separable: xj �→i vi(j), ∀j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Anonymous: formulas φk, k ∈ [n] such that a subset of variables ξ satisfies φk iff |ξ| = k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' φk �→i vi(k), ∀k ∈ [n] are known to exist and to have polynomial length in the number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' For further details see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Fractional: assuming φk, k ∈ [n] as above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' xj ∧ φk �→i vi(j)/k for 2 ≤ k ≤ n, ∀j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' W-games: denote by ij the j-th player in i’s prefer- ence list (in descending order);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' xn−1 �→i vi(in−1), xn−2∧¬xn−1 �→i vi(in−2), xn−3∧¬xn−1∧¬xn−2 �→i vi(in−3) and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' B-games: denote by ij the j-th player in i’s preferences (in descending order);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' x1 �→i vi(i1), x2 ∧¬x1 �→i vi(i2) and so on, xj �→i −α, ∀j ̸= i, where α sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We observe that, as stated in Section 4, we can assume to know the formulas a priori for the first three classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' This is so, because the rules are symmetric and we can freely asso- ciate variables to players in every Ri, and write φ indepen- dently from β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' For W and B-games, the formulas clearly depend on the ordered preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Thus, it is crucial that the variables xj are associated to the players in the right order (which is un- known).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' C Expressing Anonymous Preferences as Hedonic Coalition Nets Anonymous HGs admit a compact HCN representation, by assuming formulas φk, k ∈ [n] such that a subset of vari- ables ξ satisfies φk iff |ξ| = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Here, we briefly and infor- mally describe how we know that such formulas exist and why we do not state them explicitly, as for the other HGs classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' For further details, we refer the interested reader to Paterson (1976) and Wegener (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let us denote by Bn the set of n-argument Boolean func- tions {f : {0, 1}n → {0, 1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Functions in Bn are to be computed by acyclic circuits over the basis B2, containing all 16 Boolean functions with two arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' An acyclic circuit may be represented as a finite directed acyclic graph with n input nodes and one output node, where each input node corresponds to one of the arguments and each intermediate node is associated to an element of B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Furthermore, the indegree of the input nodes is zero, while every intermediate node has an ordered pair of incoming arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' An immediate complexity measure of circuits is the cir- cuit size, c, which counts the number of intermediate nodes, which we refer to as logical gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Another parameter, mo- tivated by the fact that if each logical gate in the circuit re- quires the same execution time, the bottleneck in parallel computation will be the circuit depth, d, which counts the maximum number of logical gates on any path from an in- put node to the output node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Now, if we want to represent a Boolean function as a lin- ear expression over the input variables with function sym- bols corresponding to elements of B2, we need to build an acyclic circuit where all the logical gates have outdegree one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' This transformation can be easily done by replicating inputs multiple times, as we do not longer allow for using a result of any logical gate computation more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Such circuits are then called formulas, and the size of the formula, ℓ, is equal to the number of its internal nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' It is easy to see that for any Boolean function f, it holds that c(f) ≤ ℓ(f) ≤ 2d(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A Boolean function is called symmetric, if its value de- pends only of the number of 1s in the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' It is known that ℓ(Sn) ∈ O(n), where Sn ⊂ Bn is the class of all symmet- ric Boolean functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The proof of this claim easily follows from a two-stage construction of the formula, in which the first stage is to construct a circuit which for any input vector (xi)i∈[n] computes the binary representation of � i xi and in the second stage, the required function is computed from the ⌈log(n + 1)⌉ results of the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The first stage can be computed by using a recursive procedure in which the bi- nary representations of the two halves of the argument set are first computed separately and then added together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Such a circuit has depth O(log n), and each node performs an ad- dition of two ⌈log(n + 1)⌉ bit numbers, which can naively be done with a circuit of depth O(log n) but is also possible to do with a circuit with depth O(log⌈log(n + 1)⌉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Further- more, for the second stage, since it is also known that the circuit depth of Bm is bounded by m + 1, we arrive at the claimed bound, as the number of arguments in the second stage is ⌈log(n + 1)⌉ and by the fact that ℓ(f) ≤ 2d(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' All the claims above were made for the logical gates corresponding to elements of B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We say that a basis Ω is complete if any Boolean function can be computed in an Ω-circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Already some smaller bases, as for instance {∧, ∨, ¬}, are complete, and so is the base consisting of all the conventional Boolean operators and truth constants, that we assume in HCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Since the complexity and depth of Boolean functions can increase only by a constant fac- tor when switching from one complete basis to another, we were free to restrict our attention to B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The specific symmetric function that fits the purpose of representing Anonymous HGs is the exactly-k-function, En k (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , xn) = 1 ⇐⇒ � i xi = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Alternatively, as implicitly proposed by Elkind and Wooldridge (2009), one could use the threshold function, T n k (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , xn) = 1 ⇐⇒ � i xi ≥ k, together with a construction similar to the one for W-games for building the HCN representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' There are many further known results regarding the for- mula size both for general and specific symmetric functions that can be used to derive the existence of a compact rep- resentation for Anonymous HGs via HCNs (see also Dunne (1988)), but all of them, to the best of our knowledge, in- clude constructions that result in formulas that are much more complicated to write down than what was necessary for any of the other considered HGs classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Finally, we remark that it is not possible to represent Anonymous HGs as k-DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' To be able to recognize a coali- tion of size m, the DL cannot contain only conjunctions of size strictly less than m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Otherwise, the output would depend on less than m variables, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' D Omitted Proofs from Section 4 Here we report the proofs of the results in Section 4, starting with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let H be a class of HGs that admits a com- pact HCN representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Suppose that for every set of rules Ri, we know the corresponding set of formulas Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Then, H is efficiently PAC learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='2, we know that the pseudo- dimension of the class of valuation functions of every player is polynomial in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Now, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='1, if we can show that for any sample S we can give a hypothesis ⃗v∗ consis- tent with the sample, we obtain the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The following procedure infers the real values βj to associate to each rule φj ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' To every S ∈ S ∩ Ni, by defining φj(S) = 1 if S |= φj and 0 otherwise, we associate a linear equation of the form vi(S) = � φj∈Φ βjφj(S), where βj are the unknown variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We obtain a linear sys- tem given by all the equations associated to S ∈ S ∩Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' De- pending on the rank of this system induced by the sample, we can either solve it exactly or get multiple solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We conclude with the proof regarding the learnability of HCNs whose formulas are k-DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' For completeness, we also report the procedure LEARN-k-DL(k, S) given by Rivest (1987), which is employed by Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Algorithm 5: LEARN-k-DL (Rivest 1987) 1 Input: k ∈ N, S = {(Sj, bj))}m j=1 2 Output: L, a k-DL consistent with S 1: L ← ∅ 2: while S ̸= ∅ do 3: for κ ∈ Cn k do 4: if every set in K = {S ∈ S : S |= κ} has the same label b then 5: L ← L ∪ {(κ, b)} 6: S ← S \\ K 7: break 8: return L Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let H be a class of HGs that admits a HCN representation such that (i) every coalition S ∈ Ni satisfies exactly one rule in Ri, (ii) every rule is of the form L �→ β, where L is a k-DL with k constant, and β unique, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', no pair of distinct rules have the same value β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Then, H is efficiently PAC learnable by Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let us first show that the pseudo-dimension of the class of players’ valuations is polynomial in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Note that in this setting we assume the exact formulas to be unknown, thus we cannot use Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' By (i), every coalition S satisfies only one rule and its value is unique by (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' As a consequence, if the number of rules equals r, then no sample of size r(n) + 1 can be pseudo-shattered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Indeed, any sam- ple of size r(n) + 1 contains two coalitions with the same associated value, thus every labeling containing different la- bels for these coalitions cannot be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The largest set of rules which respects our assumption is the one in which every DL contains all of the possible conjunctions of size at most k, and only one of them returns 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' In this case, the num- ber of different rules r ∈ Θ(nk), so generally it holds that Pdim(H) = O(nk), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=', polynomial in n since k is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Now, let us show that Algorithm 2 produces a hypothe- sis consistent with the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The learning problem in this case is to associate the correct rule to each different β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' By the second assumption, every rule is represented as a k-DL and Algorithm 5 is known to return a k-DL consistent with a binary labeled sample (Rivest 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Algorithm 2, thus, pro- ceeds by assigning binary labels for every possible β sepa- rately, setting the label to 1 for coalitions whose value equals β and 0 for all others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Then, it calls Algorithm 5 as a sub- routine for determining the DL associated to this particular value β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The correctness of Algorithm 2 therefore follows from the correctness of Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' E Omitted Proofs from Section 5 Here we report the proofs of the results in Section 5 starting with Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' If H is PAC stabilizable, then H has SRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Assume that H does not satisfy SRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' This means that there exists a sample S∗ such that there exist H1, H2 ∈ H[S∗] such that CS∗(H1) ̸= ∅ and CS∗(H1) ∩ CS∗(H2) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' If we now define a distribution D with positive probability mass only on the elements of S∗, we will be able to conclude that H is not PAC stabilizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Indeed, no algorithm can dis- tinguish between H1 and H2 when sampling from D and, thus, cannot return a correct answer on both instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' The following are the missing proofs for the part on PAC stability of W-games with no ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let ε > 0 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' If we denote by A(i) j the event that a sampled coalition S satisfies i, j ∈ S and S \\ {i, j} ⊆ {j + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , n − 1}, and by B(i) j the event that a sampled coalition S satisfies i, j ∈ S and S \\ {i, j} ⊆ �� log2 1 ε � + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , n − 1 � , it holds that Pr S∼D � A(i) j � ≥ ε 2λ for 1 ≤ j ≤ � log2 1 ε � , and Pr S∼D � B(i) j � ≥ ε 4λ for j > � log2 1 ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let j ≤ � log2 1 ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' By Equation (2) we have that Pr S∼D � A(i) j � ≥ 2n−j−1 λ2n = 1 λ2j+1 ≥ 1 λ2⌊log2 1 ε⌋+1 ≥ ε 2λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Similarly, if j > � log2 1 ε � , we see that Pr S∼D � B(i) j � ≥ 2n−⌊log2 1 ε⌋−2 λ2n = 1 λ2⌊log2 1 ε⌋+2 ≥ ε 4λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let ε, δ > 0 and S be a sample of size m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' If m ≥ 2λ ε log n2 δ , LEARN-W-GAMES returns an ε-estimate ⃗v∗ of ⃗v with confidence 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Consider a fixed player i, let A(i) j , B(i) j be the events defined as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='6 and Sij as defined in Algorithm ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='. First, let j ≤ � log2 1 ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Observe that, if A(i) j holds for at least one S ∈ S, then v∗ i (j) = vi(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Indeed, in that case v∗ i (j) = maxS∈Sij vi(S) = vi(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='6, Pr S∼D � A(i) j � ≤ � 1 − ε 2λ �m ≤ e−mε/2λ ≤ δ n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Now, let j > � log2 1 ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Observe that, if B(i) j holds for at least one S ∈ S, then v∗ i (j) > vi( � log2 1 ε � ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Indeed, v∗ i (j) = maxS∈Sij vi(S) ≥ vi(⌊log2 1 ε⌋ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Again by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='6, Pr S∼D � B(i) j � ≤ � 1 − ε 4λ �m ≤ e−mε/4λ ≤ � δ n2 �2 ≤ δ n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Now, let A(i) = � 1≤j≤⌊log2 1 ε⌋ A(i) j and B(i) = � j>⌊log2 1 ε⌋ B(i) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' From the previously shown bounds, Pr S∼D � A(i) ∪ B(i) � ≤ (n − 1) δ n2 ≤ δ n, which means that the probability that v∗ i is not a ε-estimate of vi is less then δ/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Considering the whole ⃗v, by using a union bound, we obtain that ⃗v∗ is an ε-estimate of ⃗v with confidence 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let π be the output of Algorithm 3 and let us call a player i green if it is not in a coalition with one of his � log2 1 ε � least preferred choices according to v∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Then, a) for i green, PrS∼D [ i ∈ S ∧ vi(S) > vi(π(i)) ] < λε, b) for ε ≥ 3� λ2 2n , PrS∼D [ S does not contain green i ] < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' To prove a), it is enough to notice that for green i Pr S∼D [ i ∈ S ∧ vi(S) > vi(π(i)) ] ≤ Pr S∼D � S \\ {i} ⊆ �� log2 1 ε � + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' , n − 1 � � ≤2n−⌊log2 1 ε⌋−2λ 2n − 1 < λ 2⌊log2 1 ε⌋+1 ≤ λε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' To prove b), we start by observing that at least (n − � log2 1 ε � + 2)/2 players are green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Indeed, notice that at the beginning of iteration t there are exactly n − 2t + 2 players left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' A player picked by the algorithm is associated with the remaining player ranked the highest in their preference list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' If t < (n− � log2 1 ε � +2)/2, the number of remaining nodes is > � log2 1 ε � , so the player picked at iteration t is green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Now, we arrive at the statement by observing that Pr S∼D [ S does not contain green i ] ≤ λ 2n − 1(2n−(n−⌊log2 1 ε⌋+2)/2 − 1) < λ2n/2−1+⌊log2 1 ε⌋/2 2n−1 ≤ λ � 1/ε 2n/2 ≤ ε where the first inequality follows from the bound on the number of green players and Equation (2), and the last one from the assumption on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' We conclude with the proof of the last result of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' If T (H) = ω(poly(2n)) for a HG class H, then H is not efficiently PAC stabilizable, even under the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Let D be the uniform distribution on 2N and let us assume that a PAC stabilizing algorithm exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Given ε = 1 2n+1 , the PAC stabiliz ing algorithm must solve the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Indeed, the PAC stabilizing algorithm cannot return a partition that is not core stable, as the probability of sam- pling a blocking coalition is at least 1 2n−1 > ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' By defini- tion, the running time of the PAC stabilizing algorithm is polynomial in 1/ε and n, implying that its time complexity is O(poly(2n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFST4oBgHgl3EQfPzg1/content/2301.13756v1.pdf'} +page_content=' Therefore, we reached a contradiction.' metadata={'source': 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a/sNE3T4oBgHgl3EQf9AvA/content/tmp_files/2301.04813v1.pdf.txt b/sNE3T4oBgHgl3EQf9AvA/content/tmp_files/2301.04813v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f7bb6e12782d70fa21cfd8fc2515c0eb5d98a8df --- /dev/null +++ b/sNE3T4oBgHgl3EQf9AvA/content/tmp_files/2301.04813v1.pdf.txt @@ -0,0 +1,1473 @@ +arXiv:2301.04813v1 [math.AG] 12 Jan 2023 +ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF +COMPLEMENTS +GUODU CHEN, JINGJUN HAN, AND JIHAO LIU +Abstract. We study the relationship between Iitaka fibrations and the conjecture on the +existence of complements, assuming the good minimal model conjecture. In one direction, +we show that the conjecture on the existence of complements implies the effective log Iitaka +fibration conjecture. As a consequence, the effective log Iitaka fibration conjecture holds +in dimension 3. In the other direction, for any Calabi-Yau type variety X such that −KX +is nef, we show that X has an n-complement for some universal constant n depending only +on the dimension of X and two natural invariants of a general fiber of an Iitaka fibration +of −KX. We also formulate the decomposable Iitaka fibration conjecture, a variation of +the effective log Iitaka fibration conjecture which is closely related to the structure of +ample models of pairs with non-rational coefficients, and study its relationship with the +forestated conjectures. +Contents +1. +Introduction +1 +2. +Preliminaries +5 +3. +Canonical bundle formulas +12 +4. +Non-vanishing orders, middle Betti numbers, and complements +17 +5. +Effective Iitaka fibrations +21 +6. +Existence of decomposable Iitaka fibrations +22 +References +24 +1. Introduction +We work over the field of complex numbers C. +Let X be a smooth projective variety with non-negative Kodaira dimension. +By a +well-known construction of Iitaka, there exists a birational morphism X∞ → X from a +smooth projective variety X∞, and a contraction f∞ : X∞ → Z∞ onto a projective variety +Z∞, such that a very general fiber of f∞ is smooth with Kodaira dimension zero, and +dim Z∞ = κ(X, KX). The morphism f∞ : X∞ → Z∞ is referred to as an Iitaka fibration of +KX (see Definition 2.20). It is conjectured that the pluricanonical system |mKX| defines a +Date: January 13, 2023. +2020 Mathematics Subject Classification. 14C20,14E05,14E30,14J30. +1 + +2 +GUODU CHEN, JINGJUN HAN, AND JIHAO LIU +map which is birational to an Iitaka fibration whenever the positive integer m is divisible +by a positive integer depending only on the dimension of X: +Conjecture 1.1 (Effective Iitaka fibration, cf. +[HM06, Conjecture 1.7])). Let d be a +positive integer. Then there exists a positive integer md depending only on d, such that +for any smooth projective variety X of dimension d with non-negative Kodaira dimension, +|mKX| defines an Iitaka fibration for any positive integer m divisible by md. +Conjecture 1.1 was proved when KX is big [HM06, Tak06] (see also [Tsu99]), when +d = 2 due to Enriques (see also [Iit70]), and when d = 3 [Mor85, Kaw86, FM00, VZ07]. An +important progress towards Conjecture 1.1 is made in [BZ16], where the authors showed +that there exists a positive integer m depending only on d and two natural invariants of the +very general fibers of an Iitaka fibration of KX (the non-vanishing order and the middle +Betti number), such that |mKX| defines an Iitaka fibration. Unfortunately, we don’t know +the boundedness of the middle Betti numbers in dimension ≥ 3, which leaves Conjecture +1.1 open in dimension ≥ 4. +In practice, it is also natural to consider the following generalized version of Conjecture +1.1, which is known as the effective log Iitaka fibration conjecture. +Conjecture 1.2 (Effective log Iitaka fibration, cf. [TX09, Conjecture 1.1]). Let d be a +positive integer and Γ ⊂ [0, 1] a DCC set. Then there exists a positive integer m depending +only on d and Γ satisfying the following. +Assume that (X, B) is an lc pair of dimension d such that B ∈ Γ and κ(X, KX +B) ≥ 0. +Then |⌊m(KX + B)⌋| defines a map which is birational to an Iitaka fibration of KX + B. +Conjecture 1.2 was proved when KX + B is big [HMX14, Theorem 1.3]. When Γ ⊂ +[0, 1] ∩ Q and (X, B) is klt, Conjecture 1.2 was proved when d ≤ 2 [Tod10], when d ≤ 3 +and κ(X, KX) > 0 [TX09, Tod10], and when d = 4 and κ(X, KX) = 2 [TX09]. +Recently, the theory of complements, which was introduced by Shokurov in the study +of the existence of flips for threefolds [Sho92], has gradually become one of the major +topics in birational geometry. This theory has played an important role in the proof of the +BAB conjecture [Bir19, Bir21], the proof of the singular Yau-Tian-Donaldson conjecture +and the stable degeneration conjecture [Xu20, BLX22, LXZ22, XZ22], and recent studies +on Shokurov’s ascending chain condition conjecture for minimal log discrepancies [Liu18, +HLS19, HL22b, HLL22]. For other related work, we refer the readers to [Sho20, CGN21, +CZ21, CHX22, CX22, Liu22]. Although the existence of n-complements [Bir19, HLS19] is +settled for Fano type varieties, it is natural to consider the existence of n-complements for +pairs admitting an lc Calabi-Yau structure, that is, R-complementary varieties. We have +the following conjecture: +Conjecture 1.3 (Existence of complements). Let d be a positive integer and Γ ⊂ [0, 1] a +DCC set. Then there exists a positive integer n depending only on d and Γ satisfying the +following. +Assume that (X/Z ∋ z, B) is an R-complementary pair of dimension d such that B ∈ Γ, +then (X/Z ∋ z, B) has an n-complement (X/Z ∋ z, B+). Moreover, if the closure of Γ +belongs to [0, 1] ∩ Q, then we can pick B+ ≥ B. + +ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS +3 +Relationship between the conjectures. It is interesting to ask whether we can establish +some connections between Conjectures 1.1, 1.2 and Conjecture 1.3. At first glance, it is +difficult to observe their relationships, as these conjectures are considering structures of +varieties and pairs with completely different positivity properties: Conjectures 1.1, 1.2 are +concentrating on pairs with positive canonical bundle (i.e., KX + B is effective), while +Conjecture 1.3 is concentrating on varieties with negative canonical bundle (i.e., −(KX + +B) is effective). +Surprisingly, we have the following theorems, which show that these +conjectures are actually deeply related with each other in multiple directions. +From complements to Iitaka fibrations. +First, we prove that Conjecture 1.3 implies +Conjecture 1.2 assuming the good minimal model conjecture. +Theorem 1.4. Let d be a positive integer. Assume that the good minimal model conjecture +(Conjecture 2.11) and the existence of complements (Conjecture 1.3) hold in dimension d. +Then the effective log Iitaka fibration conjecture (Conjecture 1.2) holds in dimension d. +As an immediate corollary, we prove Conjecture 1.2 in dimension ≤ 3: +Corollary 1.5. Conjecture 1.2 holds when d ≤ 3. +From Iitaka fibrations to complements. Next, we show that the existence of n-complements +is deeply related to some invariants associated to Iitaka fibrations. We have the following +theorem: +Theorem 1.6. Let d, b, and β be three positive integers. Assume that the good minimal +model conjecture (Conjecture 2.11) holds in dimension d. +Then there exists a positive +integer n depending only on d, b, and β satisfying the following. +Assume that X is a Q-factorial normal projective variety of dimension d, f∞ : X∞ → Z∞ +is an Iitaka fibration of −KX, h : X∞ → X is the induced birational morphism, and F is +a general fiber of f∞. Suppose that +(1) −KX is nef, +(2) X has a klt R-complement, +(3) κ(X, −KX) ≥ 0, and b is the non-vanishing order of −h∗KX|F , i.e., +b = min {a ∈ Z>0 | | − ah∗KX|F | ̸= ∅} , +and +(4) β := dim Hdim F ( ˜F, C), where ˜F is a smooth model of the cover of F associated to +the unique divisor of | − bh∗KX|F |. +Then X has an n-complement. +We remark that the assumptions on b and β in Theorem 1.6 are natural assumptions, +and are exactly the additional assumptions in [BZ16, Theorem 1.2] on the effective Iitaka +fibration conjecture. We also note that, modulo the good minimal model conjecture, the +boundedness of b follows immediately from the effective log Iitaka conjecture (Conjecture +1.2) for log pairs with Iitaka dimension 0. +Decomposable Iitaka fibrations. For any semi-ample divisor D, the ample model of D +is clearly birational to an Iitaka fibration of D. However, if D is only assumed to be an + +4 +GUODU CHEN, JINGJUN HAN, AND JIHAO LIU +R-divisor, then it is possible that D does not have any Iitaka fibration although the ample +model of D exists [CHL22, Example 1.2]. An approach to resolve this issue is to define the +invariant Iitaka fibration (see Definition 6.1). A question arises naturally: do we expect +any kind of uniform effectively on invariant Iitaka fibrations for log pairs, similar to the +effective log Iitaka fibration conjecture? +The question is expected to have a positive answer. More precisely, suppose that (X, B) +is an lc pair such that KX + B induces a map X ��� Z that is birational to an invariant +Iitaka fibration of KX + B. Then we expect that X ��� Z will actually be birational to an +Iitaka fibration of KX + B′ after we (uniformly) perturb the coefficients of the boundary +B to get a new boundary B′. Therefore, the effective log Iitaka fibration conjecture should +induce some kind of uniform effectivity on an invariant Iitaka fibration induced by KX +B. +The difficulty is to show that we can make a uniform perturbation to switch the +boundary B to a new boundary B′. In [CHL22, Theorem 1.1], the authors prove a weaker +version, which shows that, modulo the non-vanishing conjecture, there exists such uniform +perturbation such that an Iitaka dimension of KX + B′ is equal to an invariant Iitaka +dimension of KX + B. In this paper, we will show that an Iitaka fibration of KX + B′ is +actually equal to an Iitaka fibration as KX + B. To make our statements more clear, we +introduce the concept of decomposable Iitaka fibrations. +Definition 1.7 (Decomposable Iitaka fibrations). Let Γ0 := {a1, . . . , ak} ⊂ (0, 1] be a +finite set such that �k +i=1 ai = 1, and Γ′ ⊂ [0, 1] a set. Assume that (X, B) is an lc pair +such that κι(X, KX + B) ≥ 0. +We say that (X, B) has a (Γ0, Γ′)-decomposable Iitaka +fibration if there exist R-divisors B1, . . . , Bk ∈ Γ′ such that +(1) B = �k +i=1 aiBi, +(2) (X, Bi) is lc for any 1 ≤ i ≤ k, and +(3) any invariant Iitaka fibration of KX + B is an Iitaka fibration of KX + Bi for any +1 ≤ i ≤ k. +In addition, if +(4) the map defined by |⌊m(KX + Bi)⌋| is birational to any invariant Iitaka fibration +of KX + B for any integer 1 ≤ i ≤ k, +then we say that (X, B) has an (m, Γ0, Γ)-decomposable Iitaka fibration. +As an analogue of the effective log Iitaka fibration conjecture, we propose the following +conjecture on the existence of decomposable Iitaka fibrations. +Roughly speaking, the +conjecture indicates that we can uniformly perturb an invariant Iitaka fibration to get +an effective log Iitaka fibration. +Conjecture 1.8 (Decomposable Iitaka fibrations). Let d be a positive integer and Γ ⊂ [0, 1] +a DCC set. Then there exist a positive integer m, a finite set Γ0 ⊂ (0, 1], and a DCC set +Γ′ ⊂ [0, 1] depending only on d and Γ satisfying the following. Assume that (X, B) is a +Q-factorial lc pair of dimension d such that B ∈ Γ and κι(X, KX + B) ≥ 0. Then: +(1) (Weak version) (X, B) has a (Γ0, Γ′)-decomposable Iitaka fibration. +(2) (Strong version) (X, B) has an (m, Γ0, Γ′)-decomposable Iitaka fibration. + +ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS +5 +Conjecture 1.8 will help us to understand the structure of good minimal models and +their ample models for pairs with real coefficients. We show that Conjecture 1.8 follows +from the non-vanishing conjecture (Conjecture 2.12) and the effective log Iitaka fibration +conjecture (Conjecture 1.8(2)). +Theorem 1.9. Let d be a positive integer. +Assume that the non-vanishing conjecture +(Conjecture 2.12) holds in dimension d. Then: +(1) Conjecture 1.8(1) holds in dimension d. +(2) Assume that Conjecture 1.2 holds in dimension d. Then Conjecture 1.8(2) holds +in dimension d. +As an immediate corollary, we have: +Corollary 1.10. Conjecture 1.8 holds when d ≤ 3. +Combining Theorems 1.4 and 1.9, we show that Conjecture 1.8 follows from the good +minimal model conjecture and the existence of complements. +Theorem 1.11. Let d be a positive integer. Assume that the good minimal model conjecture +(Conjecture 2.11) and the existence of complements (Conjecture 1.3) hold in dimension d. +Then Conjecture 1.8 holds in dimension d. +Finally, we remark that we expect all theorems in our paper to hold in the relative setting. +That is, instead of considering Iitaka fibrations for projective varieties and projective pairs, +we may also consider Iitaka fibrations in the relative case (cf. [Li22, Definition 3.19]). +Structure of the paper. In Section 2, we introduce some notation and tools which will +be used in this paper. In Section 3, we recall the canonical bundle formulas and prove +Proposition 3.1. In Section 5, we prove Theorem 1.4. In Section 4, we prove Theorem 1.6. +In Section 6, we introduce invariant Iitaka fibrations and prove Theorems 1.9 and 1.11. +Acknowledgement. The second and the third named authors began this work when they +worked on [HL20] in Summer 2020. Part of this work was done while the first named author +visited Chuyu Zhou at EPFL in Summer 2022. He would like to thank their hospitality. +The authors would like to thank Qianyu Chen, Christopher D. Hacon, Fei Hu, Chen Jiang, +Junpeng Jiao, Zhan Li, Haidong Liu, Wenfei Liu, Yuchen Liu, Yujie Luo, Fanjun Meng, +Lingyao Xie, Qingyuan Xue, and Chuyu Zhou for valuable discussions and suggestions. +The authors would like to thank Enrica Floris for answering questions about Theorem 1.6. +The first named author was supported by the China post-doctoral grants BX2021269 and +2021M702925. The second named author was supported by National Key Research and +Development Program of China (Grant No. 2020YFA0713200). The second named author +is a member of LMNS, Fudan University. +2. Preliminaries +We adopt the standard notation and definitions in [KM98, BCHM10] and will freely use +them. + +6 +GUODU CHEN, JINGJUN HAN, AND JIHAO LIU +2.1. Divisors. +Definition 2.1. Let Γ ⊂ R be a set. We say that Γ satisfies the ascending chain condition +(ACC) if any increasing sequence in Γ stabilizes. We say that Γ satisfies the descending +chain condition (DCC) if any decreasing sequence in Γ stabilizes. +Definition 2.2 ([PS09, 3.2], [Bir19, 2.2]). Let R ⊂ [0, 1] ∩ Q be a finite set, we define +Φ(R) := +� +1 − γ +n | γ ∈ R, n ∈ Z≥1 +� +. +We say that a set Γ ⊂ [0, 1] is a hyperstandard set if there exists a finite set R ⊂ [0, 1] ∩ Q +such that 0, 1 ∈ R and Γ = Φ(R). +Definition 2.3. We say f : X → Z is a contraction if π is a projective morphism, and +f∗OX = OZ. We say that a birational map φ : X ��� Y is a birational contraction if φ is +projective and φ−1 does not contract any divisors. +Definition 2.4. Let Γ ⊂ R be a set, X a variety, and B := �s +i=1 biBi an R-divisor on X, +where Bi are the irreducible components of B. We write B ∈ Γ if bi ∈ Γ for every i. We +define +|B| := max +1≤i≤s |bi|, ⌊B⌋ := +s +� +i=1 +⌊bi⌋Bi, and {B} := +s +� +i=1 +{bi}Bi. +We may denote by K(X) the rational function field of X. +Let f : X → Z be a contraction between normal quasi-projective varieties and D an +R-divisor on X. We say that D is vertical over Z if f(SuppD) is a proper subset of Z. +We say that D is horizontal over Z if D is not vertical over Z. We may uniquely write +D = Dh + Dv such that Dv is vertical over Z and each component of Dh is horizontal over +Z. We call Dh the horizontal/Z part of D and Dv the vertical/Z part of D. +Lemma 2.5. Let f : X → Z be a contraction between normal quasi-projective varieties +and L a Cartier divisor on X such that L ∼Z 0. Suppose either L is vertical over Z or +L ≥ 0. Then there is a Cartier divisor LZ on Z such that L = f ∗LZ. +Proof. If L ≥ 0, then L is vertical over Z. Thus it suffices to prove the statement under the +condition that L is vertical over Z. By assumption there exist a rational function s ∈ K(X) +and a Cartier divisor L′ +Z on Z such that L − f ∗L′ +Z = (s). Since L is vertical over Z, we +may find an open subset V ⊂ Z such that (s)|U ≥ 0, where U := f −1V ⊂ X. In particular, +s ∈ OX(U) and thus s ∈ OZ(V ) as f is a contraction and OZ(V ) = (f∗OX)(V ) = OX(U). +Hence s ∈ K(V ) = K(Z) ֒→ K(X) and φ(s) = s, where φ : K(Z) ֒→ K(X). +Thus +f ∗(s) = (φ(s)) = (s). Set LZ := L′ +Z + (s) and we are done. +□ +Lemma 2.6. Let f : X → Z be a contraction between normal quasi-projective varieties, +D a divisor on X, and U1, U2 ⊂ Z two open subsets such that D|f−1(Ui) ∼Ui 0 for i = 1, 2. +Then D|f−1(U1∪U2) ∼U1∪U2 0. +Proof. Replacing Z by U1 ∪ U2, we may assume that Z = U1 ∪ U2. By assumption, +D|f−1(Ui) = f ∗ +i Di + (si) + +ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS +7 +for some Cartier divisor Di on Ui and si ∈ K(X), where fi := f|f−1(Ui) for i = 1, 2. Let +f12 := f|f−1(U1∩U2). Then (f12)∗(D1 − D2) = (s2 +s1) on f −1(U1 ∩ U2). By the projection +formula, we have +OU1∩U2 ∼= (f12)∗ +� +(s2 +s1 +)OX|f−1(U1∩U2) +� += (f12)∗(f12)∗OU1∩U2(D1 − D2) += OU1∩U2(D1 − D2). +Thus (D1 − D2)|U1∩U2 = (sZ) and (f12)∗(sZ) = (s2 +s1) over U1 ∩ U2 for some sZ ∈ K(Z). In +particular, (s2 +s1) − f ∗(sZ) is vertical over Z. Note that (s2 +s1 ) − f ∗(sZ) ∼Z 0. By Lemma 2.5, +there exists a Cartier divisor LZ on Z such that +(s2 +s1 +) − f ∗(sZ) = f ∗LZ. +Moreover, we have SuppLZ ∩ (U1 ∩ U2) = ∅, and LZ + (sZ) = D1 − D2 on U1 ∩ U2 as +(D1 − D2)|U1∩U2 = (sZ). Hence there exists a Cartier divisor D′ on Z, such that +D′ = D1 on U1 and D′ = D2 + (sZ) + LZ on U2. +It follows that f ∗D′ = f ∗D1 = D − (s1) over U1 and f ∗D′ = f ∗(D2 + (sZ) + LZ) = +D − (s2) + (s2 +s1) = D − (s1) over U2. Hence D = f ∗D′ − (s1), and thus D ∼Z 0. +□ +2.2. Pairs and singularities. +Definition 2.7 (Pairs, cf. [CH21, Definition 2.2]). A pair (X/Z ∋ z, B) consists of a +contraction f : X → Z between normal quasi-projective varieties, a (not necessarily closed) +point z ∈ Z, and an R-divisor B ≥ 0 on X, such that KX + B is R-Cartier over a +neighborhood of z. We may also call it an R-pair. If B ∈ Q, then we call (X/Z ∋ z, B) a +Q-pair. If f = id and z = x ∈ X, then we may use (X ∋ x, B) instead of (X/Z ∋ z, B). If +(X/Z ∋ z, B) (resp., (X ∋ x, B)) is a pair for any point z ∈ Z (resp., x ∈ X), then we call +(X/Z, B) (resp., (X, B)) a pair. +Definition 2.8 (Singularities of pairs). Let (X/Z ∋ z, B) be a pair associated with the +contraction f : X → Z, and let E be a prime divisor over X such that z ∈ f(centerX E). +Let g : Y → X be a log resolution of (X, B) such that centerY E is a divisor, and suppose +that KY + BY = g∗(KX + B) over a neighborhood of z. We define a(E, X, B) := 1 − +multE BY to be the log discrepancy of E with respect to (X, B). +We say that a prime divisor E is over X/Z ∋ z if E is a prime divisor E over X and +f(centerX E) = ¯z. We say that (X/Z ∋ z, B) is lc (resp., klt) if a(E, X, B) ≥ 0 (resp., > 0) +for any prime divisor E over X/Z ∋ z. We say that (X/Z, B) is lc (resp., klt) if (X ∋ x, B) +is lc (resp., klt) for any codimension ≥ 1 point x ∈ X. We say that (X/Z, B) is dlt if there +exists a log resolution g : Y → X of (X, B) such that a(E, X, B) > 0 for any g-exceptional +prime divisor E. +Lemma 2.9. Assume that (X/Z, B) is an lc pair and D is an R-Cartier R-divisor on X +such that KX + B ∼R,Z D and D is vertical over Z. Suppose that ψ : X ��� X′ is a + +8 +GUODU CHEN, JINGJUN HAN, AND JIHAO LIU +partial (KX + B)-MMP over Z. Then there exists a proper open subset U ⊂ Z such that +Xz is isomorphic to X′ +z for any z ∈ U, where Xz (resp., X′ +z) is the fiber of X → Z (resp., +X′ → Z) over z. +Proof. Let f : X → Z be the associated morphism. Since KX +B ∼R 0 over Z\f(SuppD), +ψ is an isomorphism over Z\f(SuppD), so we may choose U := Z\f(SuppD). +□ +Lemma 2.10. Assume that (X/Z ∋ z, B) is an lc pair such that Z is a curve, KX+B ∼R 0 +over a neighborhood of z, Bh ∈ Q, and z is a closed point. Then B + sf ∗z is a Q-divisor +over a neighborhood of z, where s := lct(X/Z ∋ z, B; f ∗z) and f : X → Z. +Proof. Possibly shrinking Z we may assume that KX + B ∼R,Z 0 and f(SuppBv) = {z}. +There exist real numbers r1, . . . , rc, Q-linear functions s1, . . . , sm : Rc+1 → R, and Weil +divisors B1, . . . , Bm on X, such that 1, r1, . . . , rc are linearly independent over Q, and +B = �m +i=1 si(1, r0)Bi, where r0 := (r1, . . . , rc). +Let B(v) := �m +i=1 si(1, v)Bi for any +v ∈ Rc. Since Bh ∈ Q, Bh = B(v)h ∈ Q for any v ∈ Rc. +By [HLS19, Lemma 5.3], KX + B(v) ∼R,Z 0 for any v ∈ Rc and thus B(v) − B ∼R,Z 0. +Moreover, as f(Supp(B(v) − B)) = f(SuppBv) = {z}, we see that B − B(v) = lvf ∗z for +some real number lv. Pick v0 ∈ Qc, then +s + lv0 = lct(X/Z ∋ z, B(v0); f ∗z) ∈ Q +as B(v0) ∈ Q. Since +B + sf ∗z = B(v0) + (s + lv0)f ∗z, +B + sf ∗z is a Q-divisor. This finishes the proof. +□ +Conjecture 2.11 (Good minimal model conjecture). Let d be a positive integer. Assume +that (X/Z, B) is an lc pair of dimension d such that KX + B is pseudo-effective over Z. +Then (X/Z, B) has a good minimal model over Z. +Conjecture 2.12 (Non-vanishing conjecture). Let d be a positive integer. Assume that +(X/Z, B) is an lc pair of dimension d such that KX + B is pseudo-effective over Z. Then +|(KX + B)/Z|R ̸= ∅. +2.3. Complements. +Definition 2.13. Let n be a positive integer, Γ ⊂ (0, 1] a set, and (X/Z ∋ z, B) and +(X/Z ∋ z, B+) two pairs. We say that (X/Z ∋ z, B+) is an R-complement of (X/Z ∋ z, B) +if (X, B+) is lc, B+ ≥ B, and KX + B+ ∼R 0 over a neighborhood of z. We say that +(X/Z ∋ z, B) is R-complementary if (X/Z ∋ z, B) has an R-complement. +We say that (X/Z ∋ z, B+) is an n-complement of (X/Z ∋ z, B) if +• (X/Z ∋ z, B+) is lc, +• nB+ ≥ ⌊(n + 1){B}⌋ + n⌊B⌋, and +• n(KX + B+) ∼ 0 over a neighborhood of z. +We say that (X/Z ∋ z, B+) is an (n, Γ)-decomposable R-complement of (X/Z ∋ z, B) if +there exist real numbers a1, . . . , ak ∈ Γ, and Q-divisors B+ +1 , . . . , B+ +k on X, such that +• �k +i=1 ai = 1 and �k +i=1 aiB+ +i = B+, + +ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS +9 +• (X/Z ∋ z, B+) is an R-complement of (X/Z ∋ z, B), and +• (X/Z ∋ z, B+ +i ) is an n-complement of itself for any integer 1 ≤ i ≤ k. +Conjecture 1.3 holds true when d = 3. More precisely, we have the following. +Theorem 2.14. Let l be a positive integer and Γ ⊂ [0, 1] a DCC set. Then there exists a +positive integer n which is divisible by l depending only on l and Γ satisfying the following. +Assume that (X/Z ∋ z, B) is an R-complementary pair of dimension 3 with B ∈ Γ. +Then (X/Z ∋ z, B) has an n-complement (X/Z ∋ z, B+). Moreover, if SpanQ≥0(¯Γ\Q) ∩ +(Q\{0}) = ∅, then we can pick B+ ≥ B. +Proof. This follows from [FMX19, Theorem 1] and [HLS19, Theorem 8.25]. +□ +2.4. Iitaka dimensions and invariant Iitaka dimensions. +Definition 2.15 (Iitaka dimensions, cf. [Nak04, II 3.2 Definition]). Let X be a normal +projective variety and D an R-divisor on X. For any positive integer m such that |⌊mD⌋| ̸= +∅, we denote Φm : X ��� P(H0(X, ⌊mD⌋)). The Iitaka dimension κ(X, D) of D is defined +in the following way. If |⌊mD⌋| ̸= ∅ for some positive integer m, then +κ(X, D) := max{dim Φm(X) | m ∈ Z>0, |⌊mD⌋| ̸= ∅}. +Otherwise, let κ(X, D) := −∞. Note that if |⌊mD⌋| ̸= ∅, then by [Nak04, II 3.8 Corollary], +κ(X, D) = max +� +k ∈ Z>0 | lim sup +m→+∞ +dim H0(X, ⌊mD⌋) +km +> 0 +� +. +Definition 2.16 (Invariant Iitaka dimensions, cf. [Cho08, Definition 2.2.1]). Let X be +a normal projective variety and D an R-divisor on X. The invariant Iitaka dimension +κι(X, D) of D is defined as follows. If |D|R ̸= ∅, then we define +κι(X, D) := κ(X, D′) +for some R-divisor D′ ∈ |D|R. Otherwise, we let κι(X, D) := −∞. Note that κι(X, D) is +independent of the choice of D′ [Cho08, Corollary 2.1.4]. +We gather some basic properties of κ and κι which will be used in the rest of this paper. +Proposition 2.17 ([Cho08, Propositions 2.1.2, 2.2.2, Corollary 2.1.4]). Let X be a normal +projective variety, and D ∼R D′ two R-Cartier R-divisors on X. Then +(1) κ(X, D) ≤ κι(X, D) = κι(X, D′), and κ(X, D) < κι(X, D) if and only if κ(X, D) = +−∞ and κι(X, D) ≥ 0. +(2) If D′ ≥ 0, then κ(X, D) ≤ κ(X, D′). +[CHL22, Example 2.10] shows that we may have strict inequality in Proposition 2.17(1). +Lemma 2.18 ([Sho03, Proposition 3.20], [Nak04, II Lemma 3.11]). Let g : Y → X be a +surjective morphism between normal projective varieties and D an R-Cartier R-divisor on +X. Then +(1) κ(X, D) = κ(Y, g∗D) and κι(X, D) = κι(Y, g∗D), and +(2) if g is birational, then κ(X, D) = κ(Y, g∗D + E) and κι(X, D) = κι(Y, g∗D + E) +for any g-exceptional R-Cartier R-divisor E ≥ 0 on Y . + +10 +GUODU CHEN, JINGJUN HAN, AND JIHAO LIU +2.5. Iitaka fibrations for R-divisors. +Definition 2.19. Assume that f : X ��� Z is a rational map and f∞ : X∞ → Z∞ is a +projective morphism. We say that f is birational to f∞ if there exist a birational morphism +h : X∞ → X and a birational map g : Z∞ ��� Z′ such that f ◦h = g◦f∞, i.e., the following +diagram is commutative: +X∞ +f∞ +� +h +� X +f +�✤ +✤ +✤ +Z∞ +g +�❴ +❴ +❴ +❴ +❴ +❴ +Z. +Definition 2.20 (cf. [Li22, Definition 3.19]). Let X be a normal projective variety and +D an R-Cartier R-divisor on X such that κ(X, D) ≥ 0. +A projective morphism f∞ : +X∞ → Z∞ between quasi-projective smooth varieties is called an Iitaka fibration of D if +the following hold: +(1) dim Z∞ = κ(X, D), +(2) fm : X ��� Zm ⊂ PH0(X, ⌊mD⌋), the map associated with the complete linear +system |⌊mD⌋|, is birational to f∞ with the morphism h : X∞ → X, for any +sufficiently divisible large integer m, and +(3) for any sufficiently large integer n, we have +κ +� +F, +� +h−1 +∗ D + nE +� +|F +� += 0, +where F is a very general fiber of f∞ and E is the sum of all the h-exceptional +prime divisors. +By [Li22, Proposition 3.20], for any R-Cartier R-divisor D such that κ(X, D) ≥ 0, there +always exists an Iitaka fibration of D. The following lemmas are well-known to experts. +Lemma 2.21. Notation as in Definition 2.20. +(1) Assume that D∞ is an R-divisor on X∞ such that D∞ − h∗D ≥ 0 and is h- +exceptional, then f∞ is an Iitaka fibration of D∞. +(2) Assume that X′ +∞ → X∞ is a birational morphism from a smooth variety, then +X′ +∞ → Z∞ is an Iitaka fibration of D. +Proof. (1) By assumption, H0(X, ⌊mD⌋) = H0(X∞, ⌊mD∞⌋) and X∞ ��� Zm is also the +map associated with the complete linear system |⌊mD∞⌋|. +Then (1) follows from the +definition. +(2) We only need to show κ(F ′, ((h′)−1 +∗ D +nE′)|F ′) = 0 for any sufficiently large integer +n, where h′ : X′ +∞ → X, F ′ is a very general fiber of X′ +∞ → Z∞ and E′ is the sum of all the +h′-exceptional prime divisors. Let ψ′ +∞ : W → X′ +∞ be a resolution which resolves the map +X ��� Zm, and we denote ψ : W → X the induced morphism. By [Li22, Lemma 3.10], for +any sufficiently large integer n, we have that +κ +� +Fm, +� +(ψ)−1 +∗ +D + nEW +� +|Fm +� += 0, + +ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS +11 +where Fm is a very general fiber of W → Zm and EW is the sum of all the ψ-exceptional +prime divisors. Let E1 be the sum of all the ψ′ +∞-exceptional prime divisors. Since EW = +(ψ′ +∞)−1 +∗ E′ + E1 and +ψ−1 +∗ D + n +� +ψ′ +∞ +�−1 +∗ +E′ + (n + m)E1 − +� +ψ′ +∞ +�∗ �� +h′�−1 +∗ +D + nE′� +≥ 0 +for any sufficiently large integers n and m, we see that +κ +� +Fm, +�� +ψ′ +∞ +�∗ � +h−1 +∗ D + nE1 +�� +|Fm +� += 0. +Note that Z∞ is birational to Zm, thus there exists a birational morphism from Fm to F ′. +Therefore κ(F ′, ((h′)−1 +∗ D + nE′)|F ′) = 0. This finishes the proof. +□ +Lemma 2.22. Notation as in Definition 2.20. Suppose that φ : X′ → X is a birational +morphism from a normal projective variety and D′ is an R-Cartier R-divisor on X′ such +that D′ − φ∗D ≥ 0 and is φ-exceptional. Then +(1) possibly replacing X∞ with a high model, f∞ is an Iitaka fibration of D′, and +(2) any Iitaka fibration of D′ is an Iitaka fibration of D. +Proof. By our assumption, H0(X, ⌊mD⌋) = H0(X∞, ⌊mD∞⌋) and X ��� Zm is the map +associated with the complete linear system |⌊mD⌋| if and only if X′ ��� Zm is the map +associated with the complete linear system |⌊mD′⌋|. +(1) By Lemma 2.21(2), possibly replacing X∞ with a high model, we may assume that +the map X∞ ��� X′ is a morphism and we denote it by h0. It suffices to prove that for +any sufficiently large integer n, +κ +� +F, +� +(h0)−1 +∗ D′ + nE0 +� +|F +� += 0, +where E0 is the sum of all the h0-exceptional prime divisors. This follows from the fact +that +(h0)−1 +∗ D′ + nE0 ≤ h−1 +∗ D + nE +for any sufficiently large integer n. +(2) Suppose that f ′ +∞ : X′ +∞ → Z′ +∞ is an Iitaka fibration of D′. +Let W → X′ +∞ and +W → X∞ be a common resolution of X∞ and X′ +∞. By Lemma 2.21(2), W → Z∞ is also +an Iitaka fibration of D. It follows that X′ +∞ → Z∞ is also an Iitaka fibration of D. Since +Z∞ is birational to Z′ +∞, X′ +∞ → Z′ +∞ is an Iitaka fibration of D. +□ +Lemma 2.23. Notation as in Definition 2.20. Assume that ψ : X ��� X′ is a D-non- +negative birational contraction. Then possibly replacing X∞ with a high model, f∞ is an +Iitaka fibration of D′. +Proof. By Lemma 2.21(2), possibly replacing X∞ with a high model, we may assume +that the map h′ : X∞ ��� X′ is a morphism. By our assumption, 0 ≤ h∗D − h′∗D′ is h′- +exceptional, and X′ ��� Zm is the map associated with the complete linear system |⌊mD′⌋|. +By Lemma 2.21(1), f∞ is an Iitaka fibration of h∗D. Then by Lemma 2.22(2), f∞ is also +an Iitaka fibration of D′. +□ +Remark 2.24. In the rest of this paper, we will use Lemmas 2.21, 2.22 and 2.23 frequently +to replace X with another birational model without citing. + +12 +GUODU CHEN, JINGJUN HAN, AND JIHAO LIU +Lemma 2.25. Assume that (X, C) is a projective klt pair such that KX +C ∼R 0. Assume +that Conjecture 2.12 holds in dimension ≤ dim X. +Suppose that D is a Q-Cartier Q- +divisor on X such that κ(X, D) ≥ 0, and f∞ : X∞ → Z∞ is an Iitaka fibration of D. +Then κ(F, h∗D|F ) = 0, where F is a general fiber of f∞ and h : X∞ → X is the induced +birational morphism. +Proof. By Shokurov type polytopes, we may assume that C ∈ Q and thus KX + C ∼Q 0. +Take a positive rational number ǫ such that (X∞, h−1 +∗ C + (1 − ǫ)E) is klt and +E′ := KX∞ + h−1 +∗ C + (1 − ǫ)E − h∗(KX + C) ≥ 0, +where E is the sum of all the h-exceptional prime divisors. Since κ(X, D) ≥ 0, one can +find a Q-Cartier Q-divisor D′ ≥ 0 such that D′ ∼Q D. Let ǫ′ be a positive rational number +such that (X∞, h−1 +∗ C + (1− ǫ)E + ǫ′h∗D′) is still klt. As f∞ is an Iitaka fibration of D and +E′ is h-exceptional, one can see that +κ +� +Fv, +� +KX∞ + h−1 +∗ C + (1 − ǫ)E + ǫ′h∗D′� +|Fv +� += κ +� +Fv, +� +E′ + ǫ′h∗D′� +|Fv +� += 0, +where Fv is a very general fiber of f∞. According to [HMX18, Corollary 1.4], we have +κ +� +F, +� +KX∞ + h−1 +∗ C + (1 − ǫ)E + ǫ′h∗D′� +|F +� += 0. +Therefore κ(F, h∗D|F ) = 0. +□ +3. Canonical bundle formulas +We refer the reader to [BZ16, Bir19, HL21a, HL22a] for the definitions and basic +properties for generalized pair (g-pair for short), and we denote by (X/Z, B + M) a g-pair +throughout this paper. We refer the reader to [Bir19, HL21b, JLX22] for the definition and +basic properties of the canonical bundle formula. To sum up, given an lc pair (X/Z, B) +and a contraction φ : X → T between normal quasi-projective normal varieties over Z such +that KX + B ∼R,T 0. Then we can find an R-divisor BT ≥ 0 and a nef over Z b-R-divisor +Mφ on T, such that (T/Z, BT + Mφ) is a glc g-pair, and +KX + B ∼R φ∗ (KT + BT + Mφ,T ) . +Here B (resp., Mφ) is called the discriminant part (resp., a moduli part) of the canonical +bundle formula for (X/Z, B) over T which is uniquely determined (resp., determined only +up to R-linear equivalence). We may also call Mφ,T the moduli part of the canonical bundle +formula for (X/Z, B) over T. Moreover, if (X/Z, B) is klt, then (T/Z, BT + Mφ) is gklt. +Here we emphase that there are many choices of Mφ, some of which could behave badly. +But we can always choose one with the required properties in the following results. +For convenience, we say that two g-pairs (X/Z, B +M) and (X′/Z, B′ +M′) are crepant +if X is birational to X′, M = M′, and p∗(KX + B + MX) = q∗(KX′ + B′ + M′ +X′) for some +common resolution p : W → X and q : W → X′. We also call (X′/Z, B′ + M) a crepant +model of (X/Z, B + M). +Proposition 3.1. Let d be a positive integer and Φ ⊂ [0, 1] a DCC set. Assume that +Conjectures 1.3 and 2.11 hold in dimension d. Then there exist a positive integer p and a +DCC set Φ′ depending only on d and Φ satisfying the following. + +ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS +13 +Assume that (X/Z, B) is an lc pair of dimension d and φ : X → T is a contraction over +Z such that dim T > 0, B ∈ Φ, KX +B ∼R,T 0, and KX +B ∼Q,T 0 over the generic point +of T. Then we can choose a moduli part Mφ of the canonical bundle formula for (X/Z, B) +over T such that BT ∈ Φ′, pMφ is b-Cartier, and +p(KX + B) ∼ pφ∗(KT + BT + Mφ,T ), +where BT is the discriminant part of the canonical bundle formula for (X/Z, B) over T. +Moreover, if Φ is a hyperstandard set, then Φ′ is a hyperstandard set. +The proof is similar to the proof of [Bir19, Proposition 6.3]. For the convenience of the +reader, we give a proof here. We also remark that the moreover part of the proposition +will not be used in this paper, but it is useful in some other situations (cf. [CHX22]). +Proof. Step 1. In this step, we construct p and make a choice of Mφ,T . Note that here +we only make a choice of Mφ,T rather than Mφ. +By [HLS19, Theorems 1.8 and 8.25], there exist a positive integer p and a finite set +Γ0 ⊂ (0, 1] depending only on d and Φ, such that for any t ∈ T, (X/T ∋ t, B) has +a (p, Γ0)-decomposable R-complement (X/T ∋ t, B + G) for some R-Cartier R-divisor +G ≥ 0, and moreover if B ∈ Γ ∩ Q, then (X/T ∋ t, B) has a monotonic p-complement. +In particular, G ∼R 0 over a neighborhood of t, and hence G is vertical over T. Since +KX + B ∼Q,T 0 over the generic point ηT of T, p(KX + B) ∼ 0 over a neighborhood of ηT , +and there exists α ∈ K(X) such that pL := p(KX +B)+(α) is zero near ηT . In particular, +pL ∼ p(KX + B) ∼R,T 0 and L is vertical over T. By [Li20, Lemma 2.11], we may find an +R-Cartier R-divisor LT on T such that L = φ∗LT . Let BT be the discriminant part of the +canonical bundle formula of (X, B) over T, and Mφ,T := LT − KT − BT . Then +p(KX + B) ∼ pL = pφ∗LT = pφ∗(KT + BT + Mφ,T ). +Step 2. In this step, we show that we can reduce to the case dim T = 1 to show the +existence of Φ′ and prove that pMφ,T is integral. +Assume dim T > 1. +Let H be a general hyperplane section of T, G := φ∗H, and +g : G → H the induced morphism. We may write KG + BG = (KX + G + B)|G. It is clear +that (G, BG) is an lc pair with KG +BG ∼Q,H 0. Suppose that BH is the discriminant part +of the canonical bundle formula for (G, BG) over H. Note that as G is a general member +of a free linear system, every lc center SG of (G, BG) is a component of S0 ∩ G for some lc +center S0 of (X, B). +We claim that multD BT = multC BH for any prime divisor D on T and any component +C of D ∩ H. Indeed, let sD be the lc threshold of φ∗D with respect to (X, B) over the +generic point of D. Then there is an lc center F of (X, B + sDφ∗D) such that φ(F) = D. +Note that F is also an lc center of (X, B +G+sDφ∗D) as G is general. Hence F ∩G is an lc +center of (G, BG+sDg∗C), by inversion of adjunction [Kaw07]. Moreover, as φ(F ∩G) = C, +we see that sD is the lc threshold of g∗C with respect to (G, BG) over the generic point of +C, and the claim holds. Since Φ is a DCC set, there is a DCC set Φ1 depending only on +Φ such that BG ∈ Φ1 (cf. [Kol+92, Corollary 16.7]). If Φ is a hyperstandard set, then we + +14 +GUODU CHEN, JINGJUN HAN, AND JIHAO LIU +can take Φ1 to be a hyperstandard set by [Bir19, Lemma 3.3]. In both case, by induction, +there is a DCC set (resp. hyperstandard set) Φ′ +1 such that BZ ∈ Φ′ +1. +Pick a general H′ ∼ H and let KH := (KT + H′)|H. Note that the restriction is well +defined as H is a general hyperplane section and KH is determined as a Weil divisor, +although KT may not be Q-Cartier. Let +Mg,H := (LT + H′)|H − (KH + BH). +Then BH + Mg,H = (BT + Mφ,T )|H, and +p(KG + BG) ∼ p(L + G)|G ∼ pg∗(LT + H′)|H ∼ pg∗(KH + BH + Mg,H). +Hence Mg,H is the moduli part of the canonical bundle formula for (G, BG) over H, and +multC(BH + Mg,H) = multD(BT + Mφ,T ) +which implies that multC Mg,H = multD Mφ,T as multC BH = multD BT . Therefore +p multD Mφ,T is integral if and only if p multC Mg,H is integral. Repeating the process +we may finish this step. +Step 3. In this step we show the existence of Φ′ and that pMφ,T is integral. Note that by +Step 2, we may assume that dim T = 1. +Step 3.1. We construct the set Φ′. If Φ is a hyperstandard set which is not a hyperstandard +set, then by [HMX14, Theorem 1.1], BT ∈ Φ′ for some DCC set Φ′ which only depends on +d and Φ. If Φ = Φ(R) is a hyperstandard set, then we show BT ∈ Φ′ := Φ(R′), where +R′ := +� +r, r +l1 +− l2 +p | r ∈ R, l1 , l2 ∈ Z>0 +� +∩ [0, 1] +is a finite set of rational numbers. +Claim 3.2. Suppose that t ∈ T is a closed point. Then (X/T ∋ t, B +sφ∗t) is a monotonic +p-complement of (X/T ∋ t, B), where s := lct(X/T ∋ t, B; φ∗t). In particular, B + sφ∗t is +a Q-divisor over a neighborhood of t. +Proof of Claim 3.2. By Lemma 2.10, Bt := B + sφ∗t is a Q-divisor over a neighborhood of +t. Possibly shrinking T around t, we may assume that Bt ∈ Q. Let (X′, B′ +t) be a Q-factorial +dlt modification of (X, Bt). Then ⌊B′ +t⌋ has a component mapping to t and KX′ +B′ +t ∼Q,T 0. +There exists a Q-divisor B′ on X′ such that B′ ∈ Φ ∩ Q, ⌊B′⌋ = ⌊B′ +t⌋, and BX′ ≤ B′ ≤ B′ +t, +where BX′ is the strict transform of B on X′. +By assumption, (X′/T ∋ t, B′) has a +monotonic p-complement (X′/T ∋ t, B′+). Let B+ be the strict transform of B′+ on X. +Then (X/T ∋ t, B+) is a monotonic p-complement of (X/T ∋ t, B). Since B+ −B ≥ 0 and +B+ − B ∼Q 0 over a neighborhood of t, B+ − B is vertical over T. Moreover, as B′+ ≥ B′, +⌊B′+⌋ has a component mapping to t, and thus (X, B+) has an lc center mapping to t. +Therefore B+ − B = sφ∗t over t. The claim holds. +□ +Pick a closed point t ∈ T. By Claim 3.2, (X/T ∋ t, B+ := B + sφ∗t) is a p-complement +of (X/T ∋ t, B), where s := lct(X/T ∋ t, B; φ∗t). +For any component S of φ∗t, let + +ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS +15 +b := multS B, b+ := multS B+ and m := multS φ∗t. Then b+ = b + sm and thus s = b+−b +m . +Since b ∈ Φ, we may write b = 1 − r/l for some r ∈ R and l ∈ Z>0. In particular, +s = b+ − 1 + r/l +m += 1 +m +�r +l − +� +1 − b+�� +. +Then multt BT = 1 − s ∈ Φ′, as b+ ∈ 1 +pZ ∩ [0, 1]. +Step 3.2. We show that pMφ,T is integral. +We may assume that p(KX + B) ∼ 0 over some non-empty open subset U0 ⊆ T such +that SuppBT ⊆ T \ U0. Let +Θ := B + +� +t∈T\U0 +stφ∗t, +where st := lct(X/T ∋ t, B; φ∗t). Let ΘT be the discriminant part of the canonical bundle +formula for (X, Θ) over T. Then +ΘT = BT + +� +t∈T\U0 +stt +which is a reduced divisor. Moreover, (X/T ∋ t, Θ) is a p-complement of (X/T ∋ t, B) for +every t ∈ T \ U0 by Claim 3.2. Hence p(KX + Θ) ∼T 0 by Lemma 2.6. Since +p(KX + Θ) = p(KX + B) + p(Θ − B) +∼ pφ∗(KT + BT + Mφ,T) + pφ∗(ΘT − BT ) += pφ∗(KT + ΘT + Mφ,T ), +p(KT + ΘT + Mφ,T ) is Cartier. It follows that pMφ,T is integral as KT + ΘT is reduced. +Step 4. In this step, we finish the proof by showing that pMφ is b-Cartier and nef. Note +that in this step we do not assume dim T = 1. +According to [Bir19, Theorem 3.6], we only need to show that pMφ is b-Cartier. Let +T ′ → T be a high resolution and Y → X a log resolution of (X, B) such that Y → T ′ is a +morphism. Let U ⊆ T be a non-empty open subset such that U ′ → U is an isomorphism +where U ′ ⊆ T ′ is the inverse image of U. Let BY be the sum of the strict transform of +B and the reduced exceptional divisor of Y → X but with all the components mapping +outside U removed. In particular, the generic point of any lc center of (Y, BY ) maps into +U. We may run an MMP on KY + BY over X ×T T ′ with scaling of some ample divisor. +By [Bir12, Theorem 1.9], the MMP terminates over U ′. In fact, we reach a model X′ such +that over U ′ the pair (X′, B′) is a Q-factorial dlt modification of (X, B), where B′ is the +strict transform of BY on X′. Hence KX′ +B′ ∼Q 0 over U ′. Now by [HX13, Theorem 1.1] +(see also [Bir12, Theorem 1.1]), we can run an MMP on KX′ +B′ over T ′ which terminates +with a good minimal model X′′ over T ′ as the generic point of every lc center of (X′, B′) is +mapped into U ′. Let B′′ be the strict transform of B′ on X′′. Then KX′′ +B′′ is semi-ample +over T ′. + +16 +GUODU CHEN, JINGJUN HAN, AND JIHAO LIU +X +Y +X′ +X′′ +T +T ′ +T ′′ +U +U ′ +U ′′ +φ +φ′′ +∼ += +∼ += +Let φ′′ : X′′ → T ′′ be the contraction defined by KX′′ + B′′ over T ′. Note that T ′′ → T ′ +is birational as KX′ + B′ ∼Q 0 over U ′. Assume that (T ′′, B′′ +T ′′ + Mφ) is the crepant model +of (T, BT +Mφ). Then LT ′′ = KT ′′ +B′′ +T ′′ +Mφ,T ′′, where LT ′′ is the pullback of LT on T ′′. +Let f : W → X and f ′′ : W → X′′ be a common resolution, KX′′ + ∆′′ := f ′′ +∗ f ∗(KX + B). +Since KX + B and KX′′ + B′′ are crepant over U, we see that B′′ − ∆′′ is vertical over T ′′. +Note that B′′ − ∆′′ ∼R,T ′′ 0, B′′ − ∆′′ = (φ′′)∗PT ′′ for some R-Cartier R-divisor PT ′′ on T ′′ +by Lemma 2.5. Denote by BT ′′ the discriminant part of the canonical bundle formula for +(X′′, B′′) over T ′′. Then BT ′′ = B′′ +T ′′ + PT ′′ and +p +� +KX′′ + B′′� += p +� +KX′′ + ∆′′ + B′′ − ∆′′� +∼ p +� +f ′′ +∗ f ∗L + B′′ − ∆′′� += p +� +φ′′�∗ (LT ′′ + PT ′′) = p +� +φ′′�∗ � +KT ′′ + BT ′′ + Mφ,T ′′� +. +Now by Step 3.2, pMφ,T ′′ is an integral divisor, hence pMφ,T ′ is integral. As T ′ is smooth, +we conclude that pMφ is b-Cartier. +□ +Lemma 3.3. Assume that X → Z is a proper morphism of varieties. +Let Xν be the +normalization of X. Then Xν +z is the normalization of Xz for any general point z ∈ Z, +where Xz (resp., Xν +z ) is the fiber over z. +In particular, if Xz is normal, then Xν +z is +isomorphic to Xz. +Proof. By assumption, Xν → X is birational and finite. Thus Xν +z → Xz is birational and +finite. Let ˜Xz be the normalization of Xz. The universal property of the normalization +implies that there is a morphism ˜Xz → Xν +z . Moreover, the morphism ˜Xz → Xν +z is birational +and finite. Since both ˜Xz and Xν +z are normal, we see that ˜Xz is isomorphic to Xν +z . +□ +Proposition 3.4. Let d, b, β be three positive integers and Γ ⊂ [0, 1] ∩ Q a finite set. Then +there exist a positive integer p and a DCC set Γ′ ⊆ [0, 1] depending only on d, b, β and Γ +such that ¯Γ′ ⊆ [0, 1] ∩ Q and satisfying the following. +Assume that (X, B) is an lc pair of dimension d, φ : X → T is a contraction between +quasi-projective normal varieties, and F is a general fiber of φ. Suppose that +(1) B ∈ Γ, and KX + B ∼Q,T 0, +(2) dim T > 0 and (X, B) is klt over the generic point of T, +(3) b is the non-vanishing order of (KX + B)|F, and +(4) β := dim Hdim F ( ˜F, C), where ˜F is a smooth model of the cover of F associated to +the unique divisor of |b(KX + B)|F |. + +ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS +17 +Then we can choose a moduli part Mφ of the canonical bundle formula for (X, B) over T +such that BT ∈ Γ′, pMφ is b-Cartier, and +p (KX + B) ∼ pφ∗ (KT + BT + Mφ,T ) , +where BT is the discriminant part of the canonical bundle formula for (X, B) over T. +Proof. Let BT be the discriminant part of the canonical bundle formula for (X, B) over T. +Then, by [HMX14, Theorem 1.11], BT ∈ Γ′ for some DCC set Γ′ which only depends on +d and Γ such that ¯Γ′ ⊂ [0, 1] ∩ Q. It suffices to prove the existence of p with the required +properties. +Let T ′ → T be a resolution, X′ the normalization of the main component of X×T T ′, and +denote by φ′ : X′ → T ′. Let (X′, B′) be the crepant model of (X, B). Then KX′+B′ ∼Q,T ′ 0 +and (X′, B′) is sub-lc and is klt over the generic point of T ′. By Lemma 3.3, the general +fiber F ′ of φ′ is isomorphic to F. Therefore it holds that +• b is the non-vanishing order of (KX′ + B′)|F ′, and +• ˜F is a smooth model of the cover of F ′ associated to the unique divisor of |b(KX′ + +B′)|F ′|. +According to [Flo14, Theorem 5.1], one can find a positive integer p0 depending only on β +and a choice of a moduli part Mφ of the canonical bundle formula for (X′, B′) over T ′ such +that p0Mφ,T ′ is integral. In particular, p0Mφ,T ′ is Cartier and thus p0Mφ is b-Cartier. +Since b is the non-vanishing order of (KX + B)|F , by the choice of Mφ again, we see that +b (KX + B) ∼ bφ∗ (KT + BT + Mφ,T) . +Therefore we may conclude that p := bp0 and Γ′ have the required properties. +□ +4. Non-vanishing orders, middle Betti numbers, and complements +In this section, we prove Theorem 1.6. More generally, we proof Theorem 4.1. +Theorem 4.1. Let d, b, β be three positive integers and Γ ⊂ [0, 1] ∩ Q a DCC set. Assume +that the good minimal model conjecture (Conjecture 2.11) holds in dimension d. Then there +exists a positive integer n depending only on d, b, β and Γ satisfying the following. +Assume that (X, B) is a Q-factorial projective pair of dimension d, f∞ : X∞ → Z∞ is +an Iitaka fibration of −(KX + B), h : X∞ → X is the induced birational morphism, and +F is a general fiber of f∞. Suppose that +(1) B ∈ Γ, +(2) −(KX + B) is nef, +(3) (X, B) has a klt R-complement, +(4) κ(X, −(KX + B)) ≥ 0, and b is the non-vanishing order of −h∗(KX + B)|F , i.e., +b = min {a ∈ Z>0 | | − ah∗(KX + B)|F | ̸= ∅} , +and +(5) β := dim Hdim F ( ˜F, C), where ˜F is a smooth model of the cover of F associated to +the unique divisor of | − bh∗(KX + B)|F |. +Then (X, B) has an n-complement. + +18 +GUODU CHEN, JINGJUN HAN, AND JIHAO LIU +We remark here that κ(F, −h∗(KX + B)|F) = 0 by Lemma 2.25. We first show the +existence of (n0, Γ0)-decomposable R-complements under the assumption of Theorem 4.1. +Theorem 4.2. Notation and assumptions as in Theorem 4.1. Then (X, B) has an (n0, Γ0)- +decomposable R-complement, where n0 is a positive integer and Γ0 ⊂ [0, 1] is a finite set +depending only on d, b, β and Γ. +Proof. Step 1. In this step, we show that −(KX + B) is semi-ample. +Let (X, C) be a klt R-complement of (X, B) for some effective R-divisor C ≥ B. Pick +some positive real number ǫ0 such that (X, C + ǫ0(C − B)) is klt. Since KX + C + ǫ0(C − +B) ∼R −ǫ0(KX + B) is nef, we see that KX + C + ǫ0(C − B) is semi-ample by [HMX18, +Lemma 2.9.1] and hence −(KX + B) is also semi-ample. Denote by f : X → Z the ample +model of −(KX + B) and F0 a general fiber of f. Moreover, by [Li22, Proposition 3.21], +f∞ is birational to f, so there exists a naturally induced morphism h : F → F0 such that +b(KX + B)|F0 ∼ 0. +If dim Z = 0, then b(KX + B) ∼ 0 and thus (X, B) has a (b, {1})-decomposable R- +complement. Therefore, we may assume that dim Z > 0. +Step 2. In this step, we construct finite sets Γ0 ⊂ (0, 1] and Γ′ +2 ⊂ [0, 1]∩Q depending only +on d, b and Γ. +Let α be a positive real number such that +α < min +� +|γ − i +b| > 0 | γ ∈ ¯Γ, i ∈ Z≥0 +� +. +By [HLS19, Theorem 5.20], there exist a finite set Γ1 depending only on d, α and Γ, and +an R-divisor ¯B on X, such that +• ¯B ∈ Γ1, +• α Supp B ≥ ¯B − B ≥ 0, and +• (X, ¯B) is R-complementary. +In particular, for any component S of B such that b multS B ∈ Z, multS ¯B = multS B. +Hence ( ¯B)h = Bh. +By [HLS19, Theorem 5.16], there exist a point v0 := (v0 +1, . . . , v0 +m) +and an open subset V of the rational envelope of v0 (that is the smallest affine subspace +containing v0 which is defined over Q) in Rm depending only on d and Γ1, and Weil divisors +B1, . . . , Bm ≥ 0 on X, such that B(v0) = ¯B, and (X, B(v)) is R-complementary for any +v ∈ V , where B(v) := �m +i=1 viBi for any v := (v1, . . . , vm) ∈ Rm. Moreover, possibly +replacing V , we may assume that +B(v) ≥ D := ¯B − B, and |B(v) − ¯B| < α +for any v ∈ V . We remark that (B(v))h = ( ¯B)h = Bh for any v ∈ V . +Pick points v1, . . . , vk ∈ V ∩Qm, such that v0 is in the interior of the convex hull spanned +by v1, . . . , vk. For any integer 1 ≤ i ≤ k, set +B(i) := B(vi) − D and ¯B(i) := B(vi). + +ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS +19 +There exist finite sets Γ0 := {a1, . . . , ak} ∪ {1} ⊂ (0, 1] and Γ′ +2 ⊂ [0, 1] ∩ Q such that +¯B(i) ∈ Γ′ +2 for any integer 1 ≤ i ≤ k, �k +i=1 ai = 1, and �k +i=1 aivi = v0. In particular, +�k +i=1 aiB(i) = B. +Step 3. In this step, we run an MMP. +Let (X, ¯B(i)+ ¯Gi) be an R-complement of (X, ¯B(i)) for some effective R-Cartier R-divisor +¯Gi for any integer 1 ≤ i ≤ k. Pick a positive real number ǫ such that (X, C + ǫ ¯Gi) is klt. +Then we may run an MMP on KX + C + ǫ ¯Gi over Z, which terminates with a model Wi +over Z such that KWi +CWi +ǫ ¯GWi is semi-ample over Z, where CWi and ¯GWi are the strict +transforms of C and ¯Gi on Wi respectively. This MMP is also an MMP on −(KX + ¯B(i)) +over Z, hence −(KWi + ¯B(i) +Wi) is semi-ample over Z. Let gi : Wi → Ti be the ample model of +−(KWi + ¯B(i) +Wi) over Z. Recall that by construction and by Lemma 5.1, B − ¯B(i) is vertical +over Z. As −(KX + ¯B(i)) ∼R,Z B − ¯B(i), one can see that F0 is isomorphic to a general +fiber of Wi → Z by Lemma 2.9, and Ti is birational to Z. +Step 4. +In this step, we construct a positive integer p, and a hyperstandard set Γ2 +depending only on d, b, β and Γ′ +2, and make a choice of a moduli part Mi of the canonical +bundle formula for (Wi, ¯B(i) +Wi) over Ti such that +p +� +KWi + ¯B(i) +Wi +� +∼ pg∗ +i (KTi + BTi + Mi,Ti) , +BTi ∈ Γ2, and pMi is b-Cartier, where BTi is the discriminant part of the canonical bundle +formula for (Wi, ¯B(i) +Wi) over Ti. +To see this, we first claim that +(i) b is the non-vanishing order of −(KWi + ¯B(i) +Wi)|F ′ +i , and b(KWi + ¯B(i) +Wi)|F ′ +i ∼ 0, and +(ii) ˜F is a smooth model of the cover of F ′ +i associated to the unique divisor of |b(KWi + +¯B(i) +Wi)|F ′ +i |, where F ′ +i is a general fiber of gi. +Assume that b0 is the non-vanishing order of (KX + B)|F0 ∼Q 0. Then b0(KX + B)|F0 ∼ 0 +and b0 ≤ b. Since h∗ +0((KX+B)|F0) = h∗(KX+B)|F, we have b0 = b and bh∗(KX+B)|F ∼ 0. +Therefore b is also the non-vanishing order of (KX + B)|F0. Note that +Spec +�b−1 +� +i=0 +OF (⌊ih∗(KX + B)|F ⌋) +� +→ F (resp., Spec +�b−1 +� +i=0 +OF0 (⌊i(KX + B)|F0⌋) +� +→ F0) +is the cover associated to the unique element of ⌊h∗(KX + B)|F ⌋, see [Kol13, §2.3]. By +[Nak04, II Lemma 2.11], there is a natural isomorphism +(h0)∗OF (⌊i(h∗(KX + B)|F )⌋) → OF0 (⌊i((KX + B)|F0)⌋) . +Thus ˜F is a smooth model of the cover of F0 associated to the unique divisor of |b(KX + +B)|F0|. Furthermore, by our construction, X is isomorphic to Wi and B = ¯B(i) over the +generic point of Z, and F ′ +i is isomorphic to F0 (see Lemma 2.9). Thus the claim holds. +Since (X, B) has a klt R-complement, (Wi, BWi) has a klt R-complement, where BWi is +the strict transform of B on Wi. Moreover, as BWi = ¯B(i) +Wi over the generic point of Ti, we + +20 +GUODU CHEN, JINGJUN HAN, AND JIHAO LIU +can see that (Wi, ¯B(i) +Wi) is klt over the generic point of Ti. Recall that Γ′ +2 ⊂ [0, 1] ∩ Q is a +finite set and ¯B(i) +Wi ∈ Γ′ +2. By Proposition 3.4, we may find a positive integer p, and a DCC +set Γ2 depending only on d, b, β and Γ′ +2 such that ¯Γ2 ⊂ [0, 1] ∩ Q, and make a choice of a +moduli part Mi of the canonical bundle formula for (Wi, ¯B(i) +Wi) over Ti, such that +p +� +KWi + ¯B(i) +Wi +� +∼ pg∗ +i (KTi + BTi + Mi,Ti) , +BTi ∈ Γ2, and pMi is b-Cartier, where BTi is the discriminant part of the canonical bundle +formula for (Wi, ¯B(i) +Wi) over Ti. Note that (Ti, BTi + Mi) is glc. +Step 5. In this step, we find an integer n0 which only depends on d, bp and Γ2 satisfying +our requirements and thus finish the proof. +We first show that Ti is of Fano type. Indeed, according to [Amb05, Theorem 0.2], there +exist klt pairs (Z, ∆) and (Ti, ∆Ti) such that +KX + B ∼R f ∗(KZ + ∆) and KWi + BWi ∼R g∗ +i (KTi + ∆Ti). +In particular, KTi + ∆Ti ∼R τ ∗ +i (KZ + ∆) where τi denotes the induced morphism Ti → Z. +Since Z is the ample model of −(KX+B), −(KZ+∆) is ample. It follows that −(KTi+∆Ti) +is big and nef as τi is a birational morphism. Thus Ti is of Fano type. +Since (Wi, ¯B(i) +Wi) is R-complementary, (Ti, BTi + Mi) is R-complementary, that is there +is an R-divisor Pi ≥ 0 such that (Ti, BTi +Pi +Mi) is glc and KTi +BTi +Pi +Mi,Ti ∼R 0. +As Ti is of Fano type, by [Bir19, Theorem 1.10] (see also [Che20, Theorem 1.3]), there +exists a positive integer n0 divisible by bp depending only on d, bp and Γ2, and a Q-divisor +B+ +Ti ≥ BTi on Ti, such that (Ti, B+ +Ti + Mi) is glc and n0(KTi + B+ +Ti + Mi,Ti) ∼ 0. +It is enough to show that (Wi, ¯B(i),+ +Wi +:= ¯B(i) +Wi + g∗ +i (B+ +Ti − BTi)) is a monotonic n0- +complement of (Wi, ¯B(i) +Wi). Indeed, by [PS09, Corollary 7.18(1)], (Wi, ¯B(i),+ +Wi +) is lc. Since +n0 +� +KWi + ¯B(i),+ +Wi +� += n0 +� +KWi + ¯B(i) +Wi +� ++ n0g∗ +i +� +B+ +Ti − BTi +� +∼ n0g∗ +i (KTi + BTi + Mi,Ti) + n0g∗ +i +� +B+ +Ti − BTi +� += n0g∗ +i +� +KTi + B+ +Ti + Mi,Ti +� +∼ 0, +one can see that (Wi, ¯B(i),+ +Wi +) is a monotonic n0-complement of (Wi, ¯B(i) +Wi). Remark that +by Step 3, X ��� Wi is −(KX + ¯B(i))-negative. Therefore (X, ¯B(i)) also has a monotonic +n0-complement (X, ¯B(i),+). This immediately implies that (X, �k +i=1 ai ¯B(i),+) is an (n0, Γ0)- +decomposable R-complement of (X, B). We may finish the proof. +□ +Proof of Theorem 4.1. By Theorem 4.2, there exist a positive integer n0 and a finite set +Γ0 ⊂ (0, 1] depending only on d, b, β, Γ, such that (X, B) has an (n0, Γ0)-decomposable +R-complement. Theorem 4.1 follows from Diophantine approximation as in the proof of +[HLS19, Theorem 1.8] (see [HLS19, Section 6] for details). +□ +Proof of Theorem 1.6. It follows from Theorem 4.1. +□ + +ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS +21 +5. Effective Iitaka fibrations +In this section, we prove Theorem 1.4. +Lemma 5.1. Assume that f : X → Z is a contraction between normal quasi-projective +varieties, and D is an R-Cartier R-divisor on X. Suppose that either +• dim Z = 0 and D ∼R 0, or +• dim Z > 0 and D ∼R f ∗DZ for some big R-Cartier R-divisor DZ on Z. +Then κ(X, D) ≥ 0 if and only if Dh is a Q-divisor. +Proof. We first assume that κ(X, D) ≥ 0. Suppose on the contrary that Dh is not a Q- +divisor. Let F be a very general fiber of f. Then {mD|F } = {mDh|F} ̸= 0 for any positive +integer m. By our assumption, D|F ∼R 0 and +⌊mD|F⌋ = mD|F − {mD|F } ∼R −{mD|F } +is not pseudo-effective for any positive integer m. Thus ⌊mD⌋ is not pseudo-effective for +any positive integer m, which implies that κ(X, D) = −∞, a contradiction. Therefore, Dh +is a Q-divisor. +Now suppose that Dh is a Q-divisor. If dim Z = 0, then D ∼Q 0 and thus κ(X, D) = 0. +Assume that dim Z > 0. Let b be a positive integer, such that bD ∼ 0 on the generic fiber of +f. Then there exists α ∈ K(X) such that bD+(α) is vertical over Z. Since bD+(α) ∼R,Z 0, +bD + (α) = f ∗D′ +Z for some R-Cartier R-divisor D′ +Z on Z by [Li20, Lemma 2.11]. Thus +D′ +Z ∼R bDZ, so D′ +Z is big. By Lemma 2.18, +κ(X, D) = κ(X, bD) = κ(X, bD + (α)) = κ(Z, D′ +Z) = dim Z > 0, +and we are done. +□ +Proof of Theorem 1.4. Let Γ ⊂ [0, 1] be a DCC set. Without loss of generality, we may +assume that 1 ∈ Γ. Assume that (X, B) is a projective lc pair of dimension d such that +κ(X, KX +B) ≥ 0 and B ∈ Γ. Since we assume Conjecture 2.11 in dimension d, (X, B) has +a good minimal model (X′, B′). Possibly replacing (X, B) with (X′, B′), we may assume +that KX + B is semi-ample. Let f : X → Z be the ample model of KX + B. +Suppose that dim Z = 0. Then B ∈ Γ0 for some finite set Γ0 ⊂ Γ∩Q which only depends +on d and Γ by the global ACC [HMX14, Theorem D] and Lemma 5.1, and KX + B ∼R 0. +As Conjecture 1.3 holds in dimension d, we may find a positive integer m0 depending only +on d and Γ0 such that m0(KX + B) ∼ 0. In what follows, we may assume that dim Z > 0. +According to Proposition 3.1, there exist a positive integer p and a DCC set Γ′ ⊂ [0, 1] +depending only on d and Γ, and a choice of the moduli part Mf of the canonical bundle +formula for (X, B) over Z, such that BZ ∈ Γ′, pMf is b-Cartier, and +p(KX + B) ∼ pf ∗ (KZ + BZ + Mf,Z) , +where BZ is the discriminant part of the canonical bundle formula for (X, B) over Z. +Recall that Z is the ample model of KX + B, hence KZ + BZ + Mf,Z is big. By [BZ16, +Theorem 1.3], there is a positive integer m1 depending only on d, p, and Γ′, such that + +22 +GUODU CHEN, JINGJUN HAN, AND JIHAO LIU +|⌊m1(KZ + BZ + Mf,Z)⌋| defines a birational map. By [Nak04, II 2.11 Lemma], we have +that +H0 (X, ⌊pm1(KX + B)⌋) = H0 (X, ⌊pm1f ∗ (KZ + BZ + Mf,Z)⌋) +∼= H0(Z, ⌊pm1(KZ + BZ + Mf,Z)⌋). +Therefore |⌊pm1(KX + B)⌋| defines a map which is birational to f∞. +Let m := pm0m1, then m satisfies our required property. +□ +Proof of Corollary 1.5. It follows from Theorem 1.4, the existence of good minimal models +in dimension ≤ 3 [KMM87, KMM94], and Theorem 2.14. +□ +6. Existence of decomposable Iitaka fibrations +In this section, we recall the definition of invariant Iitaka fibrations, which generalizes +Iitaka fibrations to the category of all pairs with non-negative invariant Iitaka dimensions. +We then show the existence of decomposable Iitaka fibrations. +Definition 6.1 (Invariant Iitaka fibrations). Let X be a normal projective variety and +D an R-Cartier R-divisor on X such that κι(X, D) ≥ 0. A morphism f∞ : X∞ → Z∞ +between smooth varieties is called an invariant Iitaka fibration of D if there exists an R- +Cartier R-divisor D′ on X, such that D ∼R D′, κ(X, D′) ≥ 0, and f∞ is an Iitaka fibration +of D′. +By [Hu20, Lemma 2.3], an invariant Iitaka fibration of D always exists, and is +independent of the choice of D′. +We propose the following conjecture, which is a little stronger than Conjecture 1.8. +Conjecture 6.2. Let d be a positive integer and Γ ⊂ [0, 1] a DCC set. Then there exist +a positive integer m, a finite set Γ0 ⊂ (0, 1], and a DCC set Γ′ ⊂ [0, 1] depending only on +d and Γ satisfying the following. Assume that (X, B) is an lc pair of dimension d such +that B ∈ Γ, κι(X, KX + B) ≥ 0, and either Γ is a finite set or all component of B are +Q-Cartier. Then +(1) (Weak version) (X, B) has a (Γ0, Γ′)-decomposable Iitaka fibration. +(2) (Strong version) (X, B) has an (m, Γ0, Γ′)-decomposable Iitaka fibration. +Theorem 6.3. Let d be a positive integer. +Assume that the non-vanishing conjecture +(Conjecture 2.12) holds in dimension d. Then: +(1) Conjecture 6.2(1) holds in dimension d. +(2) Assume the effective log Iitaka conjecture (Conjecture 1.2) holds in dimension d. +Then Conjecture 6.2(2) holds in dimension d. +Proof of Theorem 6.3. Let Γ ⊂ [0, 1] be a DCC set and possibly replacing Γ with Γ ∪ {1}, +we may assume that 1 ∈ Γ. +Assume that (X, B) is an lc pair of dimension d such that B ∈ Γ, κι(X, KX + B) ≥ 0, +and either Γ is a finite set or every component of B is Q-Cartier. Let f∞ : X∞ → Z∞ be +an invariant Iitaka fibration of KX + B, h : X∞ → X the induced morphism, and F a very + +ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS +23 +general fiber of f∞. Let B∞ := h−1 +∗ B + E, where E is the sum of all the h-exceptional +prime divisors. Then (X∞, B∞) is log smooth, κι(X∞, KX∞ + B∞) = κι(X, KX + B), and +κι(F, (KX∞ + B∞)|F ) = 0. +If Γ is a finite set, then we let ˜Γ := Γ, otherwise we let ˜Γ := ∅. +We let v0 := +(v0 +1, . . . , v0 +m), g, and V be as in [CHL22, Proposition 5.1] which depend only on d, Γ, ˜Γ. +We may write B∞ = � bjB(j) +∞ , where B(j) +∞ are the irreducible components of B∞. Then +there exist distinct Weil divisors B∞,1, . . . , B∞,m ≥ 0 on X∞, such that +• g(γ) ≥ γ for any γ ∈ ¯Γ, and g(γ′) = γ′ for any γ′ ∈ ˜Γ, +• B∞(v0) = � g(bj)B(j) +∞ , where B∞(v) := �m +i=1 viB∞,i for any v := (v1, . . . , vm) ∈ +Rm, +• both (X∞, B∞(v)) and (X∞, B∞(v) − D∞) are lc for any v ∈ V , where D∞ := +B∞(v0) − B∞ ≥ 0, and +• κ(X∞, KX∞ + B∞(v) − D∞) = κι(X∞, KX∞ + B∞) = dim Z for any v ∈ V ∩ Qm. +Let DF := D∞|F , D := h∗D∞, BF (v) := B∞(v)|F and B(v) := h∗B∞(v) for any v ∈ Rm. +By the construction of v0, g, and V , we have +• (X, B(v) − D) is lc for any v ∈ V , and +• κ(F, KF + BF(v) − DF ) = κι(F, (KX∞ + B∞)|F ) = 0 for any v ∈ V ∩ Qm. +Note that if Γ is a finite set, then D∞ = 0 and DF = 0. +By the definition of Iitaka +fibrations, f∞ is an Iitaka fibration of KX∞ + B∞(v) − D∞ for any v ∈ V ∩ Qm. As +(X, B(v) − D) is lc for any v ∈ V , +KX∞ + B∞(v) − D∞ − h∗(KX + B(v) − D) ≥ 0 +and is h-exceptional for any v ∈ V . Therefore f∞ is an Iitaka fibration of KX + B(v) − D +for any v ∈ V ∩ Qm. +Let v1, . . . , vk ∈ V ∩ Qm be rational points, such that v0 is contained in the interior +of the convex hull of v1, . . . , vk. There exist a DCC set Γ′ ∋ 1 and a finite set Γ0 := +{a1, . . . , ak} ⊂ (0, 1] depending only on d and Γ such that Bi := B(vi) − D ∈ Γ′ for any +integer 1 ≤ i ≤ k, and �k +i=1 aivi = v0. In particular, +KX + B = +k +� +i=1 +ai (KX + Bi) . +Recall that f∞ is an Iitaka fibration of KX + B(v) − D for any v ∈ V ∩ Qm. Thus f∞ is +an Iitaka fibration of KX + Bi for any integer 1 ≤ i ≤ k. We conclude that (X, B) has a +(Γ0, Γ′)-decomposable Iitaka fibration, which is (1). +Now suppose that Conjecture 1.2 holds in dimension d. Then there exists a positive +integer m depending only on d and Γ′, such that the map defined by |⌊m(KX + Bi)⌋| is +birational to f∞ for any integer 1 ≤ i ≤ k. Therefore (2) holds. +□ +Corollary 6.4. Conjecture 6.2 holds when d ≤ 3. +Proof. 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Todorov, Effective log Iitaka fibrations for surfaces and threefolds, Manuscripta Math. 133 +(2010), 183–195. +[TX09] G. T. Todorov and C. Xu, On effective log Iitaka fibration for 3-folds and 4-folds, Algebra Number +Theory 3 (2009), no. 6, 697–710. +[Tsu99] H. Tsuji, Pluricanonical systems of projective varieties of general type, arXiv:math.AG/9909021. +[VZ07] E. Viehweg and D.-Q. Zhang, Effective Iitaka fibrations, J. Algebraic Geom. 18 (2009), no. 4, +711–730. +[Xu20] C. Xu, A minimizing valuation is quasi-monomial, Ann. of Math. (2) 191 (2020), no. 3, 1003–1030. +[XZ22] C. Xu and Z. Zhuang, Stable degenerations of singularities, arXiv:2205.10915. +Institute for Theoretical Sciences, Westlake University, Hangzhou, Zhejiang, 310024, +China +Email address: chenguodu@westlake.edu.cn +Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200438, China +Email address: hanjingjun@fudan.edu.cn +Department of Mathematics, Northwestern University, 2033 Sheridan Rd, Evanston, IL +60208, USA +Email address: jliu@northwestern.edu + diff --git a/sNE3T4oBgHgl3EQf9AvA/content/tmp_files/load_file.txt b/sNE3T4oBgHgl3EQf9AvA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b3026767f3f6d32d9d907eed5245ab79c48c614a --- /dev/null +++ b/sNE3T4oBgHgl3EQf9AvA/content/tmp_files/load_file.txt @@ -0,0 +1,1335 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf,len=1334 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='04813v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='AG] 12 Jan 2023 ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS GUODU CHEN, JINGJUN HAN, AND JIHAO LIU Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We study the relationship between Iitaka fibrations and the conjecture on the existence of complements, assuming the good minimal model conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In one direction, we show that the conjecture on the existence of complements implies the effective log Iitaka fibration conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' As a consequence, the effective log Iitaka fibration conjecture holds in dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In the other direction, for any Calabi-Yau type variety X such that −KX is nef, we show that X has an n-complement for some universal constant n depending only on the dimension of X and two natural invariants of a general fiber of an Iitaka fibration of −KX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We also formulate the decomposable Iitaka fibration conjecture, a variation of the effective log Iitaka fibration conjecture which is closely related to the structure of ample models of pairs with non-rational coefficients, and study its relationship with the forestated conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Preliminaries 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Canonical bundle formulas 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Non-vanishing orders, middle Betti numbers, and complements 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Effective Iitaka fibrations 21 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Existence of decomposable Iitaka fibrations 22 References 24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Introduction We work over the field of complex numbers C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let X be a smooth projective variety with non-negative Kodaira dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By a well-known construction of Iitaka, there exists a birational morphism X∞ → X from a smooth projective variety X∞, and a contraction f∞ : X∞ → Z∞ onto a projective variety Z∞, such that a very general fiber of f∞ is smooth with Kodaira dimension zero, and dim Z∞ = κ(X, KX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' The morphism f∞ : X∞ → Z∞ is referred to as an Iitaka fibration of KX (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' It is conjectured that the pluricanonical system |mKX| defines a Date: January 13, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 14C20,14E05,14E30,14J30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 1 2 GUODU CHEN, JINGJUN HAN, AND JIHAO LIU map which is birational to an Iitaka fibration whenever the positive integer m is divisible by a positive integer depending only on the dimension of X: Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1 (Effective Iitaka fibration, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' [HM06, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='7])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let d be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then there exists a positive integer md depending only on d, such that for any smooth projective variety X of dimension d with non-negative Kodaira dimension, |mKX| defines an Iitaka fibration for any positive integer m divisible by md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1 was proved when KX is big [HM06, Tak06] (see also [Tsu99]), when d = 2 due to Enriques (see also [Iit70]), and when d = 3 [Mor85, Kaw86, FM00, VZ07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' An important progress towards Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1 is made in [BZ16], where the authors showed that there exists a positive integer m depending only on d and two natural invariants of the very general fibers of an Iitaka fibration of KX (the non-vanishing order and the middle Betti number), such that |mKX| defines an Iitaka fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Unfortunately, we don’t know the boundedness of the middle Betti numbers in dimension ≥ 3, which leaves Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1 open in dimension ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In practice, it is also natural to consider the following generalized version of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1, which is known as the effective log Iitaka fibration conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2 (Effective log Iitaka fibration, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' [TX09, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let d be a positive integer and Γ ⊂ [0, 1] a DCC set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then there exists a positive integer m depending only on d and Γ satisfying the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that (X, B) is an lc pair of dimension d such that B ∈ Γ and κ(X, KX +B) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then |⌊m(KX + B)⌋| defines a map which is birational to an Iitaka fibration of KX + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2 was proved when KX + B is big [HMX14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' When Γ ⊂ [0, 1] ∩ Q and (X, B) is klt, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2 was proved when d ≤ 2 [Tod10], when d ≤ 3 and κ(X, KX) > 0 [TX09, Tod10], and when d = 4 and κ(X, KX) = 2 [TX09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Recently, the theory of complements, which was introduced by Shokurov in the study of the existence of flips for threefolds [Sho92], has gradually become one of the major topics in birational geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' This theory has played an important role in the proof of the BAB conjecture [Bir19, Bir21], the proof of the singular Yau-Tian-Donaldson conjecture and the stable degeneration conjecture [Xu20, BLX22, LXZ22, XZ22], and recent studies on Shokurov’s ascending chain condition conjecture for minimal log discrepancies [Liu18, HLS19, HL22b, HLL22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' For other related work, we refer the readers to [Sho20, CGN21, CZ21, CHX22, CX22, Liu22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Although the existence of n-complements [Bir19, HLS19] is settled for Fano type varieties, it is natural to consider the existence of n-complements for pairs admitting an lc Calabi-Yau structure, that is, R-complementary varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We have the following conjecture: Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3 (Existence of complements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let d be a positive integer and Γ ⊂ [0, 1] a DCC set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then there exists a positive integer n depending only on d and Γ satisfying the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that (X/Z ∋ z, B) is an R-complementary pair of dimension d such that B ∈ Γ, then (X/Z ∋ z, B) has an n-complement (X/Z ∋ z, B+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Moreover, if the closure of Γ belongs to [0, 1] ∩ Q, then we can pick B+ ≥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS 3 Relationship between the conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' It is interesting to ask whether we can establish some connections between Conjectures 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2 and Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' At first glance, it is difficult to observe their relationships, as these conjectures are considering structures of varieties and pairs with completely different positivity properties: Conjectures 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2 are concentrating on pairs with positive canonical bundle (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', KX + B is effective), while Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3 is concentrating on varieties with negative canonical bundle (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', −(KX + B) is effective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Surprisingly, we have the following theorems, which show that these conjectures are actually deeply related with each other in multiple directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' From complements to Iitaka fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' First, we prove that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3 implies Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2 assuming the good minimal model conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let d be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that the good minimal model conjecture (Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='11) and the existence of complements (Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3) hold in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then the effective log Iitaka fibration conjecture (Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2) holds in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' As an immediate corollary, we prove Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2 in dimension ≤ 3: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2 holds when d ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' From Iitaka fibrations to complements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Next, we show that the existence of n-complements is deeply related to some invariants associated to Iitaka fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We have the following theorem: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let d, b, and β be three positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that the good minimal model conjecture (Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='11) holds in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then there exists a positive integer n depending only on d, b, and β satisfying the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that X is a Q-factorial normal projective variety of dimension d, f∞ : X∞ → Z∞ is an Iitaka fibration of −KX, h : X∞ → X is the induced birational morphism, and F is a general fiber of f∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Suppose that (1) −KX is nef, (2) X has a klt R-complement, (3) κ(X, −KX) ≥ 0, and b is the non-vanishing order of −h∗KX|F , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', b = min {a ∈ Z>0 | | − ah∗KX|F | ̸= ∅} , and (4) β := dim Hdim F ( ˜F, C), where ˜F is a smooth model of the cover of F associated to the unique divisor of | − bh∗KX|F |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then X has an n-complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We remark that the assumptions on b and β in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='6 are natural assumptions, and are exactly the additional assumptions in [BZ16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2] on the effective Iitaka fibration conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We also note that, modulo the good minimal model conjecture, the boundedness of b follows immediately from the effective log Iitaka conjecture (Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2) for log pairs with Iitaka dimension 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Decomposable Iitaka fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' For any semi-ample divisor D, the ample model of D is clearly birational to an Iitaka fibration of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' However, if D is only assumed to be an 4 GUODU CHEN, JINGJUN HAN, AND JIHAO LIU R-divisor, then it is possible that D does not have any Iitaka fibration although the ample model of D exists [CHL22, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' An approach to resolve this issue is to define the invariant Iitaka fibration (see Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' A question arises naturally: do we expect any kind of uniform effectively on invariant Iitaka fibrations for log pairs, similar to the effective log Iitaka fibration conjecture?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' The question is expected to have a positive answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' More precisely, suppose that (X, B) is an lc pair such that KX + B induces a map X ��� Z that is birational to an invariant Iitaka fibration of KX + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then we expect that X ��� Z will actually be birational to an Iitaka fibration of KX + B′ after we (uniformly) perturb the coefficients of the boundary B to get a new boundary B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Therefore, the effective log Iitaka fibration conjecture should induce some kind of uniform effectivity on an invariant Iitaka fibration induced by KX +B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' The difficulty is to show that we can make a uniform perturbation to switch the boundary B to a new boundary B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In [CHL22, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1], the authors prove a weaker version, which shows that, modulo the non-vanishing conjecture, there exists such uniform perturbation such that an Iitaka dimension of KX + B′ is equal to an invariant Iitaka dimension of KX + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In this paper, we will show that an Iitaka fibration of KX + B′ is actually equal to an Iitaka fibration as KX + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' To make our statements more clear, we introduce the concept of decomposable Iitaka fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='7 (Decomposable Iitaka fibrations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let Γ0 := {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , ak} ⊂ (0, 1] be a finite set such that �k i=1 ai = 1, and Γ′ ⊂ [0, 1] a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that (X, B) is an lc pair such that κι(X, KX + B) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We say that (X, B) has a (Γ0, Γ′)-decomposable Iitaka fibration if there exist R-divisors B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , Bk ∈ Γ′ such that (1) B = �k i=1 aiBi, (2) (X, Bi) is lc for any 1 ≤ i ≤ k, and (3) any invariant Iitaka fibration of KX + B is an Iitaka fibration of KX + Bi for any 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In addition, if (4) the map defined by |⌊m(KX + Bi)⌋| is birational to any invariant Iitaka fibration of KX + B for any integer 1 ≤ i ≤ k, then we say that (X, B) has an (m, Γ0, Γ)-decomposable Iitaka fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' As an analogue of the effective log Iitaka fibration conjecture, we propose the following conjecture on the existence of decomposable Iitaka fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Roughly speaking, the conjecture indicates that we can uniformly perturb an invariant Iitaka fibration to get an effective log Iitaka fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='8 (Decomposable Iitaka fibrations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let d be a positive integer and Γ ⊂ [0, 1] a DCC set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then there exist a positive integer m, a finite set Γ0 ⊂ (0, 1], and a DCC set Γ′ ⊂ [0, 1] depending only on d and Γ satisfying the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that (X, B) is a Q-factorial lc pair of dimension d such that B ∈ Γ and κι(X, KX + B) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then: (1) (Weak version) (X, B) has a (Γ0, Γ′)-decomposable Iitaka fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' (2) (Strong version) (X, B) has an (m, Γ0, Γ′)-decomposable Iitaka fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS 5 Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='8 will help us to understand the structure of good minimal models and their ample models for pairs with real coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We show that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='8 follows from the non-vanishing conjecture (Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='12) and the effective log Iitaka fibration conjecture (Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='8(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let d be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that the non-vanishing conjecture (Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='12) holds in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then: (1) Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='8(1) holds in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' (2) Assume that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2 holds in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='8(2) holds in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' As an immediate corollary, we have: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='8 holds when d ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Combining Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='9, we show that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='8 follows from the good minimal model conjecture and the existence of complements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let d be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that the good minimal model conjecture (Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='11) and the existence of complements (Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3) hold in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='8 holds in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Finally, we remark that we expect all theorems in our paper to hold in the relative setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' That is, instead of considering Iitaka fibrations for projective varieties and projective pairs, we may also consider Iitaka fibrations in the relative case (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' [Li22, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Structure of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In Section 2, we introduce some notation and tools which will be used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In Section 3, we recall the canonical bundle formulas and prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In Section 5, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In Section 4, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In Section 6, we introduce invariant Iitaka fibrations and prove Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='9 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' The second and the third named authors began this work when they worked on [HL20] in Summer 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Part of this work was done while the first named author visited Chuyu Zhou at EPFL in Summer 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' He would like to thank their hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' The authors would like to thank Qianyu Chen, Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Hacon, Fei Hu, Chen Jiang, Junpeng Jiao, Zhan Li, Haidong Liu, Wenfei Liu, Yuchen Liu, Yujie Luo, Fanjun Meng, Lingyao Xie, Qingyuan Xue, and Chuyu Zhou for valuable discussions and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' The authors would like to thank Enrica Floris for answering questions about Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' The first named author was supported by the China post-doctoral grants BX2021269 and 2021M702925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' The second named author was supported by National Key Research and Development Program of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 2020YFA0713200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' The second named author is a member of LMNS, Fudan University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Preliminaries We adopt the standard notation and definitions in [KM98, BCHM10] and will freely use them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 6 GUODU CHEN, JINGJUN HAN, AND JIHAO LIU 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let Γ ⊂ R be a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We say that Γ satisfies the ascending chain condition (ACC) if any increasing sequence in Γ stabilizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We say that Γ satisfies the descending chain condition (DCC) if any decreasing sequence in Γ stabilizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2 ([PS09, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2], [Bir19, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let R ⊂ [0, 1] ∩ Q be a finite set, we define Φ(R) := � 1 − γ n | γ ∈ R, n ∈ Z≥1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We say that a set Γ ⊂ [0, 1] is a hyperstandard set if there exists a finite set R ⊂ [0, 1] ∩ Q such that 0, 1 ∈ R and Γ = Φ(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We say f : X → Z is a contraction if π is a projective morphism, and f∗OX = OZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We say that a birational map φ : X ��� Y is a birational contraction if φ is projective and φ−1 does not contract any divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let Γ ⊂ R be a set, X a variety, and B := �s i=1 biBi an R-divisor on X, where Bi are the irreducible components of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We write B ∈ Γ if bi ∈ Γ for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We define |B| := max 1≤i≤s |bi|, ⌊B⌋ := s � i=1 ⌊bi⌋Bi, and {B} := s � i=1 {bi}Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We may denote by K(X) the rational function field of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let f : X → Z be a contraction between normal quasi-projective varieties and D an R-divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We say that D is vertical over Z if f(SuppD) is a proper subset of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We say that D is horizontal over Z if D is not vertical over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We may uniquely write D = Dh + Dv such that Dv is vertical over Z and each component of Dh is horizontal over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We call Dh the horizontal/Z part of D and Dv the vertical/Z part of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let f : X → Z be a contraction between normal quasi-projective varieties and L a Cartier divisor on X such that L ∼Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Suppose either L is vertical over Z or L ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then there is a Cartier divisor LZ on Z such that L = f ∗LZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' If L ≥ 0, then L is vertical over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Thus it suffices to prove the statement under the condition that L is vertical over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By assumption there exist a rational function s ∈ K(X) and a Cartier divisor L′ Z on Z such that L − f ∗L′ Z = (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since L is vertical over Z, we may find an open subset V ⊂ Z such that (s)|U ≥ 0, where U := f −1V ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In particular, s ∈ OX(U) and thus s ∈ OZ(V ) as f is a contraction and OZ(V ) = (f∗OX)(V ) = OX(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Hence s ∈ K(V ) = K(Z) ֒→ K(X) and φ(s) = s, where φ : K(Z) ֒→ K(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Thus f ∗(s) = (φ(s)) = (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Set LZ := L′ Z + (s) and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let f : X → Z be a contraction between normal quasi-projective varieties, D a divisor on X, and U1, U2 ⊂ Z two open subsets such that D|f−1(Ui) ∼Ui 0 for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then D|f−1(U1∪U2) ∼U1∪U2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Replacing Z by U1 ∪ U2, we may assume that Z = U1 ∪ U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By assumption, D|f−1(Ui) = f ∗ i Di + (si) ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS 7 for some Cartier divisor Di on Ui and si ∈ K(X), where fi := f|f−1(Ui) for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let f12 := f|f−1(U1∩U2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then (f12)∗(D1 − D2) = (s2 s1) on f −1(U1 ∩ U2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By the projection formula, we have OU1∩U2 ∼= (f12)∗ � (s2 s1 )OX|f−1(U1∩U2) � = (f12)∗(f12)∗OU1∩U2(D1 − D2) = OU1∩U2(D1 − D2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Thus (D1 − D2)|U1∩U2 = (sZ) and (f12)∗(sZ) = (s2 s1) over U1 ∩ U2 for some sZ ∈ K(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In particular, (s2 s1) − f ∗(sZ) is vertical over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Note that (s2 s1 ) − f ∗(sZ) ∼Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='5, there exists a Cartier divisor LZ on Z such that (s2 s1 ) − f ∗(sZ) = f ∗LZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Moreover, we have SuppLZ ∩ (U1 ∩ U2) = ∅, and LZ + (sZ) = D1 − D2 on U1 ∩ U2 as (D1 − D2)|U1∩U2 = (sZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Hence there exists a Cartier divisor D′ on Z, such that D′ = D1 on U1 and D′ = D2 + (sZ) + LZ on U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' It follows that f ∗D′ = f ∗D1 = D − (s1) over U1 and f ∗D′ = f ∗(D2 + (sZ) + LZ) = D − (s2) + (s2 s1) = D − (s1) over U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Hence D = f ∗D′ − (s1), and thus D ∼Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Pairs and singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='7 (Pairs, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' [CH21, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' A pair (X/Z ∋ z, B) consists of a contraction f : X → Z between normal quasi-projective varieties, a (not necessarily closed) point z ∈ Z, and an R-divisor B ≥ 0 on X, such that KX + B is R-Cartier over a neighborhood of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We may also call it an R-pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' If B ∈ Q, then we call (X/Z ∋ z, B) a Q-pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' If f = id and z = x ∈ X, then we may use (X ∋ x, B) instead of (X/Z ∋ z, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' If (X/Z ∋ z, B) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', (X ∋ x, B)) is a pair for any point z ∈ Z (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', x ∈ X), then we call (X/Z, B) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', (X, B)) a pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='8 (Singularities of pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let (X/Z ∋ z, B) be a pair associated with the contraction f : X → Z, and let E be a prime divisor over X such that z ∈ f(centerX E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let g : Y → X be a log resolution of (X, B) such that centerY E is a divisor, and suppose that KY + BY = g∗(KX + B) over a neighborhood of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We define a(E, X, B) := 1 − multE BY to be the log discrepancy of E with respect to (X, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We say that a prime divisor E is over X/Z ∋ z if E is a prime divisor E over X and f(centerX E) = ¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We say that (X/Z ∋ z, B) is lc (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', klt) if a(E, X, B) ≥ 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', > 0) for any prime divisor E over X/Z ∋ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We say that (X/Z, B) is lc (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', klt) if (X ∋ x, B) is lc (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', klt) for any codimension ≥ 1 point x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We say that (X/Z, B) is dlt if there exists a log resolution g : Y → X of (X, B) such that a(E, X, B) > 0 for any g-exceptional prime divisor E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that (X/Z, B) is an lc pair and D is an R-Cartier R-divisor on X such that KX + B ∼R,Z D and D is vertical over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Suppose that ψ : X ��� X′ is a 8 GUODU CHEN, JINGJUN HAN, AND JIHAO LIU partial (KX + B)-MMP over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then there exists a proper open subset U ⊂ Z such that Xz is isomorphic to X′ z for any z ∈ U, where Xz (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', X′ z) is the fiber of X → Z (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', X′ → Z) over z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let f : X → Z be the associated morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since KX +B ∼R 0 over Z\\f(SuppD), ψ is an isomorphism over Z\\f(SuppD), so we may choose U := Z\\f(SuppD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that (X/Z ∋ z, B) is an lc pair such that Z is a curve, KX+B ∼R 0 over a neighborhood of z, Bh ∈ Q, and z is a closed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then B + sf ∗z is a Q-divisor over a neighborhood of z, where s := lct(X/Z ∋ z, B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' f ∗z) and f : X → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Possibly shrinking Z we may assume that KX + B ∼R,Z 0 and f(SuppBv) = {z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' There exist real numbers r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , rc, Q-linear functions s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , sm : Rc+1 → R, and Weil divisors B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , Bm on X, such that 1, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , rc are linearly independent over Q, and B = �m i=1 si(1, r0)Bi, where r0 := (r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , rc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let B(v) := �m i=1 si(1, v)Bi for any v ∈ Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since Bh ∈ Q, Bh = B(v)h ∈ Q for any v ∈ Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By [HLS19, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3], KX + B(v) ∼R,Z 0 for any v ∈ Rc and thus B(v) − B ∼R,Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Moreover, as f(Supp(B(v) − B)) = f(SuppBv) = {z}, we see that B − B(v) = lvf ∗z for some real number lv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Pick v0 ∈ Qc, then s + lv0 = lct(X/Z ∋ z, B(v0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' f ∗z) ∈ Q as B(v0) ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since B + sf ∗z = B(v0) + (s + lv0)f ∗z, B + sf ∗z is a Q-divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='11 (Good minimal model conjecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let d be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that (X/Z, B) is an lc pair of dimension d such that KX + B is pseudo-effective over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then (X/Z, B) has a good minimal model over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='12 (Non-vanishing conjecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let d be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that (X/Z, B) is an lc pair of dimension d such that KX + B is pseudo-effective over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then |(KX + B)/Z|R ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Complements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let n be a positive integer, Γ ⊂ (0, 1] a set, and (X/Z ∋ z, B) and (X/Z ∋ z, B+) two pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We say that (X/Z ∋ z, B+) is an R-complement of (X/Z ∋ z, B) if (X, B+) is lc, B+ ≥ B, and KX + B+ ∼R 0 over a neighborhood of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We say that (X/Z ∋ z, B) is R-complementary if (X/Z ∋ z, B) has an R-complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We say that (X/Z ∋ z, B+) is an n-complement of (X/Z ∋ z, B) if (X/Z ∋ z, B+) is lc, nB+ ≥ ⌊(n + 1){B}⌋ + n⌊B⌋, and n(KX + B+) ∼ 0 over a neighborhood of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We say that (X/Z ∋ z, B+) is an (n, Γ)-decomposable R-complement of (X/Z ∋ z, B) if there exist real numbers a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , ak ∈ Γ, and Q-divisors B+ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , B+ k on X, such that �k i=1 ai = 1 and �k i=1 aiB+ i = B+, ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS 9 (X/Z ∋ z, B+) is an R-complement of (X/Z ∋ z, B), and (X/Z ∋ z, B+ i ) is an n-complement of itself for any integer 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3 holds true when d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' More precisely, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let l be a positive integer and Γ ⊂ [0, 1] a DCC set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then there exists a positive integer n which is divisible by l depending only on l and Γ satisfying the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that (X/Z ∋ z, B) is an R-complementary pair of dimension 3 with B ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then (X/Z ∋ z, B) has an n-complement (X/Z ∋ z, B+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Moreover, if SpanQ≥0(¯Γ\\Q) ∩ (Q\\{0}) = ∅, then we can pick B+ ≥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' This follows from [FMX19, Theorem 1] and [HLS19, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Iitaka dimensions and invariant Iitaka dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='15 (Iitaka dimensions, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' [Nak04, II 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2 Definition]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let X be a normal projective variety and D an R-divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' For any positive integer m such that |⌊mD⌋| ̸= ∅, we denote Φm : X ��� P(H0(X, ⌊mD⌋)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' The Iitaka dimension κ(X, D) of D is defined in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' If |⌊mD⌋| ̸= ∅ for some positive integer m, then κ(X, D) := max{dim Φm(X) | m ∈ Z>0, |⌊mD⌋| ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Otherwise, let κ(X, D) := −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Note that if |⌊mD⌋| ̸= ∅, then by [Nak04, II 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='8 Corollary], κ(X, D) = max � k ∈ Z>0 | lim sup m→+∞ dim H0(X, ⌊mD⌋) km > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='16 (Invariant Iitaka dimensions, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' [Cho08, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let X be a normal projective variety and D an R-divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' The invariant Iitaka dimension κι(X, D) of D is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' If |D|R ̸= ∅, then we define κι(X, D) := κ(X, D′) for some R-divisor D′ ∈ |D|R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Otherwise, we let κι(X, D) := −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Note that κι(X, D) is independent of the choice of D′ [Cho08, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We gather some basic properties of κ and κι which will be used in the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='17 ([Cho08, Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let X be a normal projective variety, and D ∼R D′ two R-Cartier R-divisors on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then (1) κ(X, D) ≤ κι(X, D) = κι(X, D′), and κ(X, D) < κι(X, D) if and only if κ(X, D) = −∞ and κι(X, D) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' (2) If D′ ≥ 0, then κ(X, D) ≤ κ(X, D′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' [CHL22, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='10] shows that we may have strict inequality in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='17(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='18 ([Sho03, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='20], [Nak04, II Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let g : Y → X be a surjective morphism between normal projective varieties and D an R-Cartier R-divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then (1) κ(X, D) = κ(Y, g∗D) and κι(X, D) = κι(Y, g∗D), and (2) if g is birational, then κ(X, D) = κ(Y, g∗D + E) and κι(X, D) = κι(Y, g∗D + E) for any g-exceptional R-Cartier R-divisor E ≥ 0 on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 10 GUODU CHEN, JINGJUN HAN, AND JIHAO LIU 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Iitaka fibrations for R-divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that f : X ��� Z is a rational map and f∞ : X∞ → Z∞ is a projective morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We say that f is birational to f∞ if there exist a birational morphism h : X∞ → X and a birational map g : Z∞ ��� Z′ such that f ◦h = g◦f∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', the following diagram is commutative: X∞ f∞ � h � X f �✤ ✤ ✤ Z∞ g �❴ ❴ ❴ ❴ ❴ ❴ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='20 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' [Li22, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let X be a normal projective variety and D an R-Cartier R-divisor on X such that κ(X, D) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' A projective morphism f∞ : X∞ → Z∞ between quasi-projective smooth varieties is called an Iitaka fibration of D if the following hold: (1) dim Z∞ = κ(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' D),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' (2) fm : X ��� Zm ⊂ PH0(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' ⌊mD⌋),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' the map associated with the complete linear system |⌊mD⌋|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' is birational to f∞ with the morphism h : X∞ → X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' for any sufficiently divisible large integer m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' and (3) for any sufficiently large integer n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' we have κ � F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' � h−1 ∗ D + nE � |F � = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' where F is a very general fiber of f∞ and E is the sum of all the h-exceptional prime divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By [Li22, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='20], for any R-Cartier R-divisor D such that κ(X, D) ≥ 0, there always exists an Iitaka fibration of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' The following lemmas are well-known to experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Notation as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' (1) Assume that D∞ is an R-divisor on X∞ such that D∞ − h∗D ≥ 0 and is h- exceptional, then f∞ is an Iitaka fibration of D∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' (2) Assume that X′ ∞ → X∞ is a birational morphism from a smooth variety, then X′ ∞ → Z∞ is an Iitaka fibration of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' (1) By assumption, H0(X, ⌊mD⌋) = H0(X∞, ⌊mD∞⌋) and X∞ ��� Zm is also the map associated with the complete linear system |⌊mD∞⌋|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then (1) follows from the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' (2) We only need to show κ(F ′, ((h′)−1 ∗ D +nE′)|F ′) = 0 for any sufficiently large integer n, where h′ : X′ ∞ → X, F ′ is a very general fiber of X′ ∞ → Z∞ and E′ is the sum of all the h′-exceptional prime divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let ψ′ ∞ : W → X′ ∞ be a resolution which resolves the map X ��� Zm, and we denote ψ : W → X the induced morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By [Li22, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='10], for any sufficiently large integer n, we have that κ � Fm, � (ψ)−1 ∗ D + nEW � |Fm � = 0, ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS 11 where Fm is a very general fiber of W → Zm and EW is the sum of all the ψ-exceptional prime divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let E1 be the sum of all the ψ′ ∞-exceptional prime divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since EW = (ψ′ ∞)−1 ∗ E′ + E1 and ψ−1 ∗ D + n � ψ′ ∞ �−1 ∗ E′ + (n + m)E1 − � ψ′ ∞ �∗ �� h′�−1 ∗ D + nE′� ≥ 0 for any sufficiently large integers n and m, we see that κ � Fm, �� ψ′ ∞ �∗ � h−1 ∗ D + nE1 �� |Fm � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Note that Z∞ is birational to Zm, thus there exists a birational morphism from Fm to F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Therefore κ(F ′, ((h′)−1 ∗ D + nE′)|F ′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Notation as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Suppose that φ : X′ → X is a birational morphism from a normal projective variety and D′ is an R-Cartier R-divisor on X′ such that D′ − φ∗D ≥ 0 and is φ-exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then (1) possibly replacing X∞ with a high model, f∞ is an Iitaka fibration of D′, and (2) any Iitaka fibration of D′ is an Iitaka fibration of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By our assumption, H0(X, ⌊mD⌋) = H0(X∞, ⌊mD∞⌋) and X ��� Zm is the map associated with the complete linear system |⌊mD⌋| if and only if X′ ��� Zm is the map associated with the complete linear system |⌊mD′⌋|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' (1) By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='21(2), possibly replacing X∞ with a high model, we may assume that the map X∞ ��� X′ is a morphism and we denote it by h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' It suffices to prove that for any sufficiently large integer n, κ � F, � (h0)−1 ∗ D′ + nE0 � |F � = 0, where E0 is the sum of all the h0-exceptional prime divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' This follows from the fact that (h0)−1 ∗ D′ + nE0 ≤ h−1 ∗ D + nE for any sufficiently large integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' (2) Suppose that f ′ ∞ : X′ ∞ → Z′ ∞ is an Iitaka fibration of D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let W → X′ ∞ and W → X∞ be a common resolution of X∞ and X′ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='21(2), W → Z∞ is also an Iitaka fibration of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' It follows that X′ ∞ → Z∞ is also an Iitaka fibration of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since Z∞ is birational to Z′ ∞, X′ ∞ → Z′ ∞ is an Iitaka fibration of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Notation as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that ψ : X ��� X′ is a D-non- negative birational contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then possibly replacing X∞ with a high model, f∞ is an Iitaka fibration of D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='21(2), possibly replacing X∞ with a high model, we may assume that the map h′ : X∞ ��� X′ is a morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By our assumption, 0 ≤ h∗D − h′∗D′ is h′- exceptional, and X′ ��� Zm is the map associated with the complete linear system |⌊mD′⌋|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='21(1), f∞ is an Iitaka fibration of h∗D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='22(2), f∞ is also an Iitaka fibration of D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In the rest of this paper, we will use Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='21, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='22 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='23 frequently to replace X with another birational model without citing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 12 GUODU CHEN, JINGJUN HAN, AND JIHAO LIU Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that (X, C) is a projective klt pair such that KX +C ∼R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='12 holds in dimension ≤ dim X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Suppose that D is a Q-Cartier Q- divisor on X such that κ(X, D) ≥ 0, and f∞ : X∞ → Z∞ is an Iitaka fibration of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then κ(F, h∗D|F ) = 0, where F is a general fiber of f∞ and h : X∞ → X is the induced birational morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By Shokurov type polytopes, we may assume that C ∈ Q and thus KX + C ∼Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Take a positive rational number ǫ such that (X∞, h−1 ∗ C + (1 − ǫ)E) is klt and E′ := KX∞ + h−1 ∗ C + (1 − ǫ)E − h∗(KX + C) ≥ 0, where E is the sum of all the h-exceptional prime divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since κ(X, D) ≥ 0, one can find a Q-Cartier Q-divisor D′ ≥ 0 such that D′ ∼Q D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let ǫ′ be a positive rational number such that (X∞, h−1 ∗ C + (1− ǫ)E + ǫ′h∗D′) is still klt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' As f∞ is an Iitaka fibration of D and E′ is h-exceptional, one can see that κ � Fv, � KX∞ + h−1 ∗ C + (1 − ǫ)E + ǫ′h∗D′� |Fv � = κ � Fv, � E′ + ǫ′h∗D′� |Fv � = 0, where Fv is a very general fiber of f∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' According to [HMX18, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='4], we have κ � F, � KX∞ + h−1 ∗ C + (1 − ǫ)E + ǫ′h∗D′� |F � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Therefore κ(F, h∗D|F ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Canonical bundle formulas We refer the reader to [BZ16, Bir19, HL21a, HL22a] for the definitions and basic properties for generalized pair (g-pair for short), and we denote by (X/Z, B + M) a g-pair throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We refer the reader to [Bir19, HL21b, JLX22] for the definition and basic properties of the canonical bundle formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' To sum up, given an lc pair (X/Z, B) and a contraction φ : X → T between normal quasi-projective normal varieties over Z such that KX + B ∼R,T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then we can find an R-divisor BT ≥ 0 and a nef over Z b-R-divisor Mφ on T, such that (T/Z, BT + Mφ) is a glc g-pair, and KX + B ∼R φ∗ (KT + BT + Mφ,T ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Here B (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', Mφ) is called the discriminant part (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', a moduli part) of the canonical bundle formula for (X/Z, B) over T which is uniquely determined (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', determined only up to R-linear equivalence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We may also call Mφ,T the moduli part of the canonical bundle formula for (X/Z, B) over T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Moreover, if (X/Z, B) is klt, then (T/Z, BT + Mφ) is gklt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Here we emphase that there are many choices of Mφ, some of which could behave badly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' But we can always choose one with the required properties in the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' For convenience, we say that two g-pairs (X/Z, B +M) and (X′/Z, B′ +M′) are crepant if X is birational to X′, M = M′, and p∗(KX + B + MX) = q∗(KX′ + B′ + M′ X′) for some common resolution p : W → X and q : W → X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We also call (X′/Z, B′ + M) a crepant model of (X/Z, B + M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let d be a positive integer and Φ ⊂ [0, 1] a DCC set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that Conjectures 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='11 hold in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then there exist a positive integer p and a DCC set Φ′ depending only on d and Φ satisfying the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS 13 Assume that (X/Z, B) is an lc pair of dimension d and φ : X → T is a contraction over Z such that dim T > 0, B ∈ Φ, KX +B ∼R,T 0, and KX +B ∼Q,T 0 over the generic point of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then we can choose a moduli part Mφ of the canonical bundle formula for (X/Z, B) over T such that BT ∈ Φ′, pMφ is b-Cartier, and p(KX + B) ∼ pφ∗(KT + BT + Mφ,T ), where BT is the discriminant part of the canonical bundle formula for (X/Z, B) over T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Moreover, if Φ is a hyperstandard set, then Φ′ is a hyperstandard set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' The proof is similar to the proof of [Bir19, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' For the convenience of the reader, we give a proof here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We also remark that the moreover part of the proposition will not be used in this paper, but it is useful in some other situations (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' [CHX22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In this step, we construct p and make a choice of Mφ,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Note that here we only make a choice of Mφ,T rather than Mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By [HLS19, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='8 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='25], there exist a positive integer p and a finite set Γ0 ⊂ (0, 1] depending only on d and Φ, such that for any t ∈ T, (X/T ∋ t, B) has a (p, Γ0)-decomposable R-complement (X/T ∋ t, B + G) for some R-Cartier R-divisor G ≥ 0, and moreover if B ∈ Γ ∩ Q, then (X/T ∋ t, B) has a monotonic p-complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In particular, G ∼R 0 over a neighborhood of t, and hence G is vertical over T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since KX + B ∼Q,T 0 over the generic point ηT of T, p(KX + B) ∼ 0 over a neighborhood of ηT , and there exists α ∈ K(X) such that pL := p(KX +B)+(α) is zero near ηT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In particular, pL ∼ p(KX + B) ∼R,T 0 and L is vertical over T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By [Li20, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='11], we may find an R-Cartier R-divisor LT on T such that L = φ∗LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let BT be the discriminant part of the canonical bundle formula of (X, B) over T, and Mφ,T := LT − KT − BT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then p(KX + B) ∼ pL = pφ∗LT = pφ∗(KT + BT + Mφ,T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In this step, we show that we can reduce to the case dim T = 1 to show the existence of Φ′ and prove that pMφ,T is integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume dim T > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let H be a general hyperplane section of T, G := φ∗H, and g : G → H the induced morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We may write KG + BG = (KX + G + B)|G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' It is clear that (G, BG) is an lc pair with KG +BG ∼Q,H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Suppose that BH is the discriminant part of the canonical bundle formula for (G, BG) over H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Note that as G is a general member of a free linear system, every lc center SG of (G, BG) is a component of S0 ∩ G for some lc center S0 of (X, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We claim that multD BT = multC BH for any prime divisor D on T and any component C of D ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Indeed, let sD be the lc threshold of φ∗D with respect to (X, B) over the generic point of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then there is an lc center F of (X, B + sDφ∗D) such that φ(F) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Note that F is also an lc center of (X, B +G+sDφ∗D) as G is general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Hence F ∩G is an lc center of (G, BG+sDg∗C), by inversion of adjunction [Kaw07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Moreover, as φ(F ∩G) = C, we see that sD is the lc threshold of g∗C with respect to (G, BG) over the generic point of C, and the claim holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since Φ is a DCC set, there is a DCC set Φ1 depending only on Φ such that BG ∈ Φ1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' [Kol+92, Corollary 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' If Φ is a hyperstandard set, then we 14 GUODU CHEN, JINGJUN HAN, AND JIHAO LIU can take Φ1 to be a hyperstandard set by [Bir19, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In both case, by induction, there is a DCC set (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' hyperstandard set) Φ′ 1 such that BZ ∈ Φ′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Pick a general H′ ∼ H and let KH := (KT + H′)|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Note that the restriction is well defined as H is a general hyperplane section and KH is determined as a Weil divisor, although KT may not be Q-Cartier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let Mg,H := (LT + H′)|H − (KH + BH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then BH + Mg,H = (BT + Mφ,T )|H, and p(KG + BG) ∼ p(L + G)|G ∼ pg∗(LT + H′)|H ∼ pg∗(KH + BH + Mg,H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Hence Mg,H is the moduli part of the canonical bundle formula for (G, BG) over H, and multC(BH + Mg,H) = multD(BT + Mφ,T ) which implies that multC Mg,H = multD Mφ,T as multC BH = multD BT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Therefore p multD Mφ,T is integral if and only if p multC Mg,H is integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Repeating the process we may finish this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In this step we show the existence of Φ′ and that pMφ,T is integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Note that by Step 2, we may assume that dim T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We construct the set Φ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' If Φ is a hyperstandard set which is not a hyperstandard set, then by [HMX14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1], BT ∈ Φ′ for some DCC set Φ′ which only depends on d and Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' If Φ = Φ(R) is a hyperstandard set, then we show BT ∈ Φ′ := Φ(R′), where R′ := � r, r l1 − l2 p | r ∈ R, l1 , l2 ∈ Z>0 � ∩ [0, 1] is a finite set of rational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Suppose that t ∈ T is a closed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then (X/T ∋ t, B +sφ∗t) is a monotonic p-complement of (X/T ∋ t, B), where s := lct(X/T ∋ t, B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' φ∗t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In particular, B + sφ∗t is a Q-divisor over a neighborhood of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof of Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='10, Bt := B + sφ∗t is a Q-divisor over a neighborhood of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Possibly shrinking T around t, we may assume that Bt ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let (X′, B′ t) be a Q-factorial dlt modification of (X, Bt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then ⌊B′ t⌋ has a component mapping to t and KX′ +B′ t ∼Q,T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' There exists a Q-divisor B′ on X′ such that B′ ∈ Φ ∩ Q, ⌊B′⌋ = ⌊B′ t⌋, and BX′ ≤ B′ ≤ B′ t, where BX′ is the strict transform of B on X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By assumption, (X′/T ∋ t, B′) has a monotonic p-complement (X′/T ∋ t, B′+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let B+ be the strict transform of B′+ on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then (X/T ∋ t, B+) is a monotonic p-complement of (X/T ∋ t, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since B+ −B ≥ 0 and B+ − B ∼Q 0 over a neighborhood of t, B+ − B is vertical over T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Moreover, as B′+ ≥ B′, ⌊B′+⌋ has a component mapping to t, and thus (X, B+) has an lc center mapping to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Therefore B+ − B = sφ∗t over t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' The claim holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ Pick a closed point t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2, (X/T ∋ t, B+ := B + sφ∗t) is a p-complement of (X/T ∋ t, B), where s := lct(X/T ∋ t, B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' φ∗t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' For any component S of φ∗t, let ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS 15 b := multS B, b+ := multS B+ and m := multS φ∗t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then b+ = b + sm and thus s = b+−b m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since b ∈ Φ, we may write b = 1 − r/l for some r ∈ R and l ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In particular, s = b+ − 1 + r/l m = 1 m �r l − � 1 − b+�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then multt BT = 1 − s ∈ Φ′, as b+ ∈ 1 pZ ∩ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We show that pMφ,T is integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We may assume that p(KX + B) ∼ 0 over some non-empty open subset U0 ⊆ T such that SuppBT ⊆ T \\ U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let Θ := B + � t∈T\\U0 stφ∗t, where st := lct(X/T ∋ t, B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' φ∗t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let ΘT be the discriminant part of the canonical bundle formula for (X, Θ) over T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then ΘT = BT + � t∈T\\U0 stt which is a reduced divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Moreover, (X/T ∋ t, Θ) is a p-complement of (X/T ∋ t, B) for every t ∈ T \\ U0 by Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Hence p(KX + Θ) ∼T 0 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since p(KX + Θ) = p(KX + B) + p(Θ − B) ∼ pφ∗(KT + BT + Mφ,T) + pφ∗(ΘT − BT ) = pφ∗(KT + ΘT + Mφ,T ), p(KT + ΘT + Mφ,T ) is Cartier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' It follows that pMφ,T is integral as KT + ΘT is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In this step, we finish the proof by showing that pMφ is b-Cartier and nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Note that in this step we do not assume dim T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' According to [Bir19, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='6], we only need to show that pMφ is b-Cartier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let T ′ → T be a high resolution and Y → X a log resolution of (X, B) such that Y → T ′ is a morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let U ⊆ T be a non-empty open subset such that U ′ → U is an isomorphism where U ′ ⊆ T ′ is the inverse image of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let BY be the sum of the strict transform of B and the reduced exceptional divisor of Y → X but with all the components mapping outside U removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In particular, the generic point of any lc center of (Y, BY ) maps into U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We may run an MMP on KY + BY over X ×T T ′ with scaling of some ample divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By [Bir12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='9], the MMP terminates over U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In fact, we reach a model X′ such that over U ′ the pair (X′, B′) is a Q-factorial dlt modification of (X, B), where B′ is the strict transform of BY on X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Hence KX′ +B′ ∼Q 0 over U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Now by [HX13, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1] (see also [Bir12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1]), we can run an MMP on KX′ +B′ over T ′ which terminates with a good minimal model X′′ over T ′ as the generic point of every lc center of (X′, B′) is mapped into U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let B′′ be the strict transform of B′ on X′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then KX′′ +B′′ is semi-ample over T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 16 GUODU CHEN, JINGJUN HAN, AND JIHAO LIU X Y X′ X′′ T T ′ T ′′ U U ′ U ′′ φ φ′′ ∼ = ∼ = Let φ′′ : X′′ → T ′′ be the contraction defined by KX′′ + B′′ over T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Note that T ′′ → T ′ is birational as KX′ + B′ ∼Q 0 over U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that (T ′′, B′′ T ′′ + Mφ) is the crepant model of (T, BT +Mφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then LT ′′ = KT ′′ +B′′ T ′′ +Mφ,T ′′, where LT ′′ is the pullback of LT on T ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let f : W → X and f ′′ : W → X′′ be a common resolution, KX′′ + ∆′′ := f ′′ ∗ f ∗(KX + B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since KX + B and KX′′ + B′′ are crepant over U, we see that B′′ − ∆′′ is vertical over T ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Note that B′′ − ∆′′ ∼R,T ′′ 0, B′′ − ∆′′ = (φ′′)∗PT ′′ for some R-Cartier R-divisor PT ′′ on T ′′ by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Denote by BT ′′ the discriminant part of the canonical bundle formula for (X′′, B′′) over T ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then BT ′′ = B′′ T ′′ + PT ′′ and p � KX′′ + B′′� = p � KX′′ + ∆′′ + B′′ − ∆′′� ∼ p � f ′′ ∗ f ∗L + B′′ − ∆′′� = p � φ′′�∗ (LT ′′ + PT ′′) = p � φ′′�∗ � KT ′′ + BT ′′ + Mφ,T ′′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Now by Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2, pMφ,T ′′ is an integral divisor, hence pMφ,T ′ is integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' As T ′ is smooth, we conclude that pMφ is b-Cartier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that X → Z is a proper morphism of varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let Xν be the normalization of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then Xν z is the normalization of Xz for any general point z ∈ Z, where Xz (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', Xν z ) is the fiber over z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In particular, if Xz is normal, then Xν z is isomorphic to Xz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By assumption, Xν → X is birational and finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Thus Xν z → Xz is birational and finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let ˜Xz be the normalization of Xz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' The universal property of the normalization implies that there is a morphism ˜Xz → Xν z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Moreover, the morphism ˜Xz → Xν z is birational and finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since both ˜Xz and Xν z are normal, we see that ˜Xz is isomorphic to Xν z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let d, b, β be three positive integers and Γ ⊂ [0, 1] ∩ Q a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then there exist a positive integer p and a DCC set Γ′ ⊆ [0, 1] depending only on d, b, β and Γ such that ¯Γ′ ⊆ [0, 1] ∩ Q and satisfying the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that (X, B) is an lc pair of dimension d, φ : X → T is a contraction between quasi-projective normal varieties, and F is a general fiber of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Suppose that (1) B ∈ Γ, and KX + B ∼Q,T 0, (2) dim T > 0 and (X, B) is klt over the generic point of T, (3) b is the non-vanishing order of (KX + B)|F, and (4) β := dim Hdim F ( ˜F, C), where ˜F is a smooth model of the cover of F associated to the unique divisor of |b(KX + B)|F |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS 17 Then we can choose a moduli part Mφ of the canonical bundle formula for (X, B) over T such that BT ∈ Γ′, pMφ is b-Cartier, and p (KX + B) ∼ pφ∗ (KT + BT + Mφ,T ) , where BT is the discriminant part of the canonical bundle formula for (X, B) over T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let BT be the discriminant part of the canonical bundle formula for (X, B) over T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then, by [HMX14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='11], BT ∈ Γ′ for some DCC set Γ′ which only depends on d and Γ such that ¯Γ′ ⊂ [0, 1] ∩ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' It suffices to prove the existence of p with the required properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let T ′ → T be a resolution, X′ the normalization of the main component of X×T T ′, and denote by φ′ : X′ → T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let (X′, B′) be the crepant model of (X, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then KX′+B′ ∼Q,T ′ 0 and (X′, B′) is sub-lc and is klt over the generic point of T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3, the general fiber F ′ of φ′ is isomorphic to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Therefore it holds that b is the non-vanishing order of (KX′ + B′)|F ′, and ˜F is a smooth model of the cover of F ′ associated to the unique divisor of |b(KX′ + B′)|F ′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' According to [Flo14, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1], one can find a positive integer p0 depending only on β and a choice of a moduli part Mφ of the canonical bundle formula for (X′, B′) over T ′ such that p0Mφ,T ′ is integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In particular, p0Mφ,T ′ is Cartier and thus p0Mφ is b-Cartier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since b is the non-vanishing order of (KX + B)|F , by the choice of Mφ again, we see that b (KX + B) ∼ bφ∗ (KT + BT + Mφ,T) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Therefore we may conclude that p := bp0 and Γ′ have the required properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Non-vanishing orders, middle Betti numbers, and complements In this section, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' More generally, we proof Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let d, b, β be three positive integers and Γ ⊂ [0, 1] ∩ Q a DCC set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that the good minimal model conjecture (Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='11) holds in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then there exists a positive integer n depending only on d, b, β and Γ satisfying the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that (X, B) is a Q-factorial projective pair of dimension d, f∞ : X∞ → Z∞ is an Iitaka fibration of −(KX + B), h : X∞ → X is the induced birational morphism, and F is a general fiber of f∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Suppose that (1) B ∈ Γ, (2) −(KX + B) is nef, (3) (X, B) has a klt R-complement, (4) κ(X, −(KX + B)) ≥ 0, and b is the non-vanishing order of −h∗(KX + B)|F , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', b = min {a ∈ Z>0 | | − ah∗(KX + B)|F | ̸= ∅} , and (5) β := dim Hdim F ( ˜F, C), where ˜F is a smooth model of the cover of F associated to the unique divisor of | − bh∗(KX + B)|F |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then (X, B) has an n-complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 18 GUODU CHEN, JINGJUN HAN, AND JIHAO LIU We remark here that κ(F, −h∗(KX + B)|F) = 0 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We first show the existence of (n0, Γ0)-decomposable R-complements under the assumption of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Notation and assumptions as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then (X, B) has an (n0, Γ0)- decomposable R-complement, where n0 is a positive integer and Γ0 ⊂ [0, 1] is a finite set depending only on d, b, β and Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In this step, we show that −(KX + B) is semi-ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let (X, C) be a klt R-complement of (X, B) for some effective R-divisor C ≥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Pick some positive real number ǫ0 such that (X, C + ǫ0(C − B)) is klt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since KX + C + ǫ0(C − B) ∼R −ǫ0(KX + B) is nef, we see that KX + C + ǫ0(C − B) is semi-ample by [HMX18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1] and hence −(KX + B) is also semi-ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Denote by f : X → Z the ample model of −(KX + B) and F0 a general fiber of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Moreover, by [Li22, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='21], f∞ is birational to f, so there exists a naturally induced morphism h : F → F0 such that b(KX + B)|F0 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' If dim Z = 0, then b(KX + B) ∼ 0 and thus (X, B) has a (b, {1})-decomposable R- complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Therefore, we may assume that dim Z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In this step, we construct finite sets Γ0 ⊂ (0, 1] and Γ′ 2 ⊂ [0, 1]∩Q depending only on d, b and Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let α be a positive real number such that α < min � |γ − i b| > 0 | γ ∈ ¯Γ, i ∈ Z≥0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By [HLS19, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='20], there exist a finite set Γ1 depending only on d, α and Γ, and an R-divisor ¯B on X, such that ¯B ∈ Γ1, α Supp B ≥ ¯B − B ≥ 0, and (X, ¯B) is R-complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In particular, for any component S of B such that b multS B ∈ Z, multS ¯B = multS B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Hence ( ¯B)h = Bh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By [HLS19, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='16], there exist a point v0 := (v0 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , v0 m) and an open subset V of the rational envelope of v0 (that is the smallest affine subspace containing v0 which is defined over Q) in Rm depending only on d and Γ1, and Weil divisors B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , Bm ≥ 0 on X, such that B(v0) = ¯B, and (X, B(v)) is R-complementary for any v ∈ V , where B(v) := �m i=1 viBi for any v := (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , vm) ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Moreover, possibly replacing V , we may assume that B(v) ≥ D := ¯B − B, and |B(v) − ¯B| < α for any v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We remark that (B(v))h = ( ¯B)h = Bh for any v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Pick points v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , vk ∈ V ∩Qm, such that v0 is in the interior of the convex hull spanned by v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' For any integer 1 ≤ i ≤ k, set B(i) := B(vi) − D and ¯B(i) := B(vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS 19 There exist finite sets Γ0 := {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , ak} ∪ {1} ⊂ (0, 1] and Γ′ 2 ⊂ [0, 1] ∩ Q such that ¯B(i) ∈ Γ′ 2 for any integer 1 ≤ i ≤ k, �k i=1 ai = 1, and �k i=1 aivi = v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In particular, �k i=1 aiB(i) = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In this step, we run an MMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let (X, ¯B(i)+ ¯Gi) be an R-complement of (X, ¯B(i)) for some effective R-Cartier R-divisor ¯Gi for any integer 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Pick a positive real number ǫ such that (X, C + ǫ ¯Gi) is klt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then we may run an MMP on KX + C + ǫ ¯Gi over Z, which terminates with a model Wi over Z such that KWi +CWi +ǫ ¯GWi is semi-ample over Z, where CWi and ¯GWi are the strict transforms of C and ¯Gi on Wi respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' This MMP is also an MMP on −(KX + ¯B(i)) over Z, hence −(KWi + ¯B(i) Wi) is semi-ample over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let gi : Wi → Ti be the ample model of −(KWi + ¯B(i) Wi) over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Recall that by construction and by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1, B − ¯B(i) is vertical over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' As −(KX + ¯B(i)) ∼R,Z B − ¯B(i), one can see that F0 is isomorphic to a general fiber of Wi → Z by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='9, and Ti is birational to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In this step, we construct a positive integer p, and a hyperstandard set Γ2 depending only on d, b, β and Γ′ 2, and make a choice of a moduli part Mi of the canonical bundle formula for (Wi, ¯B(i) Wi) over Ti such that p � KWi + ¯B(i) Wi � ∼ pg∗ i (KTi + BTi + Mi,Ti) , BTi ∈ Γ2, and pMi is b-Cartier, where BTi is the discriminant part of the canonical bundle formula for (Wi, ¯B(i) Wi) over Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' To see this, we first claim that (i) b is the non-vanishing order of −(KWi + ¯B(i) Wi)|F ′ i , and b(KWi + ¯B(i) Wi)|F ′ i ∼ 0, and (ii) ˜F is a smooth model of the cover of F ′ i associated to the unique divisor of |b(KWi + ¯B(i) Wi)|F ′ i |, where F ′ i is a general fiber of gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that b0 is the non-vanishing order of (KX + B)|F0 ∼Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then b0(KX + B)|F0 ∼ 0 and b0 ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since h∗ 0((KX+B)|F0) = h∗(KX+B)|F, we have b0 = b and bh∗(KX+B)|F ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Therefore b is also the non-vanishing order of (KX + B)|F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Note that Spec �b−1 � i=0 OF (⌊ih∗(KX + B)|F ⌋) � → F (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=', Spec �b−1 � i=0 OF0 (⌊i(KX + B)|F0⌋) � → F0) is the cover associated to the unique element of ⌊h∗(KX + B)|F ⌋, see [Kol13, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By [Nak04, II Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='11], there is a natural isomorphism (h0)∗OF (⌊i(h∗(KX + B)|F )⌋) → OF0 (⌊i((KX + B)|F0)⌋) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Thus ˜F is a smooth model of the cover of F0 associated to the unique divisor of |b(KX + B)|F0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Furthermore, by our construction, X is isomorphic to Wi and B = ¯B(i) over the generic point of Z, and F ′ i is isomorphic to F0 (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Thus the claim holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since (X, B) has a klt R-complement, (Wi, BWi) has a klt R-complement, where BWi is the strict transform of B on Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Moreover, as BWi = ¯B(i) Wi over the generic point of Ti, we 20 GUODU CHEN, JINGJUN HAN, AND JIHAO LIU can see that (Wi, ¯B(i) Wi) is klt over the generic point of Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Recall that Γ′ 2 ⊂ [0, 1] ∩ Q is a finite set and ¯B(i) Wi ∈ Γ′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='4, we may find a positive integer p, and a DCC set Γ2 depending only on d, b, β and Γ′ 2 such that ¯Γ2 ⊂ [0, 1] ∩ Q, and make a choice of a moduli part Mi of the canonical bundle formula for (Wi, ¯B(i) Wi) over Ti, such that p � KWi + ¯B(i) Wi � ∼ pg∗ i (KTi + BTi + Mi,Ti) , BTi ∈ Γ2, and pMi is b-Cartier, where BTi is the discriminant part of the canonical bundle formula for (Wi, ¯B(i) Wi) over Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Note that (Ti, BTi + Mi) is glc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In this step, we find an integer n0 which only depends on d, bp and Γ2 satisfying our requirements and thus finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We first show that Ti is of Fano type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Indeed, according to [Amb05, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2], there exist klt pairs (Z, ∆) and (Ti, ∆Ti) such that KX + B ∼R f ∗(KZ + ∆) and KWi + BWi ∼R g∗ i (KTi + ∆Ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In particular, KTi + ∆Ti ∼R τ ∗ i (KZ + ∆) where τi denotes the induced morphism Ti → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since Z is the ample model of −(KX+B), −(KZ+∆) is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' It follows that −(KTi+∆Ti) is big and nef as τi is a birational morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Thus Ti is of Fano type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since (Wi, ¯B(i) Wi) is R-complementary, (Ti, BTi + Mi) is R-complementary, that is there is an R-divisor Pi ≥ 0 such that (Ti, BTi +Pi +Mi) is glc and KTi +BTi +Pi +Mi,Ti ∼R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' As Ti is of Fano type, by [Bir19, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='10] (see also [Che20, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3]), there exists a positive integer n0 divisible by bp depending only on d, bp and Γ2, and a Q-divisor B+ Ti ≥ BTi on Ti, such that (Ti, B+ Ti + Mi) is glc and n0(KTi + B+ Ti + Mi,Ti) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' It is enough to show that (Wi, ¯B(i),+ Wi := ¯B(i) Wi + g∗ i (B+ Ti − BTi)) is a monotonic n0- complement of (Wi, ¯B(i) Wi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Indeed, by [PS09, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='18(1)], (Wi, ¯B(i),+ Wi ) is lc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since n0 � KWi + ¯B(i),+ Wi � = n0 � KWi + ¯B(i) Wi � + n0g∗ i � B+ Ti − BTi � ∼ n0g∗ i (KTi + BTi + Mi,Ti) + n0g∗ i � B+ Ti − BTi � = n0g∗ i � KTi + B+ Ti + Mi,Ti � ∼ 0, one can see that (Wi, ¯B(i),+ Wi ) is a monotonic n0-complement of (Wi, ¯B(i) Wi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Remark that by Step 3, X ��� Wi is −(KX + ¯B(i))-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Therefore (X, ¯B(i)) also has a monotonic n0-complement (X, ¯B(i),+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' This immediately implies that (X, �k i=1 ai ¯B(i),+) is an (n0, Γ0)- decomposable R-complement of (X, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We may finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2, there exist a positive integer n0 and a finite set Γ0 ⊂ (0, 1] depending only on d, b, β, Γ, such that (X, B) has an (n0, Γ0)-decomposable R-complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1 follows from Diophantine approximation as in the proof of [HLS19, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='8] (see [HLS19, Section 6] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' It follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS 21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Effective Iitaka fibrations In this section, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that f : X → Z is a contraction between normal quasi-projective varieties, and D is an R-Cartier R-divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Suppose that either dim Z = 0 and D ∼R 0, or dim Z > 0 and D ∼R f ∗DZ for some big R-Cartier R-divisor DZ on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then κ(X, D) ≥ 0 if and only if Dh is a Q-divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We first assume that κ(X, D) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Suppose on the contrary that Dh is not a Q- divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let F be a very general fiber of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then {mD|F } = {mDh|F} ̸= 0 for any positive integer m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By our assumption, D|F ∼R 0 and ⌊mD|F⌋ = mD|F − {mD|F } ∼R −{mD|F } is not pseudo-effective for any positive integer m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Thus ⌊mD⌋ is not pseudo-effective for any positive integer m, which implies that κ(X, D) = −∞, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Therefore, Dh is a Q-divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Now suppose that Dh is a Q-divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' If dim Z = 0, then D ∼Q 0 and thus κ(X, D) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that dim Z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let b be a positive integer, such that bD ∼ 0 on the generic fiber of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then there exists α ∈ K(X) such that bD+(α) is vertical over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since bD+(α) ∼R,Z 0, bD + (α) = f ∗D′ Z for some R-Cartier R-divisor D′ Z on Z by [Li20, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Thus D′ Z ∼R bDZ, so D′ Z is big.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='18, κ(X, D) = κ(X, bD) = κ(X, bD + (α)) = κ(Z, D′ Z) = dim Z > 0, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let Γ ⊂ [0, 1] be a DCC set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Without loss of generality, we may assume that 1 ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that (X, B) is a projective lc pair of dimension d such that κ(X, KX +B) ≥ 0 and B ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Since we assume Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='11 in dimension d, (X, B) has a good minimal model (X′, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Possibly replacing (X, B) with (X′, B′), we may assume that KX + B is semi-ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let f : X → Z be the ample model of KX + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Suppose that dim Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then B ∈ Γ0 for some finite set Γ0 ⊂ Γ∩Q which only depends on d and Γ by the global ACC [HMX14, Theorem D] and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1, and KX + B ∼R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' As Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3 holds in dimension d, we may find a positive integer m0 depending only on d and Γ0 such that m0(KX + B) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In what follows, we may assume that dim Z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' According to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1, there exist a positive integer p and a DCC set Γ′ ⊂ [0, 1] depending only on d and Γ, and a choice of the moduli part Mf of the canonical bundle formula for (X, B) over Z, such that BZ ∈ Γ′, pMf is b-Cartier, and p(KX + B) ∼ pf ∗ (KZ + BZ + Mf,Z) , where BZ is the discriminant part of the canonical bundle formula for (X, B) over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Recall that Z is the ample model of KX + B, hence KZ + BZ + Mf,Z is big.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By [BZ16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3], there is a positive integer m1 depending only on d, p, and Γ′, such that 22 GUODU CHEN, JINGJUN HAN, AND JIHAO LIU |⌊m1(KZ + BZ + Mf,Z)⌋| defines a birational map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By [Nak04, II 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='11 Lemma], we have that H0 (X, ⌊pm1(KX + B)⌋) = H0 (X, ⌊pm1f ∗ (KZ + BZ + Mf,Z)⌋) ∼= H0(Z, ⌊pm1(KZ + BZ + Mf,Z)⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Therefore |⌊pm1(KX + B)⌋| defines a map which is birational to f∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let m := pm0m1, then m satisfies our required property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' It follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='4, the existence of good minimal models in dimension ≤ 3 [KMM87, KMM94], and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Existence of decomposable Iitaka fibrations In this section, we recall the definition of invariant Iitaka fibrations, which generalizes Iitaka fibrations to the category of all pairs with non-negative invariant Iitaka dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We then show the existence of decomposable Iitaka fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1 (Invariant Iitaka fibrations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let X be a normal projective variety and D an R-Cartier R-divisor on X such that κι(X, D) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' A morphism f∞ : X∞ → Z∞ between smooth varieties is called an invariant Iitaka fibration of D if there exists an R- Cartier R-divisor D′ on X, such that D ∼R D′, κ(X, D′) ≥ 0, and f∞ is an Iitaka fibration of D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By [Hu20, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3], an invariant Iitaka fibration of D always exists, and is independent of the choice of D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We propose the following conjecture, which is a little stronger than Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let d be a positive integer and Γ ⊂ [0, 1] a DCC set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then there exist a positive integer m, a finite set Γ0 ⊂ (0, 1], and a DCC set Γ′ ⊂ [0, 1] depending only on d and Γ satisfying the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that (X, B) is an lc pair of dimension d such that B ∈ Γ, κι(X, KX + B) ≥ 0, and either Γ is a finite set or all component of B are Q-Cartier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then (1) (Weak version) (X, B) has a (Γ0, Γ′)-decomposable Iitaka fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' (2) (Strong version) (X, B) has an (m, Γ0, Γ′)-decomposable Iitaka fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let d be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that the non-vanishing conjecture (Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='12) holds in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then: (1) Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2(1) holds in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' (2) Assume the effective log Iitaka conjecture (Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2) holds in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2(2) holds in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let Γ ⊂ [0, 1] be a DCC set and possibly replacing Γ with Γ ∪ {1}, we may assume that 1 ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that (X, B) is an lc pair of dimension d such that B ∈ Γ, κι(X, KX + B) ≥ 0, and either Γ is a finite set or every component of B is Q-Cartier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let f∞ : X∞ → Z∞ be an invariant Iitaka fibration of KX + B, h : X∞ → X the induced morphism, and F a very ON EFFECTIVE LOG IITAKA FIBRATIONS AND EXISTENCE OF COMPLEMENTS 23 general fiber of f∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let B∞ := h−1 ∗ B + E, where E is the sum of all the h-exceptional prime divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then (X∞, B∞) is log smooth, κι(X∞, KX∞ + B∞) = κι(X, KX + B), and κι(F, (KX∞ + B∞)|F ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' If Γ is a finite set, then we let ˜Γ := Γ, otherwise we let ˜Γ := ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We let v0 := (v0 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , v0 m), g, and V be as in [CHL22, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='1] which depend only on d, Γ, ˜Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We may write B∞ = � bjB(j) ∞ , where B(j) ∞ are the irreducible components of B∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then there exist distinct Weil divisors B∞,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , B∞,m ≥ 0 on X∞, such that g(γ) ≥ γ for any γ ∈ ¯Γ, and g(γ′) = γ′ for any γ′ ∈ ˜Γ, B∞(v0) = � g(bj)B(j) ∞ , where B∞(v) := �m i=1 viB∞,i for any v := (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , vm) ∈ Rm, both (X∞, B∞(v)) and (X∞, B∞(v) − D∞) are lc for any v ∈ V , where D∞ := B∞(v0) − B∞ ≥ 0, and κ(X∞, KX∞ + B∞(v) − D∞) = κι(X∞, KX∞ + B∞) = dim Z for any v ∈ V ∩ Qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let DF := D∞|F , D := h∗D∞, BF (v) := B∞(v)|F and B(v) := h∗B∞(v) for any v ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By the construction of v0, g, and V , we have (X, B(v) − D) is lc for any v ∈ V , and κ(F, KF + BF(v) − DF ) = κι(F, (KX∞ + B∞)|F ) = 0 for any v ∈ V ∩ Qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Note that if Γ is a finite set, then D∞ = 0 and DF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' By the definition of Iitaka fibrations, f∞ is an Iitaka fibration of KX∞ + B∞(v) − D∞ for any v ∈ V ∩ Qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' As (X, B(v) − D) is lc for any v ∈ V , KX∞ + B∞(v) − D∞ − h∗(KX + B(v) − D) ≥ 0 and is h-exceptional for any v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Therefore f∞ is an Iitaka fibration of KX + B(v) − D for any v ∈ V ∩ Qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , vk ∈ V ∩ Qm be rational points, such that v0 is contained in the interior of the convex hull of v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' There exist a DCC set Γ′ ∋ 1 and a finite set Γ0 := {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' , ak} ⊂ (0, 1] depending only on d and Γ such that Bi := B(vi) − D ∈ Γ′ for any integer 1 ≤ i ≤ k, and �k i=1 aivi = v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' In particular, KX + B = k � i=1 ai (KX + Bi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Recall that f∞ is an Iitaka fibration of KX + B(v) − D for any v ∈ V ∩ Qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Thus f∞ is an Iitaka fibration of KX + Bi for any integer 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' We conclude that (X, B) has a (Γ0, Γ′)-decomposable Iitaka fibration, which is (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Now suppose that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2 holds in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then there exists a positive integer m depending only on d and Γ′, such that the map defined by |⌊m(KX + Bi)⌋| is birational to f∞ for any integer 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Therefore (2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2 holds when d ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' It follows from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3 and the existence of good minimal models in dimension ≤ 3 [KMM87, KMM94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ 24 GUODU CHEN, JINGJUN HAN, AND JIHAO LIU Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Let d be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Assume that the good minimal model conjecture (Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='11) and the existence of complements (Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3) hold in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Then Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='2 holds in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' It follows from Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' It is a special case of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' It is a special case of Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' It is a special case of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' □ References [Amb05] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Ambro, The moduli b-divisor of an lc-trivial fibration, Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 141 (2005), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 2, 385– 403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' [Bir12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Birkar, Existence of log canonical flips and a special LMMP, Pub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' IHES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 115 (2012), 325–368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' [Bir19] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Birkar, Anti-pluricanonical systems on Fano varieties, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 190 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 2, 345–463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' [Bir21] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Birkar, Singularities of linear systems and boundedness of Fano varieties, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 193 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 2, 347–405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' [BCHM10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Birkar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Cascini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Hacon and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' McKernan, Existence 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Birkar and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Zhang, Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs, Pub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' IHES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content=' 123 (2016), 283–331.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='cn Department of Mathematics, Northwestern University, 2033 Sheridan Rd, Evanston, IL 60208, USA Email address: jliu@northwestern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE3T4oBgHgl3EQf9AvA/content/2301.04813v1.pdf'} diff --git a/wNE0T4oBgHgl3EQfswEr/content/tmp_files/2301.02582v1.pdf.txt b/wNE0T4oBgHgl3EQfswEr/content/tmp_files/2301.02582v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2baa96d81b1e36d1e0271c185cd1d8cd75d92fa4 --- /dev/null +++ b/wNE0T4oBgHgl3EQfswEr/content/tmp_files/2301.02582v1.pdf.txt @@ -0,0 +1,2981 @@ +IMMERSED BOUNDARY METHOD FOR THE COMPLETE ELECTRODE +MODEL IN ELECTRICAL IMPEDANCE TOMOGRAPHY +J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS +Abstract. We propose an immersed boundary scheme for the numerical resolution of the Complete +Electrode Model in Electrical Impedance Tomography, that we use as a main ingredient in the +resolution of inverse problems in medical imaging. +Such method allows to use a Cartesian mesh +without accurate discretization of the boundary, which is useful in situations where the boundary is +complicated and/or changing. We prove the convergence of our method, and illustrate its efficiency +with two dimensional direct and inverse problems. +1. Introduction +This work is dedicated to the numerical resolution of direct and inverse problems related to Electrical +Impedance Tomography (EIT) using an immersed boundary method. The aim of EIT is to reconstruct +the electrical conductivity distribution inside a domain by imposing electrical currents on the boundary +of this domain, and measuring the resulting voltages on the same boundary. It has several applications +in the medical field, in particular in lung monitoring and stroke detection [1, 8, 30]. Mathematically, +the problem, known as Calder´on problem or inverse conductivity problem, is a severely ill-posed inverse +problem. We refer to [5, 9, 49] and the references therein for an overview on the inverse conductivity +problem. +In practical experiments, the currents are driven inside the body of interest through a collection +of surface electrodes, no current being driven between the electrodes. For each current pattern, the +potential differences between the electrodes are measured. This practical setting is accurately modeled +by the Complete Electrode Model (CEM) [11, 48]. It takes into account the shape of the electrodes +as well as the shunting effect, that is the thin resistive layer that appears at the interface between +the electrodes and the object during the measurements. +The CEM is known to correctly predict +experimental data, and therefore is widely used in the numerical resolution of both direct and inverse +problems related to EIT. +Usually, Finite Element Methods (FEM) are used to compute solutions of the CEM. This leads to +fast and efficient numerical methods in simple geometries. However, when the geometry is complicated, +the FEM can become very expensive, especially in three-dimensional problems. This is due to the need +of an accurate discretization of the geometry, in particular on the electrodes where the potential varies +rapidly in space, imposing the use of very refined meshes. +Another difficulty could arise when the geometry is not perfectly known, which usually leads to +highly incorrect reconstructions [22, 42]. Iterative Newton-type algorithms that reconstruct both the +conductivity and the geometry have been proposed to tackle this difficulty [15, 16]. In such algorithms, +the geometry changes from one iteration to the other. Therefore, a new mesh is created at each step, +leading again to a costly numerical implementation. +In the present paper, our objective is to propose an alternative to the numerical resolution of CEM +with classical mesh-adapted numerical methods, by using Immersed Boundary Methods (IBM), with +the aim of reducing the cost of computation in complex and moving geometries. In such approach, +the domain of interest is included in a larger domain with a simple geometry, typically a square in +two-dimensional settings, or a cube in three-dimensional settings. The geometry of the domain of +Date: January 6, 2023. +2010 Mathematics Subject Classification. 65N21, 65N25, 65N85, 35R30. +1 + +2 +J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS +interest is encoded as the zero level-set of a known function, and is not discretized exactly. Instead, +the larger domain is discretized using typically a Cartesian mesh. Of course, this approach requires a +particular attention to include the boundary conditions of the initial problem. +The first method on Cartesian grids for elliptic problems was designed by Mayo in 1984 [39], and +developed further in [40] and [38]. In that work an integral equation was derived to solve elliptic +interface problems with piecewise coefficients to second order accuracy in maximum norm. +Then +LeVeque and Li (1994) [36] devised the very well known Immersed Interface Method (IIM), that relies +on Taylor expansions of the solution on each side of the interface, with a local coordinate transformation +near the interface to express the jump conditions in an appropriate frame. The elliptic operator is +discretized on each grid point near the interface with formulas accounting for the jumps across the +interface. In order to find these formulas, a linear system with six unknowns needs to be solved for +each of the concerned grid points. Numerous developments of the IIM have been performed thereafter, +among them [3, 37, 52]. Another class of Cartesian method was introduced by Zhou et al. is the +Matched Interface and Boundary (MIB) method [53]. The solution on each side of the interface is +extended on fictitious points on the other side. These fictitious values are computed by iteratively +enforcing the lowest order interface jump conditions. Other classes of Cartesian methods also exist, +less accurate in the case of interface problems, but probably simpler to implement: Gibou et al. [23, +24] developed methods inspired by Fedkiw’s Ghost-Fluid Method [19] for multiphase flows. These +methods are second order accurate for Dirichlet boundary conditions on arbitrary domains, but only +first order accurate for interface problems. Then new methods were proposed to increase the accuracy +in case of interface problems [4, 18]. +In this paper, we approximate the CEM with a finite-difference scheme on a Cartesian mesh, and +we follow the methodology of [13], where additional unknowns were defined at the intersections of +the interface with the grid. These additional unknowns were used in the discretization of the elliptic +operator near the interface, which avoided deriving specific finite differences formulas containing jump +terms, or corrective terms. In order to solve the interface unknowns, the flux jump conditions were +discretized and added to the linear system to solve. This use of additionnal unknowns at the inter- +face between two domains, applied in the context of electrical impedance tomography, simplifies the +discretization because the values at the boundary of the domain are directly involved in the equations +modeling the input currents on the electrodes. +As stated above, the main difficulties for the discretization lie in the limit conditions. In particular, +the limit conditions related to the input currents are integrals on the electrodes, which are non-standard +from a finite differences point of view, and therefore need particular meticulousness. Another difficulty +appears at the boundary of the electrodes, where the limit condition changes abruptly, leading to +singularities in the solution. We tackle this difficulty by using the smoothened Complete Electrode +Model [28], in which the transition between the conditions on electrodes and the condition in between +electrodes in smooth, leading to smooth solutions. +Finally, as the problem is typically of Neumann type in order to accurately model the physical +measurements setting, one has to deal with a compatibility condition on the data which is not easy to +verify at the numerical level. We circumvent this last problem with a slight modification of the model, +which allows to get rid of the compatibility condition without changing the solution of interest: this +might be of interest on its own, as example to simplify numerical implementation of the CEM model +by finite elements. +1.1. Outline of the paper. The paper is organized as follows. In Section 2, we present the Complete +Electrode Model and its variations of interest for our study. In Section 3, we present the numerical +scheme, and prove a linear rate of convergence. We present numerical results in a two-dimensional +setting in Section 4, for both the direct problem of Electrical Impedance Tomography, the inverse +conductivity problem, and the electrodes location problem. Some needed and technical computations +are presented in the Appendix A. + +IMMERSED BOUNDARY METHOD FOR CEM IN EIT +3 +2. The Complete Electrode Model for Electrical Impedance Tomography +2.1. Geometrical and functional settings. Here and in the following, we suppose that all functions +and vectors are real-valued. Everything can be straightforwardly extended to the complex case. +Let Ω be a bounded domain of Rd, d = 2 or 3, with a smooth boundary ∂Ω. We denote by ν the +exterior unit normal of ∂Ω. Let (Em)m=1,...,M be M ≥ 2 mutually disjoint connected subdomains of +∂Ω with positive Lebesgue-measure. We set E = �M +m=1 Em, and Ec = ∂Ω \ E. +We denote RM +⋄ +the subspace of mean-free vectors of RM, +RM +⋄ = +� +V ∈ RM, +M +� +m=1 +Vm = 0 +� +, +and RM ++ the subset of RM of vectors with positive components, +RM ++ = +� +V ∈ RM, Vm > 0 +∀m ∈ {1, . . . , M} +� +. +For s ∈ R, we introduce the quotient space Hs = (Hs(Ω) × RM)/R. Two elements (u, U) and (v, V ) +of Hs(Ω) × RM are identified in Hs if they differ by an additive constant, in the following sense: +(2.1) +(u, U) ∼ (v, V ) ⇔ ∃c ∈ R, u = v + c, U = V + c[1, . . . , 1]. +Classicaly, Hs endowed with the norm +∥u, U∥Hs = inf +c∈R(∥u − c∥Hs(Ω) + ∥U − c[1, . . . , 1]∥RM ), +is a Banach space. +For all m in {1, . . . , M}, let em ⊂ Em be of positive Lebesgue measure. We set +b : ((u, U), (v, V )) ∈ (H1(Ω) × RM)2 �→ +� +Ω +∇u · ∇v dx + +M +� +m=1 +� +em +(u − Um)(v − Vm) ds(x), +and, for all (v, V ) ∈ H1(Ω) × RM, +∥(v, V )∥2 +b = b ((v, V ), (v, V )) . +Lemma 2.1. The bilinear form b defines a scalar product on H1, with corresponding norm ∥.∥b +equivalent to the norm ∥.∥H1. +As a consequence, (H1, b) is a Hilbert space. +Proof. This result is standard, but we recall its proof for the reader’s convenience. Firstly, we note +that if (u1, U1) ∼ (u2, U2) and (v1, V1) ∼ (v2, V2) in the sense of (2.1), then b ((u1, U1), (v1, V1)) = +b ((u2, U2), (v2, V2)), so the Lemma makes sense. +Clearly b is bilinear, symmetric and positive in H1 ×H1. Let (v, V ) ∈ H1 be such that ∥v, V ∥b = 0. +Then ∇v = 0 in Ω, and there exists c ∈ R such that v = c a.e. in Ω. Furthermore +� +em +(v − Vm)2ds(x) = 0, ∀m ∈ {1, . . . , M} , +we obtain that V = c [1, . . . , 1] , for the same constant c. Therefore, (v, V ) ∼ (0, 0), and b is a scalar +product on H1. +Using the continuity of the trace application from H1(Ω) to L2(∂Ω), it is readily seen that ∥.∥b ≲ +∥.∥H1. +Conversely, let (vn, Vn) be a sequence in H1 such that ∥vn, Vn∥H1 = 1 and ∥vn, Vn∥b goes to zero +as n goes to infinity. Necessarily, there exists a sequence (˜vn, ˜Vn) in H1(Ω) × RM such that +• (vn, Vn) ∼ (˜vn, ˜Vn) in the sense of (2.1), and therefore ∥˜vn, ˜Vn∥b goes to zero, +• 1 ≤ ∥˜vn∥2 +H1(Ω) + ∥ ˜Vn∥2 +RM ≤ 2, for all n ∈ N. + +4 +J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS +As (˜vn) is a bounded sequence in H1(Ω) and ( ˜Vn) is a bounded sequence in RM, there exists a +subsequence (still denoted (˜vn, ˜Vn)) and an element (˜v∞, ˜V∞) ∈ H1(Ω) × RM such that ˜vn weakly +converges to ˜v∞ in H1(Ω) and ˜Vn strongly converges to ˜V∞ in RM. As ∇˜vn strongly converges to 0, +˜vn actually strongly converges to ˜v∞ in H1(Ω), ∇˜v∞ = 0 and there exists a constant c∞ such that +˜v∞ = c∞ a.e. in Ω. +Now, as for all m ∈ {1, . . . , M}, +0 ≤ +� +em +(˜vn − ˜Vn,m)2 ds(x) ≤ ∥˜vn, ˜Vn∥2 +b, +taking the limit as n goes to infinity leads to ˜V∞,m = ˜v∞|em = c∞, or equivalently ˜V∞ = c∞[1, . . . , 1]. +But as (vn, Vn) ∼ (˜vn, ˜Vn), we have +1 = ∥vn, Vn∥H1 = ∥˜vn, ˜Vn∥H1 ≤ ∥˜vn − c∞∥H1(Ω) + ∥ ˜Vn − c∞[1, . . . , 1]∥RM , +leading to a contradiction. Therefore ∥.∥H1 ≲ ∥.∥b, and the result follows. +□ +2.2. Generalized Complete Electrode Model. We are now in position to introduce the generalized +Complete Electrode Model we will use in the following. This model differs slightly from the standard +Complete Electrode Model appearing in the context of Electrical Impedance Tomography, as we allow +interior and boundary source terms, and we also allow spatially varying surface admittivities on the +electrodes. Both features will be useful in the following study. +In our geometrical setting, Ω is the domain of interest, and each open set Em represents one of +the electrodes. We denote σ ∈ L∞(Ω) the conductivity of the domain, which is assumed to verify the +standard ellipticity condition +σ ≥ c > 0 a.e. in Ω. +In the standard Complete Electrode Model, the thin resistive layer appearing at each electrode-body +interface is represented by the vector of contact impedances z ∈ RM ++ : one contact impedance zm per +electrode Em. The corresponding contact admittivity on electrode Em is then constant, equal to +1 +zm . +In our generalized model, we allow spatially varying admittivities. To do so, we consider ξm ∈ Zm, +with +Zm = {ξ ∈ L∞(Em), ξ ≥ 0, ξ ̸≡ 0} . +These varying admittivities have been introduced in [28] as a way to obtain regularity on the solutions +of the Complete Electrode Model. The standard Complete Electrode Model corresponds to the choice +ξm = +1 +zm , which is indeed in Zm. +Let I ∈ RM represent the input currents imposed on the electrodes, and (f, g) ∈ L2(Ω) × L2(∂Ω). +The boundary value problem corresponding to the Complete Electrode Model reads: find (u, U) ∈ H1 +such that +(2.2) +� +� +� +� +� +� +� +� +� +� +� +−∇ · (σ∇u) = f +in Ω, +σ∇u · ν = g +on Ec, +σ∇u · ν + ξm(u − Um) = g +on Em, m = 1, . . . , M, +� +Em +σ∇u · ν ds(x) = Im, +m = 1, . . . , M. +Before proving that problem (2.2) is well-posed, we make several comments: +• A simple application of the divergence theorem shows that for problem (2.2) to admit a solution, +it is necessary that I, f and g satisfy the compatibility condition +(2.3) +M +� +m=1 +Im + +� +Ω +f dx + +� +Ec +g ds(x) = 0. +Of course, this is simply the well-known current conservation law applied to our system of +equations. In the case of practical Electrical Impedance Tomography, where the source terms + +IMMERSED BOUNDARY METHOD FOR CEM IN EIT +5 +f and g are null, we retrieve the mean-free condition on the input currents +M +� +m=1 +Im = 0 ⇔ I ∈ RM +⋄ . +• Suppose that (u, U) ∈ H1(Ω) × RM satisfies the system of equations (2.2). +Then for any +constant c ∈ R, (u + c, U + c[1, . . . , 1]) also satisfies (2.2). This is natural as from a physics +point of view, only potential differences can be measured. Therefore we work in the quotient +space H1 to identify all elements of H1(Ω) × RM that differ up to an additive constant. As +we shall see, this restores uniqueness. +• In the spirit of [28], the limit conditions (2.22) and (2.23) can be written at once: indeed, +denoting by 1Em the operator that maps a function ξ ∈ L∞(Em) to its extension by zero on +∂Ω, and defining the operator +Ξ : (ξ1, . . . , ξm) ∈ +M +� +m=1 +L∞(Em) �→ +M +� +m=1 +1Emξm ∈ L∞(∂Ω), +then (2.22) and (2.23) are equivalent to +σ∇u · ν + Ξ(ξ1, . . . , ξM)(u − Ξ(U1, . . . , UM)) = g on ∂Ω. +• In practical applications in Electrical Impedance Tomography, the source terms f and g are +both null, in which case the compatibility condition (2.3) becomes the usual condition I ∈ RM +⋄ . +Nevertheless, non-null source terms will be used in our study in the numerical tests, to ensure +smoothness on u and catch the convergence rates of our numerical scheme (see subsection +”Regularity results” below). +• In problem (2.2), the current configuration I is imposed, and the potential configuration U +is measured. From a mathematical point of view, it would be equivalent to impose U, and +measure I. To apply Immersed Boundary Method to our problem, it would be simpler to +impose U, as the limit conditions are then standard. Nevertheless, in practice, it is always the +current that is imposed and the potential that is measured, which breaks the symmetry as the +noise is greater on the measured data then on the imposed one. Therefore we focus our study +on problem (2.2). +Well-posedness. We now prove that problem (2.2) is well-posed. We follow the standard steps used in +the case f = 0 and g = 0, based on the variational formulation of problem (2.2), that we adapt to our +context. +Proposition 2.2. Suppose f, I and g satisfy (2.3). Then problem (2.2) admits an unique solution +(u, U) ∈ H1. +Proof. Consider the variational problem: find (u, U) ∈ H1 such that, for all (v, V ) in H1, +(2.4) +B +� +(u, U), (v, V ) +� += L +� +(v, V ) +� +, +with +B +� +(u, U), (v, V ) +� += +� +Ω +σ∇u · ∇v dx + +M +� +m=1 +� +Em +ξm(u − Um)(v − Vm)ds(x), +and +L +� +(v, V ) +� += +� +Ω +f v dx + +M +� +m=1 +� +Em +g (v − Vm) ds(x) + +� +Ec +g v ds(x) + I · V. +It is clear that B is bilinear and continuous on H1 × H1. By definition of Zm, there exists c > 0 such +that for all m in {1, . . . , M}, there exists em ⊂ Em of positive Lebesgue measure such that ξm ≥ c on +em. This property combined with the assumption on σ and Lemma 2.1 easily implies the coercivity + +6 +J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS +of B. On the other hand, as for all (v, V ) ∈ H1(Ω) × RM and all constant c, we have due to the +compatibility condition (2.3) +L ((v, V )) = L ((v − c, V − c[1, . . . , 1])) , +L is a continuous linear form on H1. Therefore, Lax-Migram theorem [6, Corollary 5.8] implies the +existence and uniqueness of (u, U) ∈ H1 satisfying (2.4). +Remains to prove that (2.4) is equivalent to (2.2). The fact that (2.2) implies (2.4) is standard, +so we focus on the reverse implication, and prove that (u, U) ∈ H1 solution of (2.4) satisfies (2.2). +Choosing (v, V ) ∼ (ϕ, 0RM ) in (2.4), with ϕ ∈ C∞ +c (Ω), immediately implies that −∇ · (σ∇u) = f in +Ω. Choosing then (v, V ) ∼ (v0, 0RM ) with v0 ∈ H1(Ω), we obtain +� +Ω +(∇ · σ∇u v0 + σ∇u · ∇v0) dx = +M +� +m=1 +� +Em +(g − ξm(u − Um)) v0 ds(x) + +� +Ec +g v0 ds(x), +which implies by surjectivity of the trace operator v0 ∈ H1(Ω) �→ v0|∂Ω ∈ H1/2(∂Ω) and the definition +of the conormal derivative σ∇u · ν that u satisfies (2.2)2 and (2.2)3. In particular σ∇u · ν belongs to +L2(∂Ω). +Finally, choosing (v, V ) ∼ (0, V0) with V0 ∈ RM yields +I · V0 = +M +� +m=1 +� +Em +(g − ξm(u − Um)) V0,m ds(x) = +M +� +m=1 +� +Em +σ∇u · ν V0,m ds(x) = W · V0, +with W ∈ RM defined by Wm = +� +Em σ∇u · ν ds(x), which ends the proof. +□ +Regularity results. The convergence results we obtain for the numerical scheme we develop in the next +section require smooth solutions to our problem of interest. Unfortunately, in lots of situations, and +in particular in the relevant ones for application, the solutions of (2.2) fail to be smooth. +Indeed, suppose that σ is smooth. Due to the mixed boundary conditions (2.2)2, (2.2)3 and (2.2)4, +the potential u satisfying system (2.2) might not be as smooth as in standard elliptic problem. +More precisely, for the standard Complete Electrode Model in which ξm is a positive constant, and +in practical settings where the source terms f and g are null, it is known that (u, U) solution of (2.2) +belongs to H2−ε for all ε > 0, but fails to be in H2 for all input currents I except for the null one [17, +28]. +The smoothened Complete Electrode Model has been introduced in [28] precisely to overcome this +difficulty. It consists in replacing ξm constant by ξm ∈ Zm, compactly supported in Em and smooth. +Then standard regularity results apply, leading to a smooth potential u. As in practice the surface +admittivity is unknown, and there is no clear evidence supporting the idea that it is constant, such +change of model is reasonable. Obviously, its main drawback is more complex parametrization of the +contact admittivities. Our convergence results always apply to the smoothened Complete Electrode +Model. +Nevertheless, regardless of the choice of ξm, there is no explicit solution for problem (2.2) when +f = g = 0, except in very particular geometric configurations. This is why we introduce the source +terms f and g: they allow us to construct explicit solutions to problem (2.2) to numerically test our +convergence results. +2.3. An alternative formulation well-posed in H1(Ω) × Rd. As seen previously, problem (2.2) is +well posed in H1. In other words, the solution of (2.2) is defined up to an additive constant. From +a numerical point of view, one has to fix that constant (or in other words, choose the ground level), +which is usually done by adding a constraint on U, such that U1 = 0 or U ∈ RM +⋄ : the solution is then +searched in H1(Ω) × ˜R, with ˜R = {U ∈ Rm, U1 = 0} or ˜R = RM +⋄ , restoring uniqueness. Theoretically, +the constraint can be imposed on the potential u (as an example, one could impose u to be mean free +over Ω), but this adds computational issues when solving numerically the problem. + +IMMERSED BOUNDARY METHOD FOR CEM IN EIT +7 +In this study, we propose another approach, consisting in modifying problem (2.2) in a new problem, +which will be well-posed in H1(Ω)×Rd, and which solution is one of the solutions to problem (2.2). Note +that by construction no additional constraint is imposed on the new problem to obtain well-posedness. +Setting U1 = 0. Let ε > 0 be fixed. We consider the modified problem +(2.5) +� +� +� +� +� +� +� +� +� +� +� +−∇ · (σ∇u) = f +in Ω +σ∇u · ν = g +on Ec +σ∇u · ν + ξm(u − Um) = g +on Em, m = 1, . . . , M, +� +Em +σ∇u · ν ds(x) + εδm1Um = Im, +m = 1, . . . , M. +The boundary equation (2.5)4 is equivalent to +� +E1 +σ∇u · ν ds(x) + εU1 = I1, +� +Em +σ∇u · ν ds(x) = Im, ∀m ∈ {2, . . . , M} . +In other words, only one equation of the initial problem (2.2) is modified. +We claim the following +Proposition 2.3. Problem (2.5) admits an unique solution (u, U) ∈ H1(Ω) × Rd. Furthermore, if I, +f and g satisfy the compatibility condition (2.3), U1 = 0 and (u, U), seen as an element of H1, is the +solution of (2.2). +Proof. We endow H1(Ω) × RM with its standard scalar product, with corresponding norm +∥(u, U)∥2 +H1(Ω)×RM = ∥u∥2 +H1(Ω) + ∥U∥2 +RM . +First, adapting the proof of Proposition 2.2, it is readily seen that problem (2.5) is equivalent to the +variationnal formulation +(2.6) +B1 +� +(u, U), (v, V ) +� += L +� +(v, V ) +� +, +∀(v, V ) ∈ H1(Ω) × RM. +with +B1 +� +(u, U), (v, V ) +� += +� +Ω +σ∇u · ∇v dx + +M +� +m=1 +� +Em +ξm(u − Um)(v − Vm)ds(x) + ε U1 V1 +and +L +� +(v, V ) +� += +� +Ω +f v dx + +M +� +m=1 +� +Em +g (v − Vm) ds(x) + +� +Ec +g v ds(x) + I · V. +Similarly, a minor adaptation of the proof of Lemma 2.1 shows that +∥(u, U)∥2 +b1 = +� +Ω +|∇u|2dx + +M +� +m=1 +|u − Um|2ds(x) + U 2 +1 , +is a norm on H1(Ω) × RM equivalent to the standard one. The assumptions on σ and ξm immediately +imply the coerciveness of the continuous bilinear form B1, while the linear form L is clearly continuous +on H1(Ω)×RM. Therefore, Lax-Milgram theorem implies the existence and uniqueness of the solution +of (2.6). +We now suppose that I, f and g satisfy (2.3). Choosing v = 1 and V = [1, . . . , 1] leads to +ε U1 = +� +Ω +f dx + +� +Ec +g ds(x) + +M +� +m=1 +Im = 0. +As a consequence, U1 = 0, which immediately implies that (u, U) satisfies the system of equation (2.2), +which ends the proof. +□ + +8 +J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS +We see that solving Problem (2.5), which is a very slight variation of our initial problem (2.2), allows +to retrieve the unique solution of (2.2) satisfying the additional condition U1 = 0, without strongly +enforcing this condition. Note that this holds for any value of the positive parameter ε, as long as I, +f and g are compatible in the sense of (2.3). +As it allows to solve (2.2) without having to fix the ground level, from this point we will always +focus on problem (2.5). +Remark 2.4. The focus on the first electrode is completely arbitrary here. A simple renumbering of +the electrodes allows to use the same formulation to enforce the zero condition on any electrode. +Remark 2.5. Problem (2.5) is well-posed regardless of the compatibility of the data I, f and g. If the +data are not compatible, then (u, U) solution of (2.5) cannot solve (2.2) as the latter has no solution. +As seen in the proof of Proposition 2.3, if (u, U) is the solution of (2.2), we always have +εU1 = +� +Ω +f dx + +� +Ec +g ds(x) + +M +� +m=1 +Im. +(2.7) +As a consequence, if we set ˜I ∈ RM defined by +˜I1 = I1 − +�� +Ω +f dx + +� +Ec +g ds(x) + +M +� +m=1 +Im +� +, +˜Im = Im, m ̸= 1, +we see that (u, U) satisfies (2.2) with the original source terms (I, f, g) replaced by (˜I, f, g). This second +set of data obviously satisfies the compatibility condition (2.3). In a sense, problem (2.5) correct the +data so that they always satisfy the compatibility condition. +In the case of compatible data, the value of parameter ε has no influence on the solution of problem +(2.5), and can be freely set equal to any arbitrary value, for instance one. From a numerical point of +view, this is not the case. Indeed the numerical compatibility condition has no reason to be exactly +satisfied by the finite-difference formulation, but only up to the order of the approximation, even if the +continuous data satisfies (2.3). Therefore, due to equation (2.7), the value of ε influences the value of +U1. This causes an interplay between the two terms in the left-hand side of (2.5)4 that will have an +influence on the amplitude of the numerical error, but not on the convergence order itself. +Setting U ∈ RM +⋄ . As a final remark, we note that it is equally easy to select (u, U) solution of (2.2) +satisfying the additional mean free value condition on U, that is U ∈ RM +⋄ , without strongly enforcing +this condition (that is, with a problem posed in H1(Ω) × RM and not in H1(Ω) × RM +⋄ . +Indeed, it suffices to solve the following problem: +(2.8) +� +� +� +� +� +� +� +� +� +� +� +−∇ · (σ∇u) = f +in Ω +σ∇u · ν = g +on Ec +σ∇u · ν + ξm(u − Um) = g +on Em, m = 1, . . . , M, +� +Em +σ∇u · ν ds(x) + ε⟨U⟩ = Im, +m = 1, . . . , M. +Here we have set ⟨U⟩ = �M +k=1 Uk. The corresponding variational form is +B⟨⟩ +� +(u, U), (v, V ) +� += L +� +(v, V ) +� +, +∀(v, V ) ∈ H1(Ω) × RM. +with +B⟨⟩ +� +(u, U), (v, V ) +� += +� +Ω +σ∇u · ∇v dx + +M +� +m=1 +� +Em +ξm(u − Um)(v − Vm)ds(x) + ε ⟨U⟩ ⟨V ⟩, +the continuous linear form L being unchanged. +Following the same line of reasoning used in the +previous section, it is not difficult to see that (2.8) is well-posed in H1(Ω) × RM, and that if (I, f, g) +satisfies (2.3), (u, U) solution of (2.8) satisfies the original Complete Electrode Model equations (2.2), +with furthermore ⟨U⟩ = 0. + +IMMERSED BOUNDARY METHOD FOR CEM IN EIT +9 +Remark 2.6. From a numerical point of view, the additional term ⟨U⟩ seems not practical, as it is +non-local and therefore tends to fill the matrix corresponding to the discretization of (2.8). However, +it appears only on equation (2.8)4, and affects only the term U ∈ RM, M being the number of +electrodes on the boundary of Ω, and therefore typically less than one hundred. As the potential u is +usually discretized in spaces containing thousands of degrees of freedom, in practice it does not affect +significantly the matrix filling. +2.4. Towards the immersed boundary method. In our numerical scheme, the domain of interest +Ω is immersed into a larger domain Ωe, a square in dimension 2 and a cube in dimension 3, such that +Ω ⊂ Ωe. The domain Ωe is fully discretized with a Cartesian grid, implying that some discretization +points lie in Ωe\Ω, a part of the extended domain where there is no physical problem in our context. In +that situation, a first idea is to simply set the solution to zero on the degrees of freedom lying outside +of Ω. But this operation is cumbersome, and not appropriate in our approach, as we need smoothness +in the solution on either side of the interface ∂Ω to compute quadrature formulas on the electrodes. +We therefore choose a different approach, which consists in solving an additional partial differential +equation in Ωe \ Ω, similar to the problem posed in Ω. As a consequence, we introduce the following +additional problem +(2.9) +� +� +� +−∆ue = 0 +in Ωe \ Ω +ue = 0 +on ∂Ωe +ue = u +on ∂Ω, +the function u being the electric potential, solution of (2.5). Clearly, such problem is well-posed in +H1(Ω), and its solution depends continuously on u, and therefore on the EIT source terms (I, f, g). +Furthermore, it will automatically inherit the smoothness of u thanks to standard elliptic regularity +results. +As a conclusion, our goal is now to define an immersed boundary method to reconstruct (u, U, ue), +with (u, U) solution of (2.5) and ue solution of (2.9). +3. An immersed boundary scheme for CEM +In this section we focus on the two-dimensional case, and present the discretization of problems +(2.5) and (2.9) with an immersed boundary method on a Cartesian grid. The domain Ω is immersed +into a larger square domain Ωe, and the boundary conditions of (2.5) are taken into account by using +additional variables located on the boundary of Ω. We prove that the method converges with first +order accuracy. The convergence proof is based on a discrete maximum principle, used to provide +estimates of the coefficients of the inverse of the discretization matrix. +3.1. Classification of grid points. We consider a uniform Cartesian grid defined on the square Ωe. +The grid spacing is denoted h. Each node of this cartesian grid is called a grid point and is denoted +Mij = (xi, yj) = (i h, j h). We denote by uh +ij the approximation of the function u at the point (xi, yj). +The set of grid points located inside the domain Ωe is denoted Ωh. +To the purpose of the discretization of the CEM we need to define additional points on the boundary +∂Ω. We define the boundary point Ii+1/2,j = (xi+1/2,j, yj) as the intersection of the boundary ∂Ω and +the segment [MijMi+1j], if it exists. Similarly, the boundary point Ii,j+1/2 = (xi, yi,j+1/2) is defined +as the intersection of the boundary and the segment [MijMij+1]. At each boundary point we create +an additional unknown uh +i+1/2,j or uh +i,j+1/2. The set of boundary points is denoted δΩh. +We say that a grid point is irregular if one of its direct neighbors is a boundary point, see Figure 1. +On the contrary, grid points that are not irregular are called regular grid points. The set of irregular +grid nodes is denoted Ω∗ +h. The set of electrode values is denoted Eh. +The grid or boundary points can be denoted with letters such as P or Q rather than with indices +such as Mi,j if it is convenient. We also denote if more convenient xP and yP the coordinates of a +point P. +3.2. Numerical scheme. + +10 +J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS +◦ ◦ ◦ ◦ ◦ ◦ +◦ ◦ ◦ ◦ ◦ ◦ +◦ ◦ ◦ ◦ ◦ +◦ ◦ ◦ ◦ ◦ +◦ ◦ ◦ ◦ ◦ +◦ ◦ ◦ ◦ ◦ +◦ ◦ ◦ +◦ +◦ ◦ +◦ ◦ +◦ ◦ +◦ +◦ +◦ +◦ +◦ ◦ ◦ +◦ ◦ ◦ +◦ ◦ ◦ +◦ ◦ ◦ +◦ ◦ ◦ +◦ ◦ ◦ +◦ ◦ +◦ ◦ +◦ ◦ +◦ +◦ ◦ +◦ ◦ ◦ +◦ ◦ ◦ +◦ ◦ ◦ +• • • +• +• +• +• +• +• +• +• • +• +• +• +• +• +• +• +• +• +• +• +• • +• • +• • +• • +• +• +• +• +• +• +• +• +• +• • • +• • • +• +• +• +• +• +• +• +• +• +• • • • • +• • • • • +• +• +• • +• • +• +• +• +• +• +• +• +• +• +• +• +• • • +• +• +• +◦ +◦ +◦ +◦ ◦ ◦ +◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ +◦ ◦ ◦ ◦ ◦ +◦ ◦ ◦ +◦ ◦ ◦ ◦ ◦ +◦ ◦ ◦ +◦ ◦ ◦ +◦ ◦ +◦ ◦ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄⋄ +⋄ +⋄ +⋄ +⋄ +⋄ ⋄ +⋄ +Figure 1. Left: regular points described by circles ◦, irregular points (belonging to +Ω∗ +h) described by bullets •, right: boundary points (belonging to δΩh). +Elliptic operator. To discretize the elliptic operator (2.5)1 or (2.9)1 on each grid point, regular or not, +we use a five point stencil with the grid point Mi,j and its nearest neighbors, boundary or grid points, +in each direction. To make this explicit, we denote uh +S the value of the solution on the nearest point in +the south direction, with coordinates (xS, yS). Similarly, we define uh +N, uh +W and uh +E and the associated +coordinates (xN, yN), (xW , yW ) and (xE, yE). The discretization reads +− +� +∇.(σ∇u) +�h +i,j += +− +� +σi+1/2,j +uh +E − uh +ij +xE − xi +− σi−1/2,j +uh +ij − uh +W +xi − xW +� 1 +h +− +� +σi,j+1/2 +uh +N − uh +ij +yN − yj +− σi,j−1/2 +uh +ij − uh +S +yj − yS +� 1 +h. +with σi+1/2,j the value of σ at point (xi + xE +2 +, yj, ). The truncation error of this discretization is +second-order accurate on regular grid points, and zeroth-order on irregular grid points. +Flux boundary conditions. At each boundary point we discretize the boundary conditions (2.5)2 or +(2.5)3, depending if the boundary point belongs to an electrode or not. +For instance, on a boundary point Ii+1/2,j, the discretization reads +σ(∇u · ν)h +i+1/2,j + ξm(Ii+1/2,j) +� +uh +i+1/2,j − Um +� += 0 if Ii+1/2,j ∈ Em, +(3.1) +σ(∇u · ν)h +i+1/2,j = 0 otherwise. +(3.2) +with (∇u · ν)h +i+1/2,j denoting the discretized normal derivative. A similar discretization is applied at +each boundary point Ii,j+1/2. +The discretization of the normal derivative (∇u · ν)h +i+1/2,j depends of the local geometry of the +interface. As illustrated on Figure 2, the first intersection between the normal to the boundary and +the grid is located on a segment: either [Mi,j, Mi,j−1], or [Mi,j−1, Mi+1,j−1], or [Mi,j, Mi,j+1], or +[Mi,j+1, Mi+1,j+1]. The discrete normal derivative is computed as the normal derivative of the linear +interpolant of the numerical solution on the triangle composed of the boundary point Ii+1/2,j and the +aforementioned segment. +If we denote K this triangle, (x1, y1), (x2, y2) and (x3, y3) its vertices, and u1, u2 and u3 the +associated values, the basis functions on the vertices for the linear interpolation write +λj(x, y) = αjx + βjy + γj, j = 1, 2, 3, + +IMMERSED BOUNDARY METHOD FOR CEM IN EIT +11 +with +αj += +yk − yi +(xj − xk)(yj − yi) − (xj − xi)(yi − yk), +βj += +xi − xk +(xj − xk)(yj − yi) − (xj − xi)(yi − yk), +γj += +xkyi − xiyk +(xj − xk)(yj − yi) − (xj − xi)(yi − yk), +(νx, νy) being an approximation of the normal at the interface point. +With these notations, the +approximation of the normal derivative writes for instance for the interface point Ii+1/2,j +(∇u · ν)h +i+1/2,j = (u1 α1 + u2 α2 + u3 α3)νx + (u1 β1 + u2 β2 + u3 β3)νy. +This discretization is first-order accurate because it is based on a linear interpolation. +I +I+1 +J +J+1 +J-1 +I-1 +Figure 2. All possible stencils for the first-order flux discretization on the left side +of the interface, with points involved in the discretization signaled by black circles. +Integral boundary conditions. We now discretize (2.5)4. More precisely, using (2.5)3, it is equivalent +to: +� +Em +� +g + ξm(Um − u) +� +ds(x) + ε δm1 U1 = Im, m = 1, . . . , M. +(3.3) +From now on, we focus on the equivalent boundary condition (3.3). +We discretize this integral boundary condition (3.3) on each electrode with a first-order quadra- +ture formula based on a first-order discrete dirac function [47]. This discrete Dirac function is by +construction positive and non-zero only on the irregular grid points, therefore it can be written as +� +P ∈Ω∗ +h +ωP +� +g(P) + ξm(P)(Um − uP ) +� ++ ε δm1 U1 = Im, m = 1, . . . , M. +(3.4) + ++1+1+112 +J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS +with the coefficients ωP that are the weights of the first-order quadrature formula in [47]. The trun- +cature error of this formula is therefore first-order. +3.3. Monotonicity of the discretization matrix. Here we aim to prove that the discretization +matrix of the linear system described before, that we denote Ah, is monotone. Let us first prove the +following property which will be useful in the reasoning: +Proposition 3.1. With the convention used for the normal to the boundary, if the minimum of v is +located on a boundary point, then at this boundary point the discrete normal derivative is negative. +Moreover, if the approximated normal derivative at this point is zero, then the three points values +involved in the stencil are equal. +Proof. The approximation of the normal derivative is constant in space, because it is computed from +a linear interpolation on a triangle. Therefore, if the minimum of v is located on a boundary point, +then at this boundary point the discrete normal derivative is negative. +□ +Proposition 3.2. The discretization matrix of the linear system described before, that we denote Ah, +is monotone, that is, Ah is invertible and all values of A−1 +h +are non-negative. +Proof. Let v be an array such that all coefficients of Ahv are non-negative, which we denote Ahv ≥ 0. +We aim to prove that all coefficients of v are non-negative. To this purpose we consider the minimum +of v in the whole domain: it can be either a grid point value, or a boundary point value, or an electrode +potential. We detail all these cases hereafter. +• If the minimum is reached on a grid point: +In this case we denote (i0, j0) the indices of the smallest component of v. We assume the +grid point is a regular grid point. Otherwise the formula would have slightly different weights, +but the reasoning would be the same. Using the elliptic operator inequality on this point we +can write: +4vi0,j0 − vi0+1,j0 − vi0−1,j0 − vi0,j0+1 − vi0,j0−1 ≥ 0, +and we deduce that +vi0+1,j0 = vi0−1,j0 = vi0,j0+1 = vi0,j0−1 = vi0,j0. +Repeating recursively this reasoning on the neighbours of (i0, j0), then on the neighbours of +the neighbours etc, we deduce that all values of v corresponding to grid or boundary points are +equal to vi0,j0. Therefore this case amounts to the case of the minimum located on a boundary +point. +• If the minimum is located on a boundary point not belonging to an electrode: +Without loss of generality, we assume that the minimum is located on Ii+1/2,j. On this +interface point we have both relationships +(σ∇v · n)h +i+1/2,j +≤ 0 because the minimum is located on this point, +(σ∇v · n)h +i+1/2,j +≥ 0 because all coefficients of Ahv are non-negative, +which leads to +(σ∇v · n)h +i+1/2,j = 0 +and the values of the grid points involved in the discretization of the normal derivative are +also equal to the minimum. Using the reasoning of the previsous subsection, it means that all +grid point and boundary point values are equal to this minimum. Therefore this case finally +amounts to considering the case of the minimum on a boundary point belonging to an electrode. + +IMMERSED BOUNDARY METHOD FOR CEM IN EIT +13 +• If the minimum is located on a boundary point belonging to an electrode: +Without loss of generality, we assume that the minimum is located on Ii+1/2,j. On this +interface point we have the relationships +(σ∇v · n)h +i+1/2,j +≤ 0, +(σ∇v · n)h +i+1/2,j + ξm(Ii+1/2,j) +� +vh +i+1/2,j − Vm +� +≥ 0, +the first one because the minimum is located on this point, and the second one because all +coefficients of Ahv are non-negative. Therefore +vh +i+1/2,j − Vm +≥ 0, +which means that +Vm += vh +i+1/2,j, +so the electrode potential Vm is also the minimum of v. Therefore this case amounts to con- +sidering that the minimum of v corresponds to an electrode potential Vm. +• If the minimum corresponds to an electrode potential Vm with m ̸= 1: +On this electrode we have the relationship: +� +P ∈Ω∗ +h +ωP ξm(P) +� +Vm − vP +� +≥ 0, +therefore all boundary points belonging to Em are equal to the minimum Vm. Using the pre- +vious reasoning it means that all values of v corresponding to grid points, boundary points or +electrode potentials are equal to the minimum. +• If the minimum corresponds to the electrode potential V1: +On this electrode we have the relationship: +� +P ∈Ω∗ +h +ωP ξ1(P) +� +V1 − vP +� ++ ε V1 ≥ 0, +which means that +ε V1 ≥ 0. +The minimum value of v is non-negative, therefore all coefficients of v are non-negative. +Therefore we have proven that if Ahv is non-negative, then v is also non-negative. This property +has two implications: Ah is invertible, and all values of A−1 +h +are non-negative. +□ +3.4. Discrete Green functions. In the following, the letters P and Q represent indices in the linear +system, representing either discretization points (on the grid or on the boundary) or electrode values. +For instance, we denote u(P) the coefficient of the row of u with the same index than the point P. +Similarly, AhU(P) represents the coefficient of the P-th row of the array AhU, and Ah(P, Q) is the +coefficient of the P-th row and Q-th column of the matrix Ah. We also define by Ah(:, Q) and Ah(P, :) +the Q-th column and the P-th row of the matrix Ah. +In the spirit of [12, 51] for each Q ∈ Ωh ∪ δΩh ∪ Eh, let us define the discrete Green’s function +Gh(:, Q) = +� +Gh(P, Q) +� +P ∈Ωh∪δΩh∪Eh +as the solution of the discrete problem: +(3.5) +� +AhGh(:, Q)(P) = +� 0, +P ̸= Q +1, +P = Q + +14 +J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS +The matrix Ah being monotone, all values of Gh(:, Q) are positive. In fact, the discrete Green functions +are columns of the inverse of Ah. We can write the solution of the discretized problem as a sum of the +source terms multiplied by the values of the discrete Green functions: +uh(P) = +� +Q∈Ωh∪δΩh∪Eh +Gh(P, Q) (Ahuh)(Q), +∀P ∈ Ωh ∪ δΩh ∪ Eh. +We can obtain bounds on the discrete Green functions with the following result: +Proposition 3.3. Let S and ˜S be two subsets of points , W a discrete function, α > 0, β > 0 and i, +j ∈ N such that: +� +� +� +(AhW)(P) ≥ 0, +∀P ∈ Ωh ∪ δΩh ∪ Eh \ ˜S, +(AhW)(P) ≥ α−i, +∀P ∈ S, +(AhW)(P) ≥ −(β−j), +∀P ∈ ˜S. +Then +� +Q∈S +Gh(P, Q) ≤ αiW(P) + αiβ−j � +Q∈ ˜S +Gh(P, Q), +∀P ∈ Ωh ∪ δΩh ∪ Eh. +Proof. Using the definition of the discrete Green functions, we can write +AhW(P) ≥ Ah +� +α−i � +Q∈S +Gh(:, Q) − β−j � +Q∈ ˜S +Gh(:, Q) +� +(P), +∀P ∈ Ωh ∪ δΩh ∪ Eh. +As all coefficients of A−1 +h +are non-negative, it leads to +W(P) − α−i � +Q∈S +Gh(P, Q) + β−j � +Q∈ ˜S +Gh(P, Q) ≥ 0, +∀P ∈ Ωh ∪ δΩh ∪ Eh, +and finally we obtain the following bound: +� +Q∈S +Gh(P, Q) ≤ αiW(P) + αiβ−j � +Q∈ ˜S +Gh(P, Q), +∀P ∈ Ωh ∪ δΩh ∪ Eh. +□ +3.5. Estimates of discrete Green functions and convergence result. In this section, we obtain +upper bounds for the discrete Green functions corresponding to the matrix arising from the discretiza- +tion of (2.5)-(2.9), and deduce from them the convergence order of the numerical scheme. We make +the assumption that the functions ξm are all smooth enough so that the solution u has the required +regularity for the following analysis, in particular for the truncation error estimation presented in +Subsection 3.2 to be valid. +Proposition 3.4. For h small enough, the following upper bounds hold: +� +Q∈Ωh\Ω∗ +h +Gh(:, Q) ≤ O(1), +� +Q∈δΩh∪Eh +Gh(:, Q) ≤ O(1), +� +Q∈Ω∗ +h +Gh(:, Q) ≤ O(h). +Proof. We use several discrete functions in the context of proposition 3.3 in order to obtain bounds +for the different blocks of the inverse matrix A−1 +h . + +IMMERSED BOUNDARY METHOD FOR CEM IN EIT +15 +• We define the array W such that W ≡ 1. Therefore, all expressions of the linear system vanish +excepted the quadrature formula involving the electrode E1: +− +� +∇.(σ∇W) +�h +i,j = 0 +∀Mi,j ∈ Ωh, +(σ∇W · ν)h +i+1/2,j − ξm(Ii+1/2,j) +� +Wm − W h +i+1/2,j +� += 0 +∀ Ii+1/2,j ∈ δΩh, +(σ∇W · ν)h +i,j+1/2 − ξm(Ii,j+1/2) +� +Wm − W h +i,j+1/2 +� += 0, +∀ Ii,j+1/2 ∈ δΩh, +� +P ∈Em +ωp ξm(P) +� +Wm − ˜WP +� += 0 +∀ m ̸= 1, +� +P ∈Em +ωp ξm(P) +� +Wm − ˜WP +� ++ εW1 = ε. +Gh(:, E1) representing the discrete Green function associated with the electrode E1, we +obtain +Gh(:, E1) = 1 +ε. +(3.6) +• We consider the exact solution ¯u of (2.5)-(2.9), with f = 1, g = 1, Im = 1 +∀ m ̸= 1, and +I1 such that the compatibility condition (2.3) is satisfied. +We define the array ¯W as the +discretisation of ¯u on the grid points, boundary points and electrode values. +The discretization of the elliptic operator and the fluxes is consistent at least with first-order +accuracy, excepted for the grid points in Ω∗ +h where it is only zeroth-order accurate. Thus for +h small enough, we can write that +− +� +∇.(σ ¯W) +�h +i,j ≥ 1 +2, +∀Mi,j ∈ Ωh \ Ω∗ +h, +− +� +∇.(σ ¯W) +�h +i,j ≥ −C1, +∀Mi,j ∈ Ω∗ +h, +(σ∇ ¯W · ν)h +i+1/2,j − ξm(Ii+1/2,j) +� +¯Wm − ¯Wi+1/2,j +� +≥ 1 +2, +∀Ii+1/2,j ∈ δΩh, +(σ∇ ¯W · ν)h +i,j+1/2 − ξm(Ii,j+1/2) +� +¯Wm − ¯Wi,j+1/2 +� +≥ 1 +2, +∀Ii,j+1/2 ∈ δΩh +� +P ∈Em +ωp ξm(P) +� +¯Wm − ¯WP +� +≥ 1 +2, +∀ m ̸= 1 +� +P ∈E1 +ωp ξm(P) +� +¯W1 − ¯WP +� ++ ε ¯W1 ≥ −2|I1|, +with C1 a strictly positive constant. It can also be re-written as: +(Ah ¯W)(P) ≥ 1 +2, +∀P ∈ (Ωh \ Ω∗ +h) ∪ (Eh \ E1), +(Ah ¯W)(P) ≥ 1 +2, +∀P ∈ δΩh, +(Ah ¯W)(P) ≥ −C1, +∀P ∈ Ω∗ +h, +(Ah ¯W)(E1) ≥ −2|I1|, +and it leads to: +� +Q∈(Ωh\Ω∗ +h)∪(Eh\E1) +Gh(:, Q) + +� +Q∈δΩh +Gh(:, Q) ≤ 2 ¯W + 2C1 +� +Q∈Ω∗ +h +Gh(:, Q) + 4|I1|Gh(:, E1). +(3.7) + +16 +J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS +• We define the array ˜W such that ˜W = 1 for all points in Ωh, and ˜W = 0 for all points in +δΩh ∪ Eh. Because of the jump in the values of ˜W between Ωh and δΩh ∪ Eh, when we apply +the discretization to ˜W we obtain +− +� +∇.(σ∇ ˜W) +�h +i,j = 0 +∀Mi,j ∈ Ωh \ Ω∗ +h, +− +� +∇.(σ∇ ˜W) +�h +i,j ≥ C2 +h2 +∀ Mi,j ∈ Ω∗ +h, +(σ∇ ˜W · ν)h +i+1/2,j − ξm(Ii+1/2,j) +� +˜Wm − ˜Wi+1/2,j +� +≥ −C3 +h +∀ Ii+1/2,j ∈ δΩh, +(σ∇ ˜W · ν)h +i,j+1/2 − ξm(Ii,j+1/2) +� +˜Wm − ˜Wi,j+1/2 +� +≥ −C3 +h , +∀ Ii,j+1/2 ∈ δΩh +� +P ∈Em +ωp ξm(P) +� +˜Wm − ˜WP +� ++ εδm1 ˜W1 = 0 +∀ m, +with C2 and C3 trictly positive constants. We thus obtain +C2 +h2 +� +Q∈Ω∗ +h +Gh(:, Q) ≤ ˜W + C3 +h +� +Q∈δΩh +Gh(:, Q), +and then: +� +Q∈Ω∗ +h +Gh(:, Q) ≤ h2 +C2 +˜W + hC3 +C2 +� +Q∈δΩh +Gh(:, Q). +(3.8) +Combining 3.8 with 3.7 we obtain +� +Q∈(Ωh\Ω∗ +h)∪(Eh\E1) +Gh(:, Q) + +� +Q∈δΩh +Gh(:, Q) ≤ 2 ¯W + 2C1 +h2 +C2 +˜W + 4|I1|Gh(:, E1) +(3.9) ++ 2C1 +C3 +C2 +h +� +Q∈δΩh +Gh(:, Q), +(3.10) +and for h small enough it leads to +� +Q∈(Ωh\Ω∗ +h)∪(Eh\E1) +Gh(:, Q) + +� +Q∈δΩh +Gh(:, Q) ≤ ¯C +(3.11) +with ¯C a positive constant. Re-injecting this inequality into 3.8 we obtain +� +Q∈Ω∗ +h +Gh(:, Q) ≤ h2 +C2 +˜W + hC3 +C2 +¯C. +(3.12) +The inequalities 3.6, 3.11 and 3.12 provide the bounds of the proposition. +□ +Proposition 3.5. If we denote ¯u the exact solution of (2.5)-(2.9) and uh the numerical solution, the +local error |¯u(P) − uh(P)| satisfies for h small enough: +|¯u(P) − uh(P)| ≤ O(h). +(3.13) +Therefore the numerical solution converges with first-order accuracy to the exact solution in L∞-norm. + +IMMERSED BOUNDARY METHOD FOR CEM IN EIT +17 +Proof. We denote τ the truncation error array arising from the discretization of (2.5)-(2.9). We have +|¯u(P) − uh(P)| = +������ +� +Q∈Ωh∪δΩh∪Eh +Gh(P, Q)τ(Q) +������ +≤ +������ +� +Q∈Ω∗ +h +Gh(P, Q)τ(Q) +������ ++ +������ +� +Q∈Ωh\Ω∗ +h +Gh(P, Q)τ(Q) +������ ++ +������ +� +Q∈δΩh∪Eh +Gh(P, Q)τ(Q) +������ +, +which, according to the previous results, immediately implies +|¯u(P) − uh(P)| ≤ O(1) +������ +� +Q∈Ω∗ +h +Gh(P, Q) +������ ++ O(h2) +������ +� +Q∈Ωh\Ω∗ +h +Gh(P, Q) +������ ++ O(h) +������ +� +Q∈δΩh∪Eh +Gh(P, Q) +������ +≤ O(1)O(h) + O(h2)O(1) + O(h)O(1) = O(h). +□ +4. Numerical experiments +We now study the efficiency of the immersed boundary method for Electrical Impedance Tomogra- +phy introduced in Section 3.2. More precisely, we test the method on three different two-dimensional +problems: +• the direct problem of EIT, which in our context reads: for given source terms f, g and I, +compute (u, U) solution of (2.5) (in the sense of Proposition 2.3). In that case, the geometry +(i.e. Ω and the electrodes’ position) and the conductivity are supposed to be known param- +eters. This is a well-posed problem, and our numerical method is designed to obtain a good +reconstruction of its solution without accurate discretization of the geometry. +• the inverse problem of EIT: for given input currents I ∈ RM +⋄ , and corresponding measured +voltages U, reconstruct the interior conductivity σ. In other word, knowing the couple (I, U), +we search for σ such that (u, U) = (u(σ), U(σ)) solves (2.5) with source terms f = 0, g = 0 and +I. Here again, the geometry is supposed to be known. This of course is the central problem +in Electrical Impedance Tomography, and therefore is the main validation of our method. +• a geometric inverse problem in EIT: for given input currents I ∈ RM +⋄ , and corresponding mea- +sured voltages U, reconstruct the positions of the electrodes knowing Ω and the conductivity σ. +Obviously, this problem does not make much sense from a practical point of view. But on one +hand it can be seen at a first step to tackle the real problem of imperfectly known geometrical +setting in EIT, which is known to severely deteriorate the quality of most conductivity’s recon- +struction algorithms (at the exception of the ones that are specifically designed to cope with +this problem, see [2, 15, 33, 34] and the references therein). On the second hand, it is a prob- +lem of interest in our context, as the electrodes are not exactly approximated in the immersed +boundary method we proposed. Therefore, knowing the high-sensibility of EIT-measurements +to electrodes’ position, it is a challenging problem. +4.1. Representation of the geometry. In the following experiments, the domain Ω is represented by +a polar parametrization of its boundary ∂Ω. More precisely, for a given vector α = [α0, α1, . . . , α2N] ∈ +R2N+1, we set +∂Ω = {r(θ)u(θ), θ ∈ [0, 2π]} , +with u(θ) = [cos(θ), sin(θ)], and +r(θ) = α0 + +N +� +k=1 +(αk cos(kθ) + αk+N sin(kθ)) . + +18 +J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS +The shape parameters [α0, . . . , α2N] are chosen so that 0 < r(θ) < 2 for all θ ∈ [0, 2π]. Finally, we set +Ω = +� +ρ[cos(θ), sin(θ)] ∈ R2, θ ∈ [0, 2π], 0 ≤ ρ < r(θ0) +� +. +Note that, with such parameterization of Ω, the outward unit vector at a point x = ∥x∥[cos(θx), sin(θx)] +∈ ∂Ω, denoted ν(x), is given by +ν(x) = +1 +ρ(θx) [r(θx)u(θx) − r′(θx)v(θx)] , +with v(θ) = [− sin(θ), cos(θ)] and ρ(θ) = +� +r(θ)2 + r′(θ)2. We also set +τ(x) = +1 +ρ(θx) [r′(θx)u(θx) + r(θx)v(θx)] , +the vector tangential to ∂Ω at the point x, such that (ν(x), τ(x)) is a direct basis of R2. +In the following tests, we use three different geometries for Ω: Ω1 is a disk of radius 1.5, corresponding +to α = [1.5], Ω2 corresponds to the choice α = [1.51, 0.01, 0.05, 0.2, 0.035, 0.01, 0.1], and Ω3 corresponds +to the choice α = [1.6, 0.002, 0.01, 0.003, 0.035, 0.2, 0.15]. +The M-electrodes E1, . . . , EM are parameterized by two vectors (Θ1, Θ2) ∈ RM × RM verifying +Θ1 +1 < Θ2 +1 < Θ2 +2 < . . . < Θ2 +M < Θ1 +1 + 2π, +such that +Em = +� +r(θ)u(θ), θ ∈ [Θ1 +m, Θ2 +m] +� +. +As in practical application the length of the electrodes is prescribed, we usually set Θ1, choose a length +size L > 0, and compute Θ2 by solving the equation +(4.1) +L = +� Θ2 +k +Θ1 +k +ρ(θ) dθ. +Unless otherwise specified, in the experiments below we use 16 electrodes, defined by Θ1 +k = −π+(k−1) π +8 +and L = 0.35. +Figure 3. The domains Ω1, Ω2 and Ω3 (in red) in the reference square [−2, 2]×[−2, 2] +(in blue), with our usual configuration of 16 electrodes (in bold black) +Remark 4.1. In this article we use a specific polar parametrization of the interface, which allows us +to compute explicitly all normals to the interface. However, such parametrization implies that the +domain of interest is star-shaped, which might not be the case in certain applications. +In such situations, it is usually still possible to use a description of the interface with a level-set +function. Such a description amounts to represent implicitly the interface as the zero level set of a +function. The level set method was introduced by Osher and Sethian [44]. We refer the interested + +0.5 +-0.5 +0.5 +-1 +2 +0 +1 +1.5 +-0.5 +-10.5 +-0.5 +0.5 +-1 +0 +1 +1.5 +2 +-0.5 +-1 +-1.50.5 +1.5 +-1 +-0.5 +0.5 +2 +0 +T +-0.5 +-1 +-1.5IMMERSED BOUNDARY METHOD FOR CEM IN EIT +19 +reader to [43, 45, 46].Although any smooth enough function cancelling at the interface location can be +used, in practice the signed distance to the interface is very often used: +(4.2) +ϕ(x) = +� +� +� +dist∂Ω(x) +outside of the interface ∂Ω, +0 +on the interface, +−dist∂Ω(x) +inside of the interface ∂Ω. +We recall here some properties occurring in this case: +• As recalled in [14], the level-set, being a distance function, is 1-Lipshitz and almost everywhere +differentiable. Moreover, if φ is differentiable at a point x, then it satisfies the so-called Eikonal +equation at x: +∥∇ϕ(x)∥ = 1. +• The smoothness of the level-set is in fact strongly related to the smoothness of the interface: +as proved in [25, p.355], if the interface is C2, then there exists a real r0 > 0 such that the +level-set is C2 for all x, y such that |ϕ(x, y)| < r0. +• The outward normal vector of the isoline of ϕ passing on x, denoted ν(x), can be expressed +where ϕ is differentiable as +(4.3) +ν(x) = +∇ϕ(x) +∥∇ϕ(x)∥, +4.2. Direct problems. First of all, we investigate the ability of the proposed immersed boundary +method to efficiently compute solutions of the Electrical Impedance Tomography problem (2.5). We +highlight that the convergence results announced in Section 3.5 are indeed observed in practice for +solutions smooth enough, and present several examples of reconstruction of the inner potential. +4.2.1. Convergence of the method. To verify that the convergence rates proved in Section 3.5 are indeed +observed in practice, we proceed as follows: we set Ωe = [−2, 2] × [−2, 2], u(x, y) = sin(x y), U = 0R16, +and compute the corresponding source terms f, g and I in (2.2). In particular, to obtain the vector +of input currents I, we compute the corresponding integrals appearing in (2.2)4 using a numerical +integration scheme. +We then use the immersed boundary method described in Section 3.2 to obtain an approximate +solution uh to problem (2.2) for different values of the mesh parameter h. To test also the convergence +of the derivatives, we compute an approximation of the gradient duh by numerical differentiation. +In the following, we compute uh and duh for various values of h ranging from +1 +25 to +1 +225. +The +parameter ε in (2.5)4 is chosen small, of order ∼ 10−10. The results are displayed in Figure 4: we +observe a linear convergence both for u and its gradient. +Remark 4.2. The linear convergence of uh to u was expected. The linear convergence of the gradient +is not surprising, as the construction of the numerical method is based on a first-order discretization +of fluxes near the boundary of the domain. Nonetheless, to prove it is not obvious and non-standard +within the finite difference framework. We left the study of the convergence of the gradient for future +works. +Remark 4.3. We have presented here numerical results for a given value of the parameter ε. +Let +us remark that changing the value of the parameter ϵ does not change the convergence behaviour. +However, the value of ε affects the amplitude of the error at fixed h. This is expected since the value +of U1 depends on the numerical compatibility condition, and therefore is not exactly zero. Therefore, +there is an interplay between the therm εU1 and the integral term in the formula (2.5)4 that influences +the value of both terms. +On Figure 5, we present a comparison of numerical errors for the generalized CEM, with smooth +spatially varying admittivities, and for the classical CEM with ξm = 1 +zm +constant. +The geometry +considered is a circle of radius R = 0.5 embeded in a square [−1, 1] × [−1, 1], with 4 electrodes equally +spaced.The discretization parameter h ranges from +1 +50 to +1 +250. The convergence rate displayed on the + +20 +J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS +Figure is computed with the classical CEM. We observe that the amplitude of the numerical errors for +both models as well as their rate of convergence is very close. +Finally, we present some reconstructions of the inner potential, with Ωe = [−2, 2]×[−2, 2] discretized +with a 700 × 700 Cartesian grid, and in different geometries, with different background conductivities +and various patterns applied to the electrodes, see Figures 6 and 7. +10-2 +10-3 +10-2 +10-1 +Errors for u +slope 1.02373 +slope 1 +(a) ∥uh − u∥∞ as a function of h. Ω = +Ω1. +10-2 +10-3 +10-2 +10-1 +Errors for the gradient of u +slope 0.89771 +slope 1 +(b) ∥duh,1 − ∂xu∥∞ as a function of h. +Ω = Ω1. +10-2 +10-3 +10-2 +10-1 +Errors for u +slope 0.982082 +slope 1 +(c) ∥uh − u∥∞ as a function of h. Ω = +Ω2. +10-2 +10-3 +10-2 +10-1 +Errors for the gradient of u +slope 0.84517 +slope 1 +(d) ∥duh,1 − ∂xu∥∞ as a function of h. +Ω = Ω2. +10-2 +10-3 +10-2 +10-1 +Errors for u +slope 0.981985 +slope 1 +(e) ∥uh − u∥∞ as a function of h. Ω = +Ω3. +10-2 +10-3 +10-2 +10-1 +Errors for the gradient of u +slope 0.88794 +slope 1 +(f) ∥duh,1 − ∂xu∥∞ as a function of h. +Ω = Ω3. +Figure 4. Convergence study with manufactured solution u(x, y) = sin(x y) and +U = 0R16 in each domain. + +IMMERSED BOUNDARY METHOD FOR CEM IN EIT +21 +(a) ∥uh − u∥∞ as a function of h. +(b) ∥duh,1 − ∂xu∥∞ as a function of h. +Figure 5. Convergence study with manufactured solution u(x, y) = exp(x2+y2) and +U = 0.5 exp(R2)1R4. Blue points stand for the numerical errors of classical CEM, red +crosses for the numerical errors of generalized CEM +Figure 6. Resolution of Problem (2.5) in domain Ω2 with input current Im = (−1)m. +Left: inner potential. Right: conductivity. +4.3. Inverse problems. From now on, we consider the standard Complete Electrode Model. +In +other words, in Problems (2.2) and (2.5), we set ξm = +1 +zm , zm being the constant and positive contact +impedance on each electrodes. + +X10~3 +6.5 +5.5 +5 +4.5 +3.5 +3 +2.5 +2 +1.5 +Errorsfor u (classical) +Errors for u (generalized) +slope 0.89 +slope 1 +0.004 +0.006 +0.008 +0.01 +0.012 +0.014 +0.0160.0180.020.1 +0.09 +0.08 +0.07 +0.06 +0.05 +0.04 +0.03 +Errors for ,u (classical) +Errors for a,u (generalized) +slope 0.88 +slope 1 +0.02 +0.004 +0.006 +0.008 +0.01 +0.012 +0.014 +0.0160.0180.026.1e-01 +0.55 +0.5 +0.45 +0.4 +0.35 +0.3 +0.25 +0.2 +0.15 +0.1 +0.05 +3 +-0.05 + -0.1 +-0.15 +-0.2 +-0.25 +-0.3 +-0.35 +-0.4 +-0.45 +-0.5 +-0.55 +-6.2e-012.0e+00 +1.9 +1.8 +1.7 +1.6 +1.5 +1.4 +1.3 +1.2 +Sigma(Sm-1) +1.1 +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0.0e+005.2e-01 +0.45 +0.4 +0.35 +0.3 +- 0.25 +0.2 +0.15 +- 0.1 +0.05 +-0.05 +-0.1 +-0.15 +-0.2 +-0.25 +-0.3 +-0.35 +-0.4 +-0.45 +-5.3e-012.0e+00 +1.9 +1.8 +1.7 +1.6 +1.5 +1.4 +1.3 +1.2 +1.1 +1 +0.9 + Sigma(Sm-1) +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0 +-0.1 +-0.2 +-0.3 +-0.4 +-0.5 +-6.2e-016.1e-01 +0.55 +0.5 +0.45 +0.4 +0.35 +0.3 +0.25 + 0.2 +- 0.15 + 0.1 + 0.05 +-0 +-0.05 +-0.1 +-0.15 +-0.2 +-0.25 +-0.3 +-0.35 +-0.4 +-0.45 +-0.5 +-0.55 +-6.1e-012.0e+00 +1.9 +1.8 +1.7 +1.6 +1.5 +1.4 +1.3 +1.2 +1.1 +1 +0.9 +Sigma(Sm-1) +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0 +-0.1 +-0.2 +-0.3 +-0.4 +-0.5 +-6.1e-0122 +J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS +Figure 7. Resolution of Problem (2.5) in domain Ω3 with input current Im = δm 8 − +δm 14. Left: inner potential. Right: conductivity. +4.3.1. Reconstruction of the conductivity, knowing the geometry. We now focus on the reconstruction +of the conductivity σ∗ in a practical Electrical Impedance Tomography context: for a family of input +currents (I1, . . . , IP ) ∈ RM +⋄ , P ∈ N∗, the corresponding (potentially noisy) electrodes’ potentials +U 1, . . . , U P are measured. They correspond to the second part of the solution of (2.2) with σ = σ∗ +and f = g = 0. To fix the constant, we set U p +1 = 0 for all p in {1, . . . , P}. As a consequence, the +potentials are also the second part of the solution of Problem (2.5) +To simplify notations, we gather all input currents and electrodes’ potentials corresponding to a +certain conductivity σ in M × P matrices +I = +� +I1 +. . . +IP � +, +U = +� +U 1, . . . , U P � +. +The measurement matrix U depends linearly on the input currents I and non-linearly on the unknown +conductivity σ. We will sometimes make this dependency explicit by writing U (σ, I ). We also note +Umeas ≈ U (σ∗, I ), +the matrix which would idealistically corresponds to our actual measurements, but in practice is only +close to it, as in applications measurements are always corrupted by noise. The inverse conductivity +problem can be formulated in the following concise way: knowing +� +I , Umeas +� +reconstruct σ∗. +Over time, sophisticated methods have been introduced to reconstruct either the conductivity or the +support of inclusions from (continuous or electrodes’ like) boundary measurements [7, 10, 20, 21, 26, 27, +29, 31, 35, 41, 50]. In our present study, we will use a comparatively very naive approach to reconstruct +σ∗, that is a basic least-square minimization approach combined with Tychonov regularization. We +choose this on purpose, as our primary objective is to test the ability of our direct method to give + +1.1e+00 +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0 + -0.1 +-0.2 +-0.3 + -0.4 + -0.5 + -0.6 +-0.7 + -0.8 + -0.9 +-1 +-1.1e+002.0e+00 +1.8 + 1.6 +1.4 +1.2 +-1 +0.8 +Sigma(Sm-1) +0.6 +0.4 +0.2 +0 + -0.2 + -0.4 +-0.6 + -0.8 +-1.1e+007.5e-01 +0.7 + 0.6 + 0.5 + 0.4 + 0.3 + 0.2 + 0.1 +- 0 + -0.1 + -0.2 + -0.3 +-0.4 +-0.5 +-0.6 +-6.9e-012.0e+00 +1.8 +1.6 +1.4 +1.2 +1 +0.8 +Sigma(Sm-1) +0.6 +0.4 +0.2 +0 +-0.2 +-0.4 +-0.6 +-0.8 +-1.1e+001.0e+00 +0.9 +0.8 +0.7 +0.6 +- 0.5 +- 0.4 + 0.3 +- 0.2 + 0.1 +- 0 +-0.1 +-0.2 +-0.3 +-0.4 +-0.5 +-0.6 +-0.7 +-0.8 +-0.9 +-1 +-1.1e+002.0e+00 +1.8 +1.6 +1.4 +1.2 +-1 +0.8 +Sigma(Sm-1) +0.6 +0.4 +0.2 +0 +-0.2 + -0.4 +-0.6 +-0.8 +-1.1e+00IMMERSED BOUNDARY METHOD FOR CEM IN EIT +23 +sufficiently good approximation of the solution to problem (2.5) to be efficiently used in an standard +algorithm for the inverse conductivity problem. +More precisely, we seek to minimize the discrepancy between the solution of the direct problem +(2.5), corresponding to a guessed conductivity σ, and the measured data, i.e., we seek σ such that the +functional +(4.4) +F(σ) = ∥U (σ, I) − Umeas∥2 +2 +is minimal. The problem being ill-posed, a regularization is needed. As a consequence, we actually +minimize the functional +(4.5) +F(σ) = ∥U (σ, I) − Umeas∥2 +RM + ϵ +2R(σ) +where R is a properly chosen regularisation functional and ϵ > 0 a regularisation parameter. In other +words, we use a deterministic Tychonov regularization of our problem. +From now on, we assume that σ∗ ∈ L∞ ∩H1(Ω). We also assume that there exists a known conduc- +tivity σ⋆ ∈ L∞ ∩ H1(Ω) such that σ∗ − σ⋆ ∈ H1 +0(Ω). This is typically the case in applications where +one searches perturbations of a known background conductivity σ⋆, assuming that the perturbations +do not touch the boundary of the domain (in that situation, the pertubations are σ∗ − σ⋆). It is then +natural to seek to minimize the following functional +(4.6) +F(σ) = 1 +2∥U (σ, I) − Umeas∥2 +RM + ϵ +2∥σ − σ⋆∥2 +H1(Ω). +The strategy of minimization is simple: for a conductivity σ and a current pattern I ∈ RM +⋄ , we +denote (u(σ, I), U(σ, I)) the solution of (2.5) with f = g = 0. Then, for a given conductivity σn, define +(dxun, dyun) by +dxun = +� +∂u(σn,I1) +∂x +, . . . , ∂u(σn,IP ) +∂x +� +∈ L2(Ω)d, +dyun = +� +∂u(σn,I1) +∂y +, . . . , ∂u(σn,IP ) +∂y +� +∈ L2(Ω)d. +Now, with Umeas = +� +U 1 +meas, . . . , U P +meas +� +, we define (dxwn, dywn) by +dxwn = +� +∂u(σn,U(σn,I1)−U 1 +meas) +∂x +, . . . , ∂u(σn,U(σn,IP )−U P +meas) +∂x +� +and +dywn = +� +∂u(σn,U(σn,I1)−U 1 +meas) +∂y +, . . . , ∂u(σn,U(σn,IP )−U P +meas) +∂y +� +. +Finally, we define δσn as the unique v ∈ H1 +0(Ω) such that +−∆v + v = ϵ +� +∆(σn − σ⋆) − (σn − σ⋆) +� ++ dxuT +ndxwn + dyuT +ndywn. +Then it is not difficult to prove the following result. +Lemma 4.4. Suppose that δσn belongs to L∞(Ω). There exists Tn > 0 such that for all t ∈ (0, Tn), +F(σn + tδσn) ≤ F(σn). +The proof of this point is standard, and is given in the Appendix. +Our algorithm of minimization is then the following: +• Choose σ0 ∈ L∞ ∩ H1(Ω) (for example, σ0 = σ⋆), +• Until the following stopping criteria is reached: +|F(σn + tnδσn) − F(σn)| < τ|δσn|, with τ small enough +(1) For a given conductivity σn, compute δσn as described above, +(2) Define σn+1 = σn + tnδσn with tn = arg mint∈(0,Tn) F(σn + tδσn). +• End. + +24 +J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS +Note that, at each step n of this algorithm, we have to solve P problems (2.5) to compute dxun and +dyun, then again P problems (2.5) to compute dwwn and dywn, and a single Laplace-type problem +to compute δσn, that is 2P + 1 elliptic problems. Luckily, the first 2P problems are exactly (2.5) with +a fixed given conductivity σn, and only the right-hand side changing, meaning that the corresponding +system to inverse is the same for the 2P problems. Hence the computation of δσn is actually very fast. +Our numerical experiments show that, for t close to 0, the function t �→ F(σn + tδσn) behave like a +quadratic function in t with a single local minimum. We therefore search for tn using a Golden-section +search. +We now present some conductivity’s reconstructions in the domains presented in Figure 3. In each +test, the background conductivity σ⋆ is equal to one, and +• first case: we place a circular inclusion in the center. +• second case : we place a similar inclusion closer to the boundary. +• third case : we place two inclusions of different amplitudes inside the domain. +The measurement matrix Umeas is obtained by solving (2.5) using our numerical method, in a very +refined mesh and for P = 15 linearly independent current patterns. The mesh used to compute Umeas +differs from the mesh used to reconstruct the conductivity in order to avoir inverse crime. To obtain +the noisy data, we then perturb the measurements with an additive Gaussian noise, scaled so that +the relative amplitude of the noise is set to δ ∈ [0, 1]. Finally, we set the regularization parameter to +ϵ = 10−4. +Figures 8 and 9 present reconstructions in the different geometrical domains with various levels of +added Gaussian noise, discretized by a 400 × 400 Cartesian grid. As can be seen, the reconstructions +are correct even with this very naive approach, meaning that our immersed boundary methods can be +efficiently used in the inverse conductivity problem. +Remark 4.5. In practical applications, the conductance ξm (or equivalently the contact impedances zm) +are unknown, and should be also reconstructed. This can be done without additional difficulties, using +that (u, U) solution of (2.5) is Frechet differentiable with respect to ξm, with an explicit expression of +the Frechet derivative. +4.3.2. Reconstruction of the electrodes’ positions, knowing the conductivity. We now focus on the geo- +metric inverse problem described in the introduction, which can be summarized as follows: from the +knowledge of (I , Umeas), find the locations of the electrodes, in other words the parameters Θ1 and +Θ2, all the other parameters (α, σ, ...) being known. +As already said, even if this problem makes little sense from a practical point of view, it is interesting +and challenging for our method as (2.2) is known to be very unstable with respect to the geometrical +setting, and our Boundary Immersed Method is, by design, not precise in terms of geometry of ∂Ω. +Adapting the notations of previous Section 4.3.1, we denote by U = U(Θ1, Θ2, I) the (second term +of the) solution of (2.5) with f = g = 0, I ∈ RM +⋄ , and electrodes’ positions determined by the angles +Θ1 and Θ2 (see Section 4.1 for the parametrization of the geometry). Now, for a set of input currents +(I1, . . . , IP ) ∈ (RM +⋄ )P , we denote +I = +� +I1, . . . , IP � +, +U = +� +U 1, . . . , U P � +. +We also set Umeas to be the following M × P possibly noisy measurements matrix, approximating the +exact measurement matrix corresponding to the exact electrodes positions Θ1 +∗ and Θ2 +∗: +Umeas ≈ U (Θ1 +∗, Θ2 +∗, I ). +Our strategy is similar as before: we seek to minimize the functional +F(Θ1, Θ2) = ∥U (Θ1, Θ2, I) − Umeas∥2 +2 + ϵ +2∥Θ1 − Θ1 +⋆∥2 +RM + ϵ +2∥Θ2 − Θ2 +⋆∥2 +RM. +As previously, Θk +⋆ has to be think as a good guess for the actual location of the k-th electrode, around +which we search the actual position of the electrode. + +IMMERSED BOUNDARY METHOD FOR CEM IN EIT +25 +Figure 8. Reconstructions in Ω1. +First column: searched conductivity. +Second +column: reconstructed conductivity with unperturbed data (δ = 0). Third column: +reconstructed conductivity with perturbed data (δ = 2.0%). +Figure 9. Reconstructions in Ω3. +First column: searched conductivity. +Second +column: reconstructed conductivity with unperturbed data (δ = 0). Third column: +reconstructed conductivity with perturbed data (δ = 2.0%). + +2.0e+00 +1.9 +1.8 +1.7 +1.6 +- 1.5 +1.4 +1.3 +1.2 +Sigma(Sm-1) +1.1 +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +1.5e-012.0e+00 +1.9 +1.8 +1.7 +- 1.6 + 1.5 +- 1.4 + 1.3 +1.2 +Sigma(Sm-1) +1.1 +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0.0e+0026 +J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS +The partial derivatives of the functional with respect to parameters Θ1 +k or Θ2 +k are easily derived +from the shape derivative obtained in [15], and in particular the sampling formula [15, Corollary 3.4]: +Lemma 4.6. Let (˜u, ˜U) be the solution of (2.5) for some electrode pattern ˜I ∈ RM +⋄ , and f = g = 0. +For k in {1, 2} and m in {1, . . . , M}, let xk +m = r(Θk +m)[cos(Θk +m), sin(Θk +m)]. +We have for m ∈ {1, . . . , M}: +(4.7) +∂U +∂Θ1m +· ˜I = ρ(Θ1 +m)(Um−u(x1 +m))( ˜ +Um−˜u(x1 +m)) and +∂U +∂Θ2m +· ˜I = ρ(Θ2 +m)(Um−u(x2 +m))( ˜ +Um−˜u(x2 +m)), +We refer to Section 4.1 for the definition of r and ρ, and postpone the proof of Lemma 4.6 to +the Appendix. The main difficulty with this sampling formula are the terms u(xk +m) and ˜u(xk +m), which +corresponds to the value of the interior potential at the end-points of the electrodes, and are difficult to +compute accurately for two reasons: first, from a theoretical point of view, this potential is continuous +at the end points of the electrodes, but no more than that, and may varies rapidly which makes it +difficult to compute accurately. Furthermore, as in our immersed method the electrodes are not exactly +discretized, such value is not directly available. We approximate it by linear interpolations from values +of the nearest interface points located inside the electrode. Nevertheless, we obtain sufficiently precise +results to reconstruct the position of the electrodes from the measurements by minimizing the functional +F using a gradient descent algorithm. +We first present reconstructions with Ω a disc of radius 0.5 covered with 4 electrodes.We either per- +turbed the first angle or the second angle of the first electrode, the other electrodes being unperturbed, +but we nevertheless let all the angles be free in our algorithm. As expected, the electrodes E2, E3 and +E4, being already at their correct positions, are almost not moved by the algorithm, while the position +of the first electrode is retrieved as shown in the following table: +Searched angles +Starting angles +Reconstructions +Θ1 +1 +Θ2 +1 +Θ1 +1 +Θ2 +1 +Θ1 +1 +Θ2 +1 +−2.35619 +− 1.85619 +−3.141592 +− 1.85619 +−2.356028 +− 1.85619 +−3.141592 +− 2.64 +−3.141592 +− 1.85619 +−3.14159 +− 2.636460 +In our final experiments, we suppose that the length of the electrodes is fixed and known. In that +case, Θ1 and Θ2 are not independent, as they are linked by relation (4.1). The method is easily adapted +to this situation: we simply decide that Θ2 is now a function of Θ1 given by relation (4.1), and U +depends only on I and Θ1. The derivatives are then obtain by simple combination of Lemma 4.6, the +chain rule, and the fact that equation (4.1) immediately implies +∂Θ2 +k +∂Θ1 +l += δkl +ρ(Θ1 +l ) +ρ(Θ2 +k). +For our final tests, we choose Ωe = [−1, 1]×[−1, 1], discretized by a 400×400 Cartesian grid, N = 4 +and α = [0.8, 0.02, 0.001, 0.05, 0.001, 0.04, 0.001]. The obtained results are displayed in the following +tables: +Searched positions +Start positions +Reconstructed positions +Θ1 +Θ2 +Θ1 +Θ2 +Θ1 +Θ2 +E1 +−2.51327 +− 1.93817 +−3.14159 +− 2.51358 +−2.55152 +− 1.97563 +E2 +−0.94247 +− 0.30516 +−1.57079 +− 0.91229 +−0.90298 +− 0.27141 +E3 +0.62831 1.24343 +0 +0.57253 +0.54018 1.14782 +E4 +2.19911 2.84837 +1.57079 2.18999 +2.08957 +2.72868 +Searched positions +Starting positions +Reconstructed positions +Θ1 +Θ2 +Θ1 +Θ2 +Θ1 +Θ2 +E1 +−2.35619 +− 1.78157 +−3.14159 +− 2.51358 +−2.24361 +− 1.66646 +E2 +−0.78539 +− 0.17053 +−1.57079 +− 0.91229 +−0.70862 +− 0.10368 +E3 +0.78539 1.41055 +0 +0.57253 +0.83745 1.46474 +E4 +2.19911 2.84837 +1.57079 2.18999 +2.23475 2.88755 + +IMMERSED BOUNDARY METHOD FOR CEM IN EIT +27 +Appendix A. +Conductivity reconstruction – Proof of Lemma 4.4. We prove Lemma 4.4 with P = 1, that +is with a single current pattern I. The extension to several current patterns is straightforward. We +therefore fix I ∈ RM +⋄ +and Umeas ∈ RM such that Umeas,1 = 0. +Let (u(σ, I), U(σ, I)) be the solution of (2.5) for a given conductivity σ ∈ L∞ ∩ H1(Ω) satisfying +σ ≥ c > 0 a.e. in Ω. It is known that the mapping +M : σ −→ (u(σ, I), U(σ, I)), +is Frechet differentiable [32]. More precisely, for all δσ in L∞ ∩ H1(Ω) such that σ + δσ ≥ ˜c > 0 a.e. +in Ω, one has +M(σ + δσ) = M(σ) + (δu, δU) + o(∥δσ∥L∞(Ω)), +where δu ∈ H1(Ω) and δU ∈ RM, δU1 = 0, are the only solution of the following variational problem: +find (u, U) ∈ H1(Ω) × RM such that for all (v, V ) ∈ H1(Ω) × RM, one has +(A.1) +� +Ω +σ∇u.∇v dx + +� +m +� +Em +ξm(u − Um)(v − Vm) ds(x) + εU1 V1 = − +� +Ω +δσ∇u(σ, I).∇v dx. +Equivalently, (δu, δU) satisfies +� +� +� +� +� +� +� +� +� +� +� +−∇ · (σ∇δu) = −∇ · (δσ∇u(σ, I)) +in Ω +σ∂ν(δu) = ξm(δUm − δu) − (δσ)∂νu(σ, I) +on Em +σ∂ν(δu) = −(δσ)∂νu(σ, I) +on ∂Ω\E, +� +Em +σ∂ν(δu) ds(x) + εδU1 = − +� +Em +(δσ)∂ν(u(σ, I)) ds(x) +for m = 1, . . . , M. +As a consequence, we have +(A.2) +F(σ + δσ) = F(σ) + (δU, U(σ, Iinput) − Umeas)RM + ϵ(δσ, σ − σ∗)H1(Ω) + o(∥δσ∥L∞∩H1(Ω)). +We now define (w, W) as the unique solution of (2.5) with f = g = 0 and input current I = +U(σ, Iinput) − Umeas, that is +w = u(σ, U(σ, Iinput) − Umeas), +W = U(σ, U(σ, Iinput) − Umeas). +Using (A.1) and the variational formulation of problem (2.5), we obtain that +(δU, U(σ, Iinput) − Umeas)RM = − +� +Ω +δσ∇u(σ, I) · ∇wdx. +As a consequence, for δσ in L∞ ∩ H1(Ω) such that σ + δσ ≥ ˜c > 0 in Ω, and t ∈ (0, 1), we have +F(σ + δσ) = F(σ) + t +� +Ω +� +δσ(ϵ(σ − σ∗) − ∇u(σ, I).∇w) + ϵ∇(δσ).∇(σ − σ∗) +� +dx + o +� +t∥δσ∥L∞∩H1(Ω) +� +Our aim is now to choose δσ such that F(σ + tδσ) ≤ F(σ) for t > 0 small enough. The previous +formula shows that it is sufficient to choose δσ such that +� +Ω +� +δσ(ϵ(σ − σ∗) − ∇u(σ, I).∇w) + ϵ∇(δσ).∇(σ − σ∗) +� +dx < 0. +Let us define v as the unique function of H1 +0(Ω) such that for all ˜v in H1 +0(Ω), +(A.3) +� +Ω +(∇v · ∇˜v + v ˜v) dx = − +� +Ω +(G · ∇˜v + f ˜v) dx, +where +G = ϵ∇(σ − σ∗), +f = ϵ(σ − σ∗) − ∇u(σ, I).∇w. +With this definition, we have +� +Ω +� +v(ϵ(σ − σ∗) − ∇u(σ, I).∇w) + ϵ∇v · ∇(σ − σ∗) +� +dx = −∥v∥2 +H1(Ω) < 0. + +28 +J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS +Therefore, assuming that v belongs to L∞(Ω), we can choose δσ = v, which ends the proof of Lemma +4.4. +Remark A.1. Clearly, v solution of (A.3) might fail to be in L∞(Ω). +However, from a numerical +point of view, we will obtain an approximation of v which will, by construction, be in L∞(Ω). Such +approximation turns out to be a good direction descent, as shown in the presented reconstructions. +Electrode position reconstruction – Proof of Lemma 4.6. Lemma 4.6 is a consequence of a +more general result, that is the Frechet derivability of the measurement map with respect to small +perturbation of the boundary of Ω, obtained in [15]. We start by very briefly recalling the results +proved in this paper. +For h ∈ C1(∂Ω, Rd), we set F[h] : x ∈ ∂Ω �→ x + h(x) and +∂Ωh = F[h](∂Ω). +For h small enough, ∂Ωh is the boundary of a smooth domain Ωh, which of course is a perturbation +of Ω. For h still small enough, Ωh is covered by M well separated electrodes Em,h defined by +Em,h = {x + h(x), x ∈ Em} . +As a consequence, for any I ∈ RM +⋄ , we can denote (u(h), U(h)) the solution of (2.5) with (Ω, Em) +replaced by (Ωh, Em,h), and f = g = 0, and define the measurement map +R : (h, I) ∈ Bd × RM +⋄ �→ U(h), +with +Bd = {h ∈ C1(∂Ω, Rn); ∥h∥C1(∂Ω,Rd) < d}, +d being a small enough fixed constant. The main result of [15] is the Frechet derivability of R with +respect to h ([15, Theorem 3.2]). A sampling formula is also provided ([15, Corollary 3.4]), allowing +to easily compute this derivative. These results are gathered in the following Theorem. +Theorem A.2. The operator R is Fr´echet differentiable at the origin with respect to the first variable. +In other words, there exists a bounded bilinear operator +R′ : C1(∂Ω; Rd) × RM +⋄ �→ RM +such that +lim +h→0 +1 +∥h∥C1 ∥R(h, ·) − R(0, ·) − R′h∥L = 0. +Furthermore, one can compute R′h component by component using the following sampling formula: +for I and ˜I in RM +⋄ , if one denotes (u, U) and (˜u, ˜U) the corresponding solutions to (2.5), then +� +(R′h)I +� +· ˜I = − +M +� +m=1 +� +∂Em +(h · ν∂Em)(Um − u)( ˜Um − ˜u)ds(x) +− +M +� +m=1 +1 +zm +� +Em +hν ((d − 1)(Um − u)H − ∂νu) ( ˜Um − ˜u)ds(x) − +� +∂Ω +hν(σ∇u)τ(∇˜u)τds(x). +Here, H is the mean curvature, ν (resp. ν∂Em) is the outward normal vector to Ω (resp. to Em), and +hν = h · ν. +Now that we have all the necessary tools, we can prove Lemma 4.6. In the following, we focus on +Θ2, the computations with Θ1 being similar. We also do the computation for a single input current +I ∈ RM +⋄ , the extension to P input currents being straightforward. Finally, for any x ∈ ∂Ω, we define +θ(x) as the unique element of [−π, π[ such that x = r(θ(x))u(θ(x)). +Let m ∈ {1, . . . , M} be fixed, and η ∈ R sufficiently small such that +Θ1 +m < Θ2 +m + η < Θ1 +m+1, + +IMMERSED BOUNDARY METHOD FOR CEM IN EIT +29 +where, to simplify notations, we set Θ1 +M+1 = Θ1 +1 + 2π. We define Θ2 +η by +Θ2 +η,k = Θ2 +k + δkm η, ∀k ∈ {1, . . . , M} . +We define, for a fixed constant c > 0, +Hη = +� +� +�h ∈ C1(∂Ω; R2), +������ +h(Θk +1) = 0 +∀k ∈ {1, . . . , M} , +h(Θk +2) = δkmη +∀k ∈ {1, . . . , M} , +∥h∥C1 ≤ cη. +� +� +� +The set Hη is non empty: indeed, take χ ∈ C∞ +c (R; [0, 1]) such that supp(χ) ∈] − 1, 1[ and χ(0) = 1 +and define +ϕη(θ) = ηχ +�θ − Θ2 +m +∆ +� +where +∆ = max +�Θ2 +m − Θ1 +m +2 +, Θ1 +m+1 − Θ2 +m +2 +� +. +Then set ωη : R �→ R 2 π-periodic, defined by +ωη : θ ∈ [Θ1 +1, Θ1 +1 + 2 π[−→ θ + ϕη(θ). +The, if the constant c is chosen sufficiently large, the vector field +h : x ∈ ∂Ω −→ r(ωη(θ(x))u(ωη(θ(x)) − r(θ)u(θ(x)), +belongs to Hη. +As, for η small enough, for hη in Hη, one has ∂Ωhη = ∂Ω, Ek,hη = Ek for all k ∈ {1, . . . , M}, k ̸= m, +and +Em,hη = +� +r(θ)u(θ), θ ∈ [Θ1 +m, Θ2 +m + η] +� +, +we have U(Θ1, Θ2 +η, I) = R(hη, I). Now, cumbersome computations show that, for all hη in Hη, for all +x in ∂Ω, one has +hη(x) = +� +θ(x + hη(x)) − θ(x) +� +ρ(θ(x))τ(x) + o(∥hη∥C1). +Applying Theorem A.2 with d = 2, immediately gives that for all ˜I ∈ RM +⋄ +and (˜u, ˜U) corresponding +solution of (2.5), one has +(R′(hη)I) · ˜I = η ρ(Θ2 +m)(Um − u(x2 +m))( ˜Um − ˜u(x2 +m)) + o(∥hη∥C1). +As +� +U(Θ1, Θ2 +η, I) − U(Θ1, Θ2, I) +� +· ˜I − η ρ(Θ2 +m)(Um − u(x2 +m))( ˜Um − ˜u(x2 +m)) += +� +R(hη, I) − R(0, I) − R′(hη)I +� +· ˜I + o(∥hη∥C1), +Theorem A.2 directly implies Lemma 4.6. +Acknowledgements. 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Wei. “High order matched interface and boundary +method for elliptic equations with discontinuous coefficients and singular sources”. In: J. Comput. +Phys. 213.1 (2006). doi: 10.1016/j.jcp.2005.07.022. +(J. Dard´e) Institut de Math´ematiques de Toulouse, UMR 5219, Universit´e de Toulouse, CNRS, UPS, 118 +route de Narbonne, 31062 Toulouse C´edex 9, France. +(N. Nasr) Univ. Bordeaux, CNRS, INRIA, Bordeaux INP, IHU-LIRYC, IMB, UMR 5251, F-33400 Talence, +France +(L. Weynans) Univ. Bordeaux, CNRS, INRIA, Bordeaux INP, IHU-LIRYC, IMB, UMR 5251, F-33400 Talence, +France +Email address, J. Dard´e: jeremi.darde@math.univ-toulouse.fr +Email address, N. Nasr: niami.nasr@u-bordeaux.fr +Email address, L. Weynans: lisl.weynans@math.u-bordeaux.fr + diff --git a/wNE0T4oBgHgl3EQfswEr/content/tmp_files/load_file.txt b/wNE0T4oBgHgl3EQfswEr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f98463551eff6619ee49269a66d68abdd84cf6e1 --- /dev/null +++ b/wNE0T4oBgHgl3EQfswEr/content/tmp_files/load_file.txt @@ -0,0 +1,2099 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf,len=2098 +page_content='IMMERSED BOUNDARY METHOD FOR THE COMPLETE ELECTRODE MODEL IN ELECTRICAL IMPEDANCE TOMOGRAPHY J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We propose an immersed boundary scheme for the numerical resolution of the Complete Electrode Model in Electrical Impedance Tomography, that we use as a main ingredient in the resolution of inverse problems in medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Such method allows to use a Cartesian mesh without accurate discretization of the boundary, which is useful in situations where the boundary is complicated and/or changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We prove the convergence of our method, and illustrate its efficiency with two dimensional direct and inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Introduction This work is dedicated to the numerical resolution of direct and inverse problems related to Electrical Impedance Tomography (EIT) using an immersed boundary method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The aim of EIT is to reconstruct the electrical conductivity distribution inside a domain by imposing electrical currents on the boundary of this domain, and measuring the resulting voltages on the same boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' It has several applications in the medical field, in particular in lung monitoring and stroke detection [1, 8, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Mathematically, the problem, known as Calder´on problem or inverse conductivity problem, is a severely ill-posed inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We refer to [5, 9, 49] and the references therein for an overview on the inverse conductivity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In practical experiments, the currents are driven inside the body of interest through a collection of surface electrodes, no current being driven between the electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' For each current pattern, the potential differences between the electrodes are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This practical setting is accurately modeled by the Complete Electrode Model (CEM) [11, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' It takes into account the shape of the electrodes as well as the shunting effect, that is the thin resistive layer that appears at the interface between the electrodes and the object during the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The CEM is known to correctly predict experimental data, and therefore is widely used in the numerical resolution of both direct and inverse problems related to EIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Usually, Finite Element Methods (FEM) are used to compute solutions of the CEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This leads to fast and efficient numerical methods in simple geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' However, when the geometry is complicated, the FEM can become very expensive, especially in three-dimensional problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This is due to the need of an accurate discretization of the geometry, in particular on the electrodes where the potential varies rapidly in space, imposing the use of very refined meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Another difficulty could arise when the geometry is not perfectly known, which usually leads to highly incorrect reconstructions [22, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Iterative Newton-type algorithms that reconstruct both the conductivity and the geometry have been proposed to tackle this difficulty [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In such algorithms, the geometry changes from one iteration to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Therefore, a new mesh is created at each step, leading again to a costly numerical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In the present paper, our objective is to propose an alternative to the numerical resolution of CEM with classical mesh-adapted numerical methods, by using Immersed Boundary Methods (IBM), with the aim of reducing the cost of computation in complex and moving geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In such approach, the domain of interest is included in a larger domain with a simple geometry, typically a square in two-dimensional settings, or a cube in three-dimensional settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The geometry of the domain of Date: January 6, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 65N21, 65N25, 65N85, 35R30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 1 2 J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS interest is encoded as the zero level-set of a known function, and is not discretized exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Instead, the larger domain is discretized using typically a Cartesian mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Of course, this approach requires a particular attention to include the boundary conditions of the initial problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The first method on Cartesian grids for elliptic problems was designed by Mayo in 1984 [39], and developed further in [40] and [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In that work an integral equation was derived to solve elliptic interface problems with piecewise coefficients to second order accuracy in maximum norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Then LeVeque and Li (1994) [36] devised the very well known Immersed Interface Method (IIM), that relies on Taylor expansions of the solution on each side of the interface, with a local coordinate transformation near the interface to express the jump conditions in an appropriate frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The elliptic operator is discretized on each grid point near the interface with formulas accounting for the jumps across the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In order to find these formulas, a linear system with six unknowns needs to be solved for each of the concerned grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Numerous developments of the IIM have been performed thereafter, among them [3, 37, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Another class of Cartesian method was introduced by Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' is the Matched Interface and Boundary (MIB) method [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The solution on each side of the interface is extended on fictitious points on the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' These fictitious values are computed by iteratively enforcing the lowest order interface jump conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Other classes of Cartesian methods also exist, less accurate in the case of interface problems, but probably simpler to implement: Gibou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' [23, 24] developed methods inspired by Fedkiw’s Ghost-Fluid Method [19] for multiphase flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' These methods are second order accurate for Dirichlet boundary conditions on arbitrary domains, but only first order accurate for interface problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Then new methods were proposed to increase the accuracy in case of interface problems [4, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In this paper, we approximate the CEM with a finite-difference scheme on a Cartesian mesh, and we follow the methodology of [13], where additional unknowns were defined at the intersections of the interface with the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' These additional unknowns were used in the discretization of the elliptic operator near the interface, which avoided deriving specific finite differences formulas containing jump terms, or corrective terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In order to solve the interface unknowns, the flux jump conditions were discretized and added to the linear system to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This use of additionnal unknowns at the inter- face between two domains, applied in the context of electrical impedance tomography, simplifies the discretization because the values at the boundary of the domain are directly involved in the equations modeling the input currents on the electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As stated above, the main difficulties for the discretization lie in the limit conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In particular, the limit conditions related to the input currents are integrals on the electrodes, which are non-standard from a finite differences point of view, and therefore need particular meticulousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Another difficulty appears at the boundary of the electrodes, where the limit condition changes abruptly, leading to singularities in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We tackle this difficulty by using the smoothened Complete Electrode Model [28], in which the transition between the conditions on electrodes and the condition in between electrodes in smooth, leading to smooth solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Finally, as the problem is typically of Neumann type in order to accurately model the physical measurements setting, one has to deal with a compatibility condition on the data which is not easy to verify at the numerical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We circumvent this last problem with a slight modification of the model, which allows to get rid of the compatibility condition without changing the solution of interest: this might be of interest on its own, as example to simplify numerical implementation of the CEM model by finite elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Outline of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In Section 2, we present the Complete Electrode Model and its variations of interest for our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In Section 3, we present the numerical scheme, and prove a linear rate of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We present numerical results in a two-dimensional setting in Section 4, for both the direct problem of Electrical Impedance Tomography, the inverse conductivity problem, and the electrodes location problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Some needed and technical computations are presented in the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' IMMERSED BOUNDARY METHOD FOR CEM IN EIT 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The Complete Electrode Model for Electrical Impedance Tomography 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Geometrical and functional settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Here and in the following, we suppose that all functions and vectors are real-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Everything can be straightforwardly extended to the complex case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Let Ω be a bounded domain of Rd, d = 2 or 3, with a smooth boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We denote by ν the exterior unit normal of ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Let (Em)m=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=',M be M ≥ 2 mutually disjoint connected subdomains of ∂Ω with positive Lebesgue-measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We set E = �M m=1 Em, and Ec = ∂Ω \\ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We denote RM ⋄ the subspace of mean-free vectors of RM, RM ⋄ = � V ∈ RM, M � m=1 Vm = 0 � , and RM + the subset of RM of vectors with positive components, RM + = � V ∈ RM, Vm > 0 ∀m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' For s ∈ R, we introduce the quotient space Hs = (Hs(Ω) × RM)/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Two elements (u, U) and (v, V ) of Hs(Ω) × RM are identified in Hs if they differ by an additive constant, in the following sense: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1) (u, U) ∼ (v, V ) ⇔ ∃c ∈ R, u = v + c, U = V + c[1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Classicaly, Hs endowed with the norm ∥u, U∥Hs = inf c∈R(∥u − c∥Hs(Ω) + ∥U − c[1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , 1]∥RM ), is a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' For all m in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M}, let em ⊂ Em be of positive Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We set b : ((u, U), (v, V )) ∈ (H1(Ω) × RM)2 �→ � Ω ∇u · ∇v dx + M � m=1 � em (u − Um)(v − Vm) ds(x), and, for all (v, V ) ∈ H1(Ω) × RM, ∥(v, V )∥2 b = b ((v, V ), (v, V )) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The bilinear form b defines a scalar product on H1, with corresponding norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='∥b equivalent to the norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As a consequence, (H1, b) is a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This result is standard, but we recall its proof for the reader’s convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Firstly, we note that if (u1, U1) ∼ (u2, U2) and (v1, V1) ∼ (v2, V2) in the sense of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1), then b ((u1, U1), (v1, V1)) = b ((u2, U2), (v2, V2)), so the Lemma makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Clearly b is bilinear, symmetric and positive in H1 ×H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Let (v, V ) ∈ H1 be such that ∥v, V ∥b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Then ∇v = 0 in Ω, and there exists c ∈ R such that v = c a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Furthermore � em (v − Vm)2ds(x) = 0, ∀m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M} , we obtain that V = c [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , 1] , for the same constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Therefore, (v, V ) ∼ (0, 0), and b is a scalar product on H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Using the continuity of the trace application from H1(Ω) to L2(∂Ω), it is readily seen that ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='∥b ≲ ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Conversely, let (vn, Vn) be a sequence in H1 such that ∥vn, Vn∥H1 = 1 and ∥vn, Vn∥b goes to zero as n goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Necessarily, there exists a sequence (˜vn, ˜Vn) in H1(Ω) × RM such that (vn, Vn) ∼ (˜vn, ˜Vn) in the sense of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1), and therefore ∥˜vn, ˜Vn∥b goes to zero, 1 ≤ ∥˜vn∥2 H1(Ω) + ∥ ˜Vn∥2 RM ≤ 2, for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 4 J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS As (˜vn) is a bounded sequence in H1(Ω) and ( ˜Vn) is a bounded sequence in RM, there exists a subsequence (still denoted (˜vn, ˜Vn)) and an element (˜v∞, ˜V∞) ∈ H1(Ω) × RM such that ˜vn weakly converges to ˜v∞ in H1(Ω) and ˜Vn strongly converges to ˜V∞ in RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As ∇˜vn strongly converges to 0, ˜vn actually strongly converges to ˜v∞ in H1(Ω), ∇˜v∞ = 0 and there exists a constant c∞ such that ˜v∞ = c∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Now, as for all m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M}, 0 ≤ � em (˜vn − ˜Vn,m)2 ds(x) ≤ ∥˜vn, ˜Vn∥2 b, taking the limit as n goes to infinity leads to ˜V∞,m = ˜v∞|em = c∞, or equivalently ˜V∞ = c∞[1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' But as (vn, Vn) ∼ (˜vn, ˜Vn), we have 1 = ∥vn, Vn∥H1 = ∥˜vn, ˜Vn∥H1 ≤ ∥˜vn − c∞∥H1(Ω) + ∥ ˜Vn − c∞[1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , 1]∥RM , leading to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Therefore ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='∥H1 ≲ ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='∥b, and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Generalized Complete Electrode Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We are now in position to introduce the generalized Complete Electrode Model we will use in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This model differs slightly from the standard Complete Electrode Model appearing in the context of Electrical Impedance Tomography, as we allow interior and boundary source terms, and we also allow spatially varying surface admittivities on the electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Both features will be useful in the following study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In our geometrical setting, Ω is the domain of interest, and each open set Em represents one of the electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We denote σ ∈ L∞(Ω) the conductivity of the domain, which is assumed to verify the standard ellipticity condition σ ≥ c > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In the standard Complete Electrode Model, the thin resistive layer appearing at each electrode-body interface is represented by the vector of contact impedances z ∈ RM + : one contact impedance zm per electrode Em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The corresponding contact admittivity on electrode Em is then constant, equal to 1 zm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In our generalized model, we allow spatially varying admittivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' To do so, we consider ξm ∈ Zm, with Zm = {ξ ∈ L∞(Em), ξ ≥ 0, ξ ̸≡ 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' These varying admittivities have been introduced in [28] as a way to obtain regularity on the solutions of the Complete Electrode Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The standard Complete Electrode Model corresponds to the choice ξm = 1 zm , which is indeed in Zm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Let I ∈ RM represent the input currents imposed on the electrodes, and (f, g) ∈ L2(Ω) × L2(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The boundary value problem corresponding to the Complete Electrode Model reads: find (u, U) ∈ H1 such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) � � � � � � � � � � � −∇ · (σ∇u) = f in Ω, σ∇u · ν = g on Ec, σ∇u · ν + ξm(u − Um) = g on Em, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M, � Em σ∇u · ν ds(x) = Im, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Before proving that problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) is well-posed, we make several comments: A simple application of the divergence theorem shows that for problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) to admit a solution, it is necessary that I, f and g satisfy the compatibility condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3) M � m=1 Im + � Ω f dx + � Ec g ds(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Of course, this is simply the well-known current conservation law applied to our system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In the case of practical Electrical Impedance Tomography, where the source terms IMMERSED BOUNDARY METHOD FOR CEM IN EIT 5 f and g are null, we retrieve the mean-free condition on the input currents M � m=1 Im = 0 ⇔ I ∈ RM ⋄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Suppose that (u, U) ∈ H1(Ω) × RM satisfies the system of equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Then for any constant c ∈ R, (u + c, U + c[1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , 1]) also satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This is natural as from a physics point of view, only potential differences can be measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Therefore we work in the quotient space H1 to identify all elements of H1(Ω) × RM that differ up to an additive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As we shall see, this restores uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In the spirit of [28], the limit conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='22) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='23) can be written at once: indeed, denoting by 1Em the operator that maps a function ξ ∈ L∞(Em) to its extension by zero on ∂Ω, and defining the operator Ξ : (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , ξm) ∈ M � m=1 L∞(Em) �→ M � m=1 1Emξm ∈ L∞(∂Ω), then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='22) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='23) are equivalent to σ∇u · ν + Ξ(ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , ξM)(u − Ξ(U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , UM)) = g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In practical applications in Electrical Impedance Tomography, the source terms f and g are both null, in which case the compatibility condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3) becomes the usual condition I ∈ RM ⋄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Nevertheless, non-null source terms will be used in our study in the numerical tests, to ensure smoothness on u and catch the convergence rates of our numerical scheme (see subsection ”Regularity results” below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2), the current configuration I is imposed, and the potential configuration U is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' From a mathematical point of view, it would be equivalent to impose U, and measure I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' To apply Immersed Boundary Method to our problem, it would be simpler to impose U, as the limit conditions are then standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Nevertheless, in practice, it is always the current that is imposed and the potential that is measured, which breaks the symmetry as the noise is greater on the measured data then on the imposed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Therefore we focus our study on problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Well-posedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We now prove that problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) is well-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We follow the standard steps used in the case f = 0 and g = 0, based on the variational formulation of problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2), that we adapt to our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Suppose f, I and g satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Then problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) admits an unique solution (u, U) ∈ H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Consider the variational problem: find (u, U) ∈ H1 such that, for all (v, V ) in H1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4) B � (u, U), (v, V ) � = L � (v, V ) � , with B � (u, U), (v, V ) � = � Ω σ∇u · ∇v dx + M � m=1 � Em ξm(u − Um)(v − Vm)ds(x), and L � (v, V ) � = � Ω f v dx + M � m=1 � Em g (v − Vm) ds(x) + � Ec g v ds(x) + I · V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' It is clear that B is bilinear and continuous on H1 × H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' By definition of Zm, there exists c > 0 such that for all m in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M}, there exists em ⊂ Em of positive Lebesgue measure such that ξm ≥ c on em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This property combined with the assumption on σ and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1 easily implies the coercivity 6 J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' On the other hand, as for all (v, V ) ∈ H1(Ω) × RM and all constant c, we have due to the compatibility condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3) L ((v, V )) = L ((v − c, V − c[1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , 1])) , L is a continuous linear form on H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Therefore, Lax-Migram theorem [6, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='8] implies the existence and uniqueness of (u, U) ∈ H1 satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Remains to prove that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4) is equivalent to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The fact that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4) is standard, so we focus on the reverse implication, and prove that (u, U) ∈ H1 solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Choosing (v, V ) ∼ (ϕ, 0RM ) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4), with ϕ ∈ C∞ c (Ω), immediately implies that −∇ · (σ∇u) = f in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Choosing then (v, V ) ∼ (v0, 0RM ) with v0 ∈ H1(Ω), we obtain � Ω (∇ · σ∇u v0 + σ∇u · ∇v0) dx = M � m=1 � Em (g − ξm(u − Um)) v0 ds(x) + � Ec g v0 ds(x), which implies by surjectivity of the trace operator v0 ∈ H1(Ω) �→ v0|∂Ω ∈ H1/2(∂Ω) and the definition of the conormal derivative σ∇u · ν that u satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2)2 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In particular σ∇u · ν belongs to L2(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Finally, choosing (v, V ) ∼ (0, V0) with V0 ∈ RM yields I · V0 = M � m=1 � Em (g − ξm(u − Um)) V0,m ds(x) = M � m=1 � Em σ∇u · ν V0,m ds(x) = W · V0, with W ∈ RM defined by Wm = � Em σ∇u · ν ds(x), which ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' □ Regularity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The convergence results we obtain for the numerical scheme we develop in the next section require smooth solutions to our problem of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Unfortunately, in lots of situations, and in particular in the relevant ones for application, the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) fail to be smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Indeed, suppose that σ is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Due to the mixed boundary conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2)2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2)3 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2)4, the potential u satisfying system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) might not be as smooth as in standard elliptic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' More precisely, for the standard Complete Electrode Model in which ξm is a positive constant, and in practical settings where the source terms f and g are null, it is known that (u, U) solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) belongs to H2−ε for all ε > 0, but fails to be in H2 for all input currents I except for the null one [17, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The smoothened Complete Electrode Model has been introduced in [28] precisely to overcome this difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' It consists in replacing ξm constant by ξm ∈ Zm, compactly supported in Em and smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Then standard regularity results apply, leading to a smooth potential u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As in practice the surface admittivity is unknown, and there is no clear evidence supporting the idea that it is constant, such change of model is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Obviously, its main drawback is more complex parametrization of the contact admittivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Our convergence results always apply to the smoothened Complete Electrode Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Nevertheless, regardless of the choice of ξm, there is no explicit solution for problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) when f = g = 0, except in very particular geometric configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This is why we introduce the source terms f and g: they allow us to construct explicit solutions to problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) to numerically test our convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' An alternative formulation well-posed in H1(Ω) × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As seen previously, problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) is well posed in H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In other words, the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) is defined up to an additive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' From a numerical point of view, one has to fix that constant (or in other words, choose the ground level), which is usually done by adding a constraint on U, such that U1 = 0 or U ∈ RM ⋄ : the solution is then searched in H1(Ω) × ˜R, with ˜R = {U ∈ Rm, U1 = 0} or ˜R = RM ⋄ , restoring uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Theoretically, the constraint can be imposed on the potential u (as an example, one could impose u to be mean free over Ω), but this adds computational issues when solving numerically the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' IMMERSED BOUNDARY METHOD FOR CEM IN EIT 7 In this study, we propose another approach, consisting in modifying problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) in a new problem, which will be well-posed in H1(Ω)×Rd, and which solution is one of the solutions to problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Note that by construction no additional constraint is imposed on the new problem to obtain well-posedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Setting U1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Let ε > 0 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We consider the modified problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) � � � � � � � � � � � −∇ · (σ∇u) = f in Ω σ∇u · ν = g on Ec σ∇u · ν + ξm(u − Um) = g on Em, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M, � Em σ∇u · ν ds(x) + εδm1Um = Im, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The boundary equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5)4 is equivalent to � E1 σ∇u · ν ds(x) + εU1 = I1, � Em σ∇u · ν ds(x) = Im, ∀m ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In other words, only one equation of the initial problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) is modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We claim the following Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) admits an unique solution (u, U) ∈ H1(Ω) × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Furthermore, if I, f and g satisfy the compatibility condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3), U1 = 0 and (u, U), seen as an element of H1, is the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We endow H1(Ω) × RM with its standard scalar product, with corresponding norm ∥(u, U)∥2 H1(Ω)×RM = ∥u∥2 H1(Ω) + ∥U∥2 RM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' First, adapting the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2, it is readily seen that problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) is equivalent to the variationnal formulation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6) B1 � (u, U), (v, V ) � = L � (v, V ) � , ∀(v, V ) ∈ H1(Ω) × RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' with B1 � (u, U), (v, V ) � = � Ω σ∇u · ∇v dx + M � m=1 � Em ξm(u − Um)(v − Vm)ds(x) + ε U1 V1 and L � (v, V ) � = � Ω f v dx + M � m=1 � Em g (v − Vm) ds(x) + � Ec g v ds(x) + I · V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Similarly, a minor adaptation of the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1 shows that ∥(u, U)∥2 b1 = � Ω |∇u|2dx + M � m=1 |u − Um|2ds(x) + U 2 1 , is a norm on H1(Ω) × RM equivalent to the standard one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The assumptions on σ and ξm immediately imply the coerciveness of the continuous bilinear form B1, while the linear form L is clearly continuous on H1(Ω)×RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Therefore, Lax-Milgram theorem implies the existence and uniqueness of the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We now suppose that I, f and g satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Choosing v = 1 and V = [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , 1] leads to ε U1 = � Ω f dx + � Ec g ds(x) + M � m=1 Im = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As a consequence, U1 = 0, which immediately implies that (u, U) satisfies the system of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2), which ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' □ 8 J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS We see that solving Problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5), which is a very slight variation of our initial problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2), allows to retrieve the unique solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) satisfying the additional condition U1 = 0, without strongly enforcing this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Note that this holds for any value of the positive parameter ε, as long as I, f and g are compatible in the sense of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As it allows to solve (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) without having to fix the ground level, from this point we will always focus on problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The focus on the first electrode is completely arbitrary here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' A simple renumbering of the electrodes allows to use the same formulation to enforce the zero condition on any electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) is well-posed regardless of the compatibility of the data I, f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' If the data are not compatible, then (u, U) solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) cannot solve (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) as the latter has no solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As seen in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3, if (u, U) is the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2), we always have εU1 = � Ω f dx + � Ec g ds(x) + M � m=1 Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='7) As a consequence, if we set ˜I ∈ RM defined by ˜I1 = I1 − �� Ω f dx + � Ec g ds(x) + M � m=1 Im � , ˜Im = Im, m ̸= 1, we see that (u, U) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) with the original source terms (I, f, g) replaced by (˜I, f, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This second set of data obviously satisfies the compatibility condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In a sense, problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) correct the data so that they always satisfy the compatibility condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In the case of compatible data, the value of parameter ε has no influence on the solution of problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5), and can be freely set equal to any arbitrary value, for instance one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' From a numerical point of view, this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Indeed the numerical compatibility condition has no reason to be exactly satisfied by the finite-difference formulation, but only up to the order of the approximation, even if the continuous data satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Therefore, due to equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='7), the value of ε influences the value of U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This causes an interplay between the two terms in the left-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5)4 that will have an influence on the amplitude of the numerical error, but not on the convergence order itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Setting U ∈ RM ⋄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As a final remark, we note that it is equally easy to select (u, U) solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) satisfying the additional mean free value condition on U, that is U ∈ RM ⋄ , without strongly enforcing this condition (that is, with a problem posed in H1(Ω) × RM and not in H1(Ω) × RM ⋄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Indeed, it suffices to solve the following problem: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='8) � � � � � � � � � � � −∇ · (σ∇u) = f in Ω σ∇u · ν = g on Ec σ∇u · ν + ξm(u − Um) = g on Em, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M, � Em σ∇u · ν ds(x) + ε⟨U⟩ = Im, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Here we have set ⟨U⟩ = �M k=1 Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The corresponding variational form is B⟨⟩ � (u, U), (v, V ) � = L � (v, V ) � , ∀(v, V ) ∈ H1(Ω) × RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' with B⟨⟩ � (u, U), (v, V ) � = � Ω σ∇u · ∇v dx + M � m=1 � Em ξm(u − Um)(v − Vm)ds(x) + ε ⟨U⟩ ⟨V ⟩, the continuous linear form L being unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Following the same line of reasoning used in the previous section, it is not difficult to see that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='8) is well-posed in H1(Ω) × RM, and that if (I, f, g) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3), (u, U) solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='8) satisfies the original Complete Electrode Model equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2), with furthermore ⟨U⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' IMMERSED BOUNDARY METHOD FOR CEM IN EIT 9 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' From a numerical point of view, the additional term ⟨U⟩ seems not practical, as it is non-local and therefore tends to fill the matrix corresponding to the discretization of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' However, it appears only on equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='8)4, and affects only the term U ∈ RM, M being the number of electrodes on the boundary of Ω, and therefore typically less than one hundred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As the potential u is usually discretized in spaces containing thousands of degrees of freedom, in practice it does not affect significantly the matrix filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Towards the immersed boundary method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In our numerical scheme, the domain of interest Ω is immersed into a larger domain Ωe, a square in dimension 2 and a cube in dimension 3, such that Ω ⊂ Ωe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The domain Ωe is fully discretized with a Cartesian grid, implying that some discretization points lie in Ωe\\Ω, a part of the extended domain where there is no physical problem in our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In that situation, a first idea is to simply set the solution to zero on the degrees of freedom lying outside of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' But this operation is cumbersome, and not appropriate in our approach, as we need smoothness in the solution on either side of the interface ∂Ω to compute quadrature formulas on the electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We therefore choose a different approach, which consists in solving an additional partial differential equation in Ωe \\ Ω, similar to the problem posed in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As a consequence, we introduce the following additional problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='9) � � � −∆ue = 0 in Ωe \\ Ω ue = 0 on ∂Ωe ue = u on ∂Ω, the function u being the electric potential, solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Clearly, such problem is well-posed in H1(Ω), and its solution depends continuously on u, and therefore on the EIT source terms (I, f, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Furthermore, it will automatically inherit the smoothness of u thanks to standard elliptic regularity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As a conclusion, our goal is now to define an immersed boundary method to reconstruct (u, U, ue), with (u, U) solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) and ue solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' An immersed boundary scheme for CEM In this section we focus on the two-dimensional case, and present the discretization of problems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='9) with an immersed boundary method on a Cartesian grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The domain Ω is immersed into a larger square domain Ωe, and the boundary conditions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) are taken into account by using additional variables located on the boundary of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We prove that the method converges with first order accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The convergence proof is based on a discrete maximum principle, used to provide estimates of the coefficients of the inverse of the discretization matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Classification of grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We consider a uniform Cartesian grid defined on the square Ωe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The grid spacing is denoted h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Each node of this cartesian grid is called a grid point and is denoted Mij = (xi, yj) = (i h, j h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We denote by uh ij the approximation of the function u at the point (xi, yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The set of grid points located inside the domain Ωe is denoted Ωh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' To the purpose of the discretization of the CEM we need to define additional points on the boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We define the boundary point Ii+1/2,j = (xi+1/2,j, yj) as the intersection of the boundary ∂Ω and the segment [MijMi+1j], if it exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Similarly, the boundary point Ii,j+1/2 = (xi, yi,j+1/2) is defined as the intersection of the boundary and the segment [MijMij+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' At each boundary point we create an additional unknown uh i+1/2,j or uh i,j+1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The set of boundary points is denoted δΩh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We say that a grid point is irregular if one of its direct neighbors is a boundary point, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' On the contrary, grid points that are not irregular are called regular grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The set of irregular grid nodes is denoted Ω∗ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The set of electrode values is denoted Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The grid or boundary points can be denoted with letters such as P or Q rather than with indices such as Mi,j if it is convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We also denote if more convenient xP and yP the coordinates of a point P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Numerical scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 10 J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ • • • • • • • • • • • • • • • • • • • • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Left: regular points described by circles ◦, irregular points (belonging to Ω∗ h) described by bullets •, right: boundary points (belonging to δΩh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Elliptic operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' To discretize the elliptic operator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5)1 or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='9)1 on each grid point, regular or not, we use a five point stencil with the grid point Mi,j and its nearest neighbors, boundary or grid points, in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' To make this explicit, we denote uh S the value of the solution on the nearest point in the south direction, with coordinates (xS, yS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Similarly, we define uh N, uh W and uh E and the associated coordinates (xN, yN), (xW , yW ) and (xE, yE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The discretization reads − � ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (σ∇u) �h i,j = − � σi+1/2,j uh E − uh ij xE − xi − σi−1/2,j uh ij − uh W xi − xW � 1 h − � σi,j+1/2 uh N − uh ij yN − yj − σi,j−1/2 uh ij − uh S yj − yS � 1 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' with σi+1/2,j the value of σ at point (xi + xE 2 , yj, ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The truncation error of this discretization is second-order accurate on regular grid points, and zeroth-order on irregular grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Flux boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' At each boundary point we discretize the boundary conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5)2 or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5)3, depending if the boundary point belongs to an electrode or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' For instance, on a boundary point Ii+1/2,j, the discretization reads σ(∇u · ν)h i+1/2,j + ξm(Ii+1/2,j) � uh i+1/2,j − Um � = 0 if Ii+1/2,j ∈ Em, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1) σ(∇u · ν)h i+1/2,j = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) with (∇u · ν)h i+1/2,j denoting the discretized normal derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' A similar discretization is applied at each boundary point Ii,j+1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The discretization of the normal derivative (∇u · ν)h i+1/2,j depends of the local geometry of the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As illustrated on Figure 2, the first intersection between the normal to the boundary and the grid is located on a segment: either [Mi,j, Mi,j−1], or [Mi,j−1, Mi+1,j−1], or [Mi,j, Mi,j+1], or [Mi,j+1, Mi+1,j+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The discrete normal derivative is computed as the normal derivative of the linear interpolant of the numerical solution on the triangle composed of the boundary point Ii+1/2,j and the aforementioned segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' If we denote K this triangle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' y1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' y2) and (x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' y3) its vertices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' and u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' u2 and u3 the associated values,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' the basis functions on the vertices for the linear interpolation write λj(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' y) = αjx + βjy + γj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' j = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' IMMERSED BOUNDARY METHOD FOR CEM IN EIT 11 with αj = yk − yi (xj − xk)(yj − yi) − (xj − xi)(yi − yk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' βj = xi − xk (xj − xk)(yj − yi) − (xj − xi)(yi − yk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' γj = xkyi − xiyk (xj − xk)(yj − yi) − (xj − xi)(yi − yk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (νx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' νy) being an approximation of the normal at the interface point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' With these notations, the approximation of the normal derivative writes for instance for the interface point Ii+1/2,j (∇u · ν)h i+1/2,j = (u1 α1 + u2 α2 + u3 α3)νx + (u1 β1 + u2 β2 + u3 β3)νy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This discretization is first-order accurate because it is based on a linear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' I I+1 J J+1 J-1 I-1 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' All possible stencils for the first-order flux discretization on the left side of the interface, with points involved in the discretization signaled by black circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Integral boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We now discretize (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' More precisely, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5)3, it is equivalent to: � Em � g + ξm(Um − u) � ds(x) + ε δm1 U1 = Im, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3) From now on, we focus on the equivalent boundary condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We discretize this integral boundary condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3) on each electrode with a first-order quadra- ture formula based on a first-order discrete dirac function [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This discrete Dirac function is by construction positive and non-zero only on the irregular grid points, therefore it can be written as � P ∈Ω∗ h ωP � g(P) + ξm(P)(Um − uP ) � + ε δm1 U1 = Im, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4) +1+1+112 J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS with the coefficients ωP that are the weights of the first-order quadrature formula in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The trun- cature error of this formula is therefore first-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Monotonicity of the discretization matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Here we aim to prove that the discretization matrix of the linear system described before, that we denote Ah, is monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Let us first prove the following property which will be useful in the reasoning: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' With the convention used for the normal to the boundary, if the minimum of v is located on a boundary point, then at this boundary point the discrete normal derivative is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Moreover, if the approximated normal derivative at this point is zero, then the three points values involved in the stencil are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The approximation of the normal derivative is constant in space, because it is computed from a linear interpolation on a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Therefore, if the minimum of v is located on a boundary point, then at this boundary point the discrete normal derivative is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The discretization matrix of the linear system described before, that we denote Ah, is monotone, that is, Ah is invertible and all values of A−1 h are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Let v be an array such that all coefficients of Ahv are non-negative, which we denote Ahv ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We aim to prove that all coefficients of v are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' To this purpose we consider the minimum of v in the whole domain: it can be either a grid point value, or a boundary point value, or an electrode potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We detail all these cases hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' If the minimum is reached on a grid point: In this case we denote (i0, j0) the indices of the smallest component of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We assume the grid point is a regular grid point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Otherwise the formula would have slightly different weights, but the reasoning would be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Using the elliptic operator inequality on this point we can write: 4vi0,j0 − vi0+1,j0 − vi0−1,j0 − vi0,j0+1 − vi0,j0−1 ≥ 0, and we deduce that vi0+1,j0 = vi0−1,j0 = vi0,j0+1 = vi0,j0−1 = vi0,j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Repeating recursively this reasoning on the neighbours of (i0, j0), then on the neighbours of the neighbours etc, we deduce that all values of v corresponding to grid or boundary points are equal to vi0,j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Therefore this case amounts to the case of the minimum located on a boundary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' If the minimum is located on a boundary point not belonging to an electrode: Without loss of generality, we assume that the minimum is located on Ii+1/2,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' On this interface point we have both relationships (σ∇v · n)h i+1/2,j ≤ 0 because the minimum is located on this point, (σ∇v · n)h i+1/2,j ≥ 0 because all coefficients of Ahv are non-negative, which leads to (σ∇v · n)h i+1/2,j = 0 and the values of the grid points involved in the discretization of the normal derivative are also equal to the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Using the reasoning of the previsous subsection, it means that all grid point and boundary point values are equal to this minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Therefore this case finally amounts to considering the case of the minimum on a boundary point belonging to an electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' IMMERSED BOUNDARY METHOD FOR CEM IN EIT 13 If the minimum is located on a boundary point belonging to an electrode: Without loss of generality, we assume that the minimum is located on Ii+1/2,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' On this interface point we have the relationships (σ∇v · n)h i+1/2,j ≤ 0, (σ∇v · n)h i+1/2,j + ξm(Ii+1/2,j) � vh i+1/2,j − Vm � ≥ 0, the first one because the minimum is located on this point, and the second one because all coefficients of Ahv are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Therefore vh i+1/2,j − Vm ≥ 0, which means that Vm = vh i+1/2,j, so the electrode potential Vm is also the minimum of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Therefore this case amounts to con- sidering that the minimum of v corresponds to an electrode potential Vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' If the minimum corresponds to an electrode potential Vm with m ̸= 1: On this electrode we have the relationship: � P ∈Ω∗ h ωP ξm(P) � Vm − vP � ≥ 0, therefore all boundary points belonging to Em are equal to the minimum Vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Using the pre- vious reasoning it means that all values of v corresponding to grid points, boundary points or electrode potentials are equal to the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' If the minimum corresponds to the electrode potential V1: On this electrode we have the relationship: � P ∈Ω∗ h ωP ξ1(P) � V1 − vP � + ε V1 ≥ 0, which means that ε V1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The minimum value of v is non-negative, therefore all coefficients of v are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Therefore we have proven that if Ahv is non-negative, then v is also non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This property has two implications: Ah is invertible, and all values of A−1 h are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Discrete Green functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In the following, the letters P and Q represent indices in the linear system, representing either discretization points (on the grid or on the boundary) or electrode values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' For instance, we denote u(P) the coefficient of the row of u with the same index than the point P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Similarly, AhU(P) represents the coefficient of the P-th row of the array AhU, and Ah(P, Q) is the coefficient of the P-th row and Q-th column of the matrix Ah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We also define by Ah(:, Q) and Ah(P, :) the Q-th column and the P-th row of the matrix Ah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In the spirit of [12, 51] for each Q ∈ Ωh ∪ δΩh ∪ Eh, let us define the discrete Green’s function Gh(:, Q) = � Gh(P, Q) � P ∈Ωh∪δΩh∪Eh as the solution of the discrete problem: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) � AhGh(:, Q)(P) = � 0, P ̸= Q 1, P = Q 14 J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS The matrix Ah being monotone, all values of Gh(:, Q) are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In fact, the discrete Green functions are columns of the inverse of Ah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We can write the solution of the discretized problem as a sum of the source terms multiplied by the values of the discrete Green functions: uh(P) = � Q∈Ωh∪δΩh∪Eh Gh(P, Q) (Ahuh)(Q), ∀P ∈ Ωh ∪ δΩh ∪ Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We can obtain bounds on the discrete Green functions with the following result: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Let S and ˜S be two subsets of points , W a discrete function, α > 0, β > 0 and i, j ∈ N such that: � � � (AhW)(P) ≥ 0, ∀P ∈ Ωh ∪ δΩh ∪ Eh \\ ˜S, (AhW)(P) ≥ α−i, ∀P ∈ S, (AhW)(P) ≥ −(β−j), ∀P ∈ ˜S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Then � Q∈S Gh(P, Q) ≤ αiW(P) + αiβ−j � Q∈ ˜S Gh(P, Q), ∀P ∈ Ωh ∪ δΩh ∪ Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Using the definition of the discrete Green functions, we can write AhW(P) ≥ Ah � α−i � Q∈S Gh(:, Q) − β−j � Q∈ ˜S Gh(:, Q) � (P), ∀P ∈ Ωh ∪ δΩh ∪ Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As all coefficients of A−1 h are non-negative, it leads to W(P) − α−i � Q∈S Gh(P, Q) + β−j � Q∈ ˜S Gh(P, Q) ≥ 0, ∀P ∈ Ωh ∪ δΩh ∪ Eh, and finally we obtain the following bound: � Q∈S Gh(P, Q) ≤ αiW(P) + αiβ−j � Q∈ ˜S Gh(P, Q), ∀P ∈ Ωh ∪ δΩh ∪ Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Estimates of discrete Green functions and convergence result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In this section, we obtain upper bounds for the discrete Green functions corresponding to the matrix arising from the discretiza- tion of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='9), and deduce from them the convergence order of the numerical scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We make the assumption that the functions ξm are all smooth enough so that the solution u has the required regularity for the following analysis, in particular for the truncation error estimation presented in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2 to be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' For h small enough, the following upper bounds hold: � Q∈Ωh\\Ω∗ h Gh(:, Q) ≤ O(1), � Q∈δΩh∪Eh Gh(:, Q) ≤ O(1), � Q∈Ω∗ h Gh(:, Q) ≤ O(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We use several discrete functions in the context of proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3 in order to obtain bounds for the different blocks of the inverse matrix A−1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' IMMERSED BOUNDARY METHOD FOR CEM IN EIT 15 We define the array W such that W ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Therefore, all expressions of the linear system vanish excepted the quadrature formula involving the electrode E1: − � ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (σ∇W) �h i,j = 0 ∀Mi,j ∈ Ωh, (σ∇W · ν)h i+1/2,j − ξm(Ii+1/2,j) � Wm − W h i+1/2,j � = 0 ∀ Ii+1/2,j ∈ δΩh, (σ∇W · ν)h i,j+1/2 − ξm(Ii,j+1/2) � Wm − W h i,j+1/2 � = 0, ∀ Ii,j+1/2 ∈ δΩh, � P ∈Em ωp ξm(P) � Wm − ˜WP � = 0 ∀ m ̸= 1, � P ∈Em ωp ξm(P) � Wm − ˜WP � + εW1 = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Gh(:, E1) representing the discrete Green function associated with the electrode E1, we obtain Gh(:, E1) = 1 ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6) We consider the exact solution ¯u of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='9), with f = 1, g = 1, Im = 1 ∀ m ̸= 1, and I1 such that the compatibility condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We define the array ¯W as the discretisation of ¯u on the grid points, boundary points and electrode values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The discretization of the elliptic operator and the fluxes is consistent at least with first-order accuracy, excepted for the grid points in Ω∗ h where it is only zeroth-order accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Thus for h small enough, we can write that − � ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (σ ¯W) �h i,j ≥ 1 2, ∀Mi,j ∈ Ωh \\ Ω∗ h, − � ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (σ ¯W) �h i,j ≥ −C1, ∀Mi,j ∈ Ω∗ h, (σ∇ ¯W · ν)h i+1/2,j − ξm(Ii+1/2,j) � ¯Wm − ¯Wi+1/2,j � ≥ 1 2, ∀Ii+1/2,j ∈ δΩh, (σ∇ ¯W · ν)h i,j+1/2 − ξm(Ii,j+1/2) � ¯Wm − ¯Wi,j+1/2 � ≥ 1 2, ∀Ii,j+1/2 ∈ δΩh � P ∈Em ωp ξm(P) � ¯Wm − ¯WP � ≥ 1 2, ∀ m ̸= 1 � P ∈E1 ωp ξm(P) � ¯W1 − ¯WP � + ε ¯W1 ≥ −2|I1|, with C1 a strictly positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' It can also be re-written as: (Ah ¯W)(P) ≥ 1 2, ∀P ∈ (Ωh \\ Ω∗ h) ∪ (Eh \\ E1), (Ah ¯W)(P) ≥ 1 2, ∀P ∈ δΩh, (Ah ¯W)(P) ≥ −C1, ∀P ∈ Ω∗ h, (Ah ¯W)(E1) ≥ −2|I1|, and it leads to: � Q∈(Ωh\\Ω∗ h)∪(Eh\\E1) Gh(:, Q) + � Q∈δΩh Gh(:, Q) ≤ 2 ¯W + 2C1 � Q∈Ω∗ h Gh(:, Q) + 4|I1|Gh(:, E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='7) 16 J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS We define the array ˜W such that ˜W = 1 for all points in Ωh, and ˜W = 0 for all points in δΩh ∪ Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Because of the jump in the values of ˜W between Ωh and δΩh ∪ Eh, when we apply the discretization to ˜W we obtain − � ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (σ∇ ˜W) �h i,j = 0 ∀Mi,j ∈ Ωh \\ Ω∗ h, − � ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (σ∇ ˜W) �h i,j ≥ C2 h2 ∀ Mi,j ∈ Ω∗ h, (σ∇ ˜W · ν)h i+1/2,j − ξm(Ii+1/2,j) � ˜Wm − ˜Wi+1/2,j � ≥ −C3 h ∀ Ii+1/2,j ∈ δΩh, (σ∇ ˜W · ν)h i,j+1/2 − ξm(Ii,j+1/2) � ˜Wm − ˜Wi,j+1/2 � ≥ −C3 h , ∀ Ii,j+1/2 ∈ δΩh � P ∈Em ωp ξm(P) � ˜Wm − ˜WP � + εδm1 ˜W1 = 0 ∀ m, with C2 and C3 trictly positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We thus obtain C2 h2 � Q∈Ω∗ h Gh(:, Q) ≤ ˜W + C3 h � Q∈δΩh Gh(:, Q), and then: � Q∈Ω∗ h Gh(:, Q) ≤ h2 C2 ˜W + hC3 C2 � Q∈δΩh Gh(:, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='8) Combining 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='8 with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='7 we obtain � Q∈(Ωh\\Ω∗ h)∪(Eh\\E1) Gh(:, Q) + � Q∈δΩh Gh(:, Q) ≤ 2 ¯W + 2C1 h2 C2 ˜W + 4|I1|Gh(:, E1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='9) + 2C1 C3 C2 h � Q∈δΩh Gh(:, Q), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='10) and for h small enough it leads to � Q∈(Ωh\\Ω∗ h)∪(Eh\\E1) Gh(:, Q) + � Q∈δΩh Gh(:, Q) ≤ ¯C (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='11) with ¯C a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Re-injecting this inequality into 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='8 we obtain � Q∈Ω∗ h Gh(:, Q) ≤ h2 C2 ˜W + hC3 C2 ¯C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='12) The inequalities 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='11 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='12 provide the bounds of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' If we denote ¯u the exact solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='9) and uh the numerical solution, the local error |¯u(P) − uh(P)| satisfies for h small enough: |¯u(P) − uh(P)| ≤ O(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='13) Therefore the numerical solution converges with first-order accuracy to the exact solution in L∞-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' IMMERSED BOUNDARY METHOD FOR CEM IN EIT 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We denote τ the truncation error array arising from the discretization of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We have |¯u(P) − uh(P)| = ������ � Q∈Ωh∪δΩh∪Eh Gh(P, Q)τ(Q) ������ ≤ ������ � Q∈Ω∗ h Gh(P, Q)τ(Q) ������ + ������ � Q∈Ωh\\Ω∗ h Gh(P, Q)τ(Q) ������ + ������ � Q∈δΩh∪Eh Gh(P, Q)τ(Q) ������ , which, according to the previous results, immediately implies |¯u(P) − uh(P)| ≤ O(1) ������ � Q∈Ω∗ h Gh(P, Q) ������ + O(h2) ������ � Q∈Ωh\\Ω∗ h Gh(P, Q) ������ + O(h) ������ � Q∈δΩh∪Eh Gh(P, Q) ������ ≤ O(1)O(h) + O(h2)O(1) + O(h)O(1) = O(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Numerical experiments We now study the efficiency of the immersed boundary method for Electrical Impedance Tomogra- phy introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' More precisely, we test the method on three different two-dimensional problems: the direct problem of EIT, which in our context reads: for given source terms f, g and I, compute (u, U) solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) (in the sense of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In that case, the geometry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Ω and the electrodes’ position) and the conductivity are supposed to be known param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This is a well-posed problem, and our numerical method is designed to obtain a good reconstruction of its solution without accurate discretization of the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' the inverse problem of EIT: for given input currents I ∈ RM ⋄ , and corresponding measured voltages U, reconstruct the interior conductivity σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In other word, knowing the couple (I, U), we search for σ such that (u, U) = (u(σ), U(σ)) solves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) with source terms f = 0, g = 0 and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Here again, the geometry is supposed to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This of course is the central problem in Electrical Impedance Tomography, and therefore is the main validation of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' a geometric inverse problem in EIT: for given input currents I ∈ RM ⋄ , and corresponding mea- sured voltages U, reconstruct the positions of the electrodes knowing Ω and the conductivity σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Obviously, this problem does not make much sense from a practical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' But on one hand it can be seen at a first step to tackle the real problem of imperfectly known geometrical setting in EIT, which is known to severely deteriorate the quality of most conductivity’s recon- struction algorithms (at the exception of the ones that are specifically designed to cope with this problem, see [2, 15, 33, 34] and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' On the second hand, it is a prob- lem of interest in our context, as the electrodes are not exactly approximated in the immersed boundary method we proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Therefore, knowing the high-sensibility of EIT-measurements to electrodes’ position, it is a challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Representation of the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In the following experiments, the domain Ω is represented by a polar parametrization of its boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' More precisely, for a given vector α = [α0, α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , α2N] ∈ R2N+1, we set ∂Ω = {r(θ)u(θ), θ ∈ [0, 2π]} , with u(θ) = [cos(θ), sin(θ)], and r(θ) = α0 + N � k=1 (αk cos(kθ) + αk+N sin(kθ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 18 J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS The shape parameters [α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , α2N] are chosen so that 0 < r(θ) < 2 for all θ ∈ [0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Finally, we set Ω = � ρ[cos(θ), sin(θ)] ∈ R2, θ ∈ [0, 2π], 0 ≤ ρ < r(θ0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Note that, with such parameterization of Ω, the outward unit vector at a point x = ∥x∥[cos(θx), sin(θx)] ∈ ∂Ω, denoted ν(x), is given by ν(x) = 1 ρ(θx) [r(θx)u(θx) − r′(θx)v(θx)] , with v(θ) = [− sin(θ), cos(θ)] and ρ(θ) = � r(θ)2 + r′(θ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We also set τ(x) = 1 ρ(θx) [r′(θx)u(θx) + r(θx)v(θx)] , the vector tangential to ∂Ω at the point x, such that (ν(x), τ(x)) is a direct basis of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In the following tests, we use three different geometries for Ω: Ω1 is a disk of radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5, corresponding to α = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5], Ω2 corresponds to the choice α = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='51, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='035, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1], and Ω3 corresponds to the choice α = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='035, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The M-electrodes E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , EM are parameterized by two vectors (Θ1, Θ2) ∈ RM × RM verifying Θ1 1 < Θ2 1 < Θ2 2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' < Θ2 M < Θ1 1 + 2π, such that Em = � r(θ)u(θ), θ ∈ [Θ1 m, Θ2 m] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As in practical application the length of the electrodes is prescribed, we usually set Θ1, choose a length size L > 0, and compute Θ2 by solving the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1) L = � Θ2 k Θ1 k ρ(θ) dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Unless otherwise specified, in the experiments below we use 16 electrodes, defined by Θ1 k = −π+(k−1) π 8 and L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The domains Ω1, Ω2 and Ω3 (in red) in the reference square [−2, 2]×[−2, 2] (in blue), with our usual configuration of 16 electrodes (in bold black) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In this article we use a specific polar parametrization of the interface, which allows us to compute explicitly all normals to the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' However, such parametrization implies that the domain of interest is star-shaped, which might not be the case in certain applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In such situations, it is usually still possible to use a description of the interface with a level-set function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Such a description amounts to represent implicitly the interface as the zero level set of a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The level set method was introduced by Osher and Sethian [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We refer the interested 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 1 2 0 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 1 0 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 2 0 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5IMMERSED BOUNDARY METHOD FOR CEM IN EIT 19 reader to [43, 45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='Although any smooth enough function cancelling at the interface location can be used, in practice the signed distance to the interface is very often used: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) ϕ(x) = � � � dist∂Ω(x) outside of the interface ∂Ω, 0 on the interface, −dist∂Ω(x) inside of the interface ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We recall here some properties occurring in this case: As recalled in [14], the level-set, being a distance function, is 1-Lipshitz and almost everywhere differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Moreover, if φ is differentiable at a point x, then it satisfies the so-called Eikonal equation at x: ∥∇ϕ(x)∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The smoothness of the level-set is in fact strongly related to the smoothness of the interface: as proved in [25, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='355], if the interface is C2, then there exists a real r0 > 0 such that the level-set is C2 for all x, y such that |ϕ(x, y)| < r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The outward normal vector of the isoline of ϕ passing on x, denoted ν(x), can be expressed where ϕ is differentiable as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3) ν(x) = ∇ϕ(x) ∥∇ϕ(x)∥, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Direct problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' First of all, we investigate the ability of the proposed immersed boundary method to efficiently compute solutions of the Electrical Impedance Tomography problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We highlight that the convergence results announced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 are indeed observed in practice for solutions smooth enough, and present several examples of reconstruction of the inner potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Convergence of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' To verify that the convergence rates proved in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 are indeed observed in practice, we proceed as follows: we set Ωe = [−2, 2] × [−2, 2], u(x, y) = sin(x y), U = 0R16, and compute the corresponding source terms f, g and I in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In particular, to obtain the vector of input currents I, we compute the corresponding integrals appearing in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2)4 using a numerical integration scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We then use the immersed boundary method described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2 to obtain an approximate solution uh to problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) for different values of the mesh parameter h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' To test also the convergence of the derivatives, we compute an approximation of the gradient duh by numerical differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In the following, we compute uh and duh for various values of h ranging from 1 25 to 1 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The parameter ε in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5)4 is chosen small, of order ∼ 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The results are displayed in Figure 4: we observe a linear convergence both for u and its gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The linear convergence of uh to u was expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The linear convergence of the gradient is not surprising, as the construction of the numerical method is based on a first-order discretization of fluxes near the boundary of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Nonetheless, to prove it is not obvious and non-standard within the finite difference framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We left the study of the convergence of the gradient for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We have presented here numerical results for a given value of the parameter ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Let us remark that changing the value of the parameter ϵ does not change the convergence behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' However, the value of ε affects the amplitude of the error at fixed h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This is expected since the value of U1 depends on the numerical compatibility condition, and therefore is not exactly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Therefore, there is an interplay between the therm εU1 and the integral term in the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5)4 that influences the value of both terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' On Figure 5, we present a comparison of numerical errors for the generalized CEM, with smooth spatially varying admittivities, and for the classical CEM with ξm = 1 zm constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The geometry considered is a circle of radius R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 embeded in a square [−1, 1] × [−1, 1], with 4 electrodes equally spaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='The discretization parameter h ranges from 1 50 to 1 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The convergence rate displayed on the 20 J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS Figure is computed with the classical CEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We observe that the amplitude of the numerical errors for both models as well as their rate of convergence is very close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Finally, we present some reconstructions of the inner potential, with Ωe = [−2, 2]×[−2, 2] discretized with a 700 × 700 Cartesian grid, and in different geometries, with different background conductivities and various patterns applied to the electrodes, see Figures 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 10-2 10-3 10-2 10-1 Errors for u slope 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='02373 slope 1 (a) ∥uh − u∥∞ as a function of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Ω = Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 10-2 10-3 10-2 10-1 Errors for the gradient of u slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='89771 slope 1 (b) ∥duh,1 − ∂xu∥∞ as a function of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Ω = Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 10-2 10-3 10-2 10-1 Errors for u slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='982082 slope 1 (c) ∥uh − u∥∞ as a function of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Ω = Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 10-2 10-3 10-2 10-1 Errors for the gradient of u slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='84517 slope 1 (d) ∥duh,1 − ∂xu∥∞ as a function of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Ω = Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 10-2 10-3 10-2 10-1 Errors for u slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='981985 slope 1 (e) ∥uh − u∥∞ as a function of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Ω = Ω3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 10-2 10-3 10-2 10-1 Errors for the gradient of u slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='88794 slope 1 (f) ∥duh,1 − ∂xu∥∞ as a function of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Ω = Ω3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Convergence study with manufactured solution u(x, y) = sin(x y) and U = 0R16 in each domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' IMMERSED BOUNDARY METHOD FOR CEM IN EIT 21 (a) ∥uh − u∥∞ as a function of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (b) ∥duh,1 − ∂xu∥∞ as a function of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Convergence study with manufactured solution u(x, y) = exp(x2+y2) and U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 exp(R2)1R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Blue points stand for the numerical errors of classical CEM, red crosses for the numerical errors of generalized CEM Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Resolution of Problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) in domain Ω2 with input current Im = (−1)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Left: inner potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Right: conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' From now on, we consider the standard Complete Electrode Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In other words, in Problems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5), we set ξm = 1 zm , zm being the constant and positive contact impedance on each electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' X10~3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 Errorsfor u (classical) Errors for u (generalized) slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='89 slope 1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='9 Sigma(Sm-1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='7 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1e-0122 J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Resolution of Problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) in domain Ω3 with input current Im = δm 8 − δm 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Left: inner potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Right: conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Reconstruction of the conductivity, knowing the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We now focus on the reconstruction of the conductivity σ∗ in a practical Electrical Impedance Tomography context: for a family of input currents (I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , IP ) ∈ RM ⋄ , P ∈ N∗, the corresponding (potentially noisy) electrodes’ potentials U 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , U P are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' They correspond to the second part of the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) with σ = σ∗ and f = g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' To fix the constant, we set U p 1 = 0 for all p in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As a consequence, the potentials are also the second part of the solution of Problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) To simplify notations, we gather all input currents and electrodes’ potentials corresponding to a certain conductivity σ in M × P matrices I = � I1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' IP � , U = � U 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , U P � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The measurement matrix U depends linearly on the input currents I and non-linearly on the unknown conductivity σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We will sometimes make this dependency explicit by writing U (σ, I ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We also note Umeas ≈ U (σ∗, I ), the matrix which would idealistically corresponds to our actual measurements, but in practice is only close to it, as in applications measurements are always corrupted by noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The inverse conductivity problem can be formulated in the following concise way: knowing � I , Umeas � reconstruct σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Over time, sophisticated methods have been introduced to reconstruct either the conductivity or the support of inclusions from (continuous or electrodes’ like) boundary measurements [7, 10, 20, 21, 26, 27, 29, 31, 35, 41, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In our present study, we will use a comparatively very naive approach to reconstruct σ∗, that is a basic least-square minimization approach combined with Tychonov regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We choose this on purpose, as our primary objective is to test the ability of our direct method to give 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1e+00 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='8 Sigma(Sm-1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1e+00IMMERSED BOUNDARY METHOD FOR CEM IN EIT 23 sufficiently good approximation of the solution to problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) to be efficiently used in an standard algorithm for the inverse conductivity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' More precisely, we seek to minimize the discrepancy between the solution of the direct problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5), corresponding to a guessed conductivity σ, and the measured data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=', we seek σ such that the functional (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4) F(σ) = ∥U (σ, I) − Umeas∥2 2 is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The problem being ill-posed, a regularization is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As a consequence, we actually minimize the functional (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) F(σ) = ∥U (σ, I) − Umeas∥2 RM + ϵ 2R(σ) where R is a properly chosen regularisation functional and ϵ > 0 a regularisation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In other words, we use a deterministic Tychonov regularization of our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' From now on, we assume that σ∗ ∈ L∞ ∩H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We also assume that there exists a known conduc- tivity σ⋆ ∈ L∞ ∩ H1(Ω) such that σ∗ − σ⋆ ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This is typically the case in applications where one searches perturbations of a known background conductivity σ⋆, assuming that the perturbations do not touch the boundary of the domain (in that situation, the pertubations are σ∗ − σ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' It is then natural to seek to minimize the following functional (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6) F(σ) = 1 2∥U (σ, I) − Umeas∥2 RM + ϵ 2∥σ − σ⋆∥2 H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The strategy of minimization is simple: for a conductivity σ and a current pattern I ∈ RM ⋄ , we denote (u(σ, I), U(σ, I)) the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) with f = g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Then, for a given conductivity σn, define (dxun, dyun) by dxun = � ∂u(σn,I1) ∂x , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , ∂u(σn,IP ) ∂x � ∈ L2(Ω)d, dyun = � ∂u(σn,I1) ∂y , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , ∂u(σn,IP ) ∂y � ∈ L2(Ω)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Now, with Umeas = � U 1 meas, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , U P meas � , we define (dxwn, dywn) by dxwn = � ∂u(σn,U(σn,I1)−U 1 meas) ∂x , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , ∂u(σn,U(σn,IP )−U P meas) ∂x � and dywn = � ∂u(σn,U(σn,I1)−U 1 meas) ∂y , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , ∂u(σn,U(σn,IP )−U P meas) ∂y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Finally, we define δσn as the unique v ∈ H1 0(Ω) such that −∆v + v = ϵ � ∆(σn − σ⋆) − (σn − σ⋆) � + dxuT ndxwn + dyuT ndywn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Then it is not difficult to prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Suppose that δσn belongs to L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' There exists Tn > 0 such that for all t ∈ (0, Tn), F(σn + tδσn) ≤ F(σn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The proof of this point is standard, and is given in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Our algorithm of minimization is then the following: Choose σ0 ∈ L∞ ∩ H1(Ω) (for example, σ0 = σ⋆), Until the following stopping criteria is reached: |F(σn + tnδσn) − F(σn)| < τ|δσn|, with τ small enough (1) For a given conductivity σn, compute δσn as described above, (2) Define σn+1 = σn + tnδσn with tn = arg mint∈(0,Tn) F(σn + tδσn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' End.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 24 J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS Note that, at each step n of this algorithm, we have to solve P problems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) to compute dxun and dyun, then again P problems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) to compute dwwn and dywn, and a single Laplace-type problem to compute δσn, that is 2P + 1 elliptic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Luckily, the first 2P problems are exactly (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) with a fixed given conductivity σn, and only the right-hand side changing, meaning that the corresponding system to inverse is the same for the 2P problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Hence the computation of δσn is actually very fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Our numerical experiments show that, for t close to 0, the function t �→ F(σn + tδσn) behave like a quadratic function in t with a single local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We therefore search for tn using a Golden-section search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We now present some conductivity’s reconstructions in the domains presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In each test, the background conductivity σ⋆ is equal to one, and first case: we place a circular inclusion in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' second case : we place a similar inclusion closer to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' third case : we place two inclusions of different amplitudes inside the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The measurement matrix Umeas is obtained by solving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) using our numerical method, in a very refined mesh and for P = 15 linearly independent current patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The mesh used to compute Umeas differs from the mesh used to reconstruct the conductivity in order to avoir inverse crime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' To obtain the noisy data, we then perturb the measurements with an additive Gaussian noise, scaled so that the relative amplitude of the noise is set to δ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Finally, we set the regularization parameter to ϵ = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Figures 8 and 9 present reconstructions in the different geometrical domains with various levels of added Gaussian noise, discretized by a 400 × 400 Cartesian grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As can be seen, the reconstructions are correct even with this very naive approach, meaning that our immersed boundary methods can be efficiently used in the inverse conductivity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In practical applications, the conductance ξm (or equivalently the contact impedances zm) are unknown, and should be also reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This can be done without additional difficulties, using that (u, U) solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) is Frechet differentiable with respect to ξm, with an explicit expression of the Frechet derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Reconstruction of the electrodes’ positions, knowing the conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We now focus on the geo- metric inverse problem described in the introduction, which can be summarized as follows: from the knowledge of (I , Umeas), find the locations of the electrodes, in other words the parameters Θ1 and Θ2, all the other parameters (α, σ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=') being known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As already said, even if this problem makes little sense from a practical point of view, it is interesting and challenging for our method as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) is known to be very unstable with respect to the geometrical setting, and our Boundary Immersed Method is, by design, not precise in terms of geometry of ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Adapting the notations of previous Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1, we denote by U = U(Θ1, Θ2, I) the (second term of the) solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) with f = g = 0, I ∈ RM ⋄ , and electrodes’ positions determined by the angles Θ1 and Θ2 (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1 for the parametrization of the geometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Now, for a set of input currents (I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , IP ) ∈ (RM ⋄ )P , we denote I = � I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , IP � , U = � U 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , U P � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We also set Umeas to be the following M × P possibly noisy measurements matrix, approximating the exact measurement matrix corresponding to the exact electrodes positions Θ1 ∗ and Θ2 ∗: Umeas ≈ U (Θ1 ∗, Θ2 ∗, I ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Our strategy is similar as before: we seek to minimize the functional F(Θ1, Θ2) = ∥U (Θ1, Θ2, I) − Umeas∥2 2 + ϵ 2∥Θ1 − Θ1 ⋆∥2 RM + ϵ 2∥Θ2 − Θ2 ⋆∥2 RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As previously, Θk ⋆ has to be think as a good guess for the actual location of the k-th electrode, around which we search the actual position of the electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' IMMERSED BOUNDARY METHOD FOR CEM IN EIT 25 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Reconstructions in Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' First column: searched conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Second column: reconstructed conductivity with unperturbed data (δ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Third column: reconstructed conductivity with perturbed data (δ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='0%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Reconstructions in Ω3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' First column: searched conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Second column: reconstructed conductivity with unperturbed data (δ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Third column: reconstructed conductivity with perturbed data (δ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='0%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='0e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2 Sigma(Sm-1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5e-012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='0e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2 Sigma(Sm-1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='0e+0026 J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS The partial derivatives of the functional with respect to parameters Θ1 k or Θ2 k are easily derived from the shape derivative obtained in [15], and in particular the sampling formula [15, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4]: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Let (˜u, ˜U) be the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) for some electrode pattern ˜I ∈ RM ⋄ , and f = g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' For k in {1, 2} and m in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M}, let xk m = r(Θk m)[cos(Θk m), sin(Θk m)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We have for m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M}: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='7) ∂U ∂Θ1m ˜I = ρ(Θ1 m)(Um−u(x1 m))( ˜ Um−˜u(x1 m)) and ∂U ∂Θ2m ˜I = ρ(Θ2 m)(Um−u(x2 m))( ˜ Um−˜u(x2 m)), We refer to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1 for the definition of r and ρ, and postpone the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6 to the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The main difficulty with this sampling formula are the terms u(xk m) and ˜u(xk m), which corresponds to the value of the interior potential at the end-points of the electrodes, and are difficult to compute accurately for two reasons: first, from a theoretical point of view, this potential is continuous at the end points of the electrodes, but no more than that, and may varies rapidly which makes it difficult to compute accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Furthermore, as in our immersed method the electrodes are not exactly discretized, such value is not directly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We approximate it by linear interpolations from values of the nearest interface points located inside the electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Nevertheless, we obtain sufficiently precise results to reconstruct the position of the electrodes from the measurements by minimizing the functional F using a gradient descent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We first present reconstructions with Ω a disc of radius 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5 covered with 4 electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='We either per- turbed the first angle or the second angle of the first electrode, the other electrodes being unperturbed, but we nevertheless let all the angles be free in our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As expected, the electrodes E2, E3 and E4, being already at their correct positions, are almost not moved by the algorithm, while the position of the first electrode is retrieved as shown in the following table: Searched angles Starting angles Reconstructions Θ1 1 Θ2 1 Θ1 1 Θ2 1 Θ1 1 Θ2 1 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='35619 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='85619 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='141592 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='85619 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='356028 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='85619 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='141592 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='64 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='141592 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='85619 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='14159 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='636460 In our final experiments, we suppose that the length of the electrodes is fixed and known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In that case, Θ1 and Θ2 are not independent, as they are linked by relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The method is easily adapted to this situation: we simply decide that Θ2 is now a function of Θ1 given by relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1), and U depends only on I and Θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The derivatives are then obtain by simple combination of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6, the chain rule, and the fact that equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1) immediately implies ∂Θ2 k ∂Θ1 l = δkl ρ(Θ1 l ) ρ(Θ2 k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' For our final tests, we choose Ωe = [−1, 1]×[−1, 1], discretized by a 400×400 Cartesian grid, N = 4 and α = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='001].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The obtained results are displayed in the following tables: Searched positions Start positions Reconstructed positions Θ1 Θ2 Θ1 Θ2 Θ1 Θ2 E1 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='51327 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='93817 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='14159 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='51358 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='55152 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='97563 E2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='94247 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='30516 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='57079 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='91229 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='90298 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='27141 E3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='62831 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='24343 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='57253 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='54018 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='14782 E4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='19911 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='84837 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='57079 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='18999 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='08957 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='72868 Searched positions Starting positions Reconstructed positions Θ1 Θ2 Θ1 Θ2 Θ1 Θ2 E1 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='35619 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='78157 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='14159 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='51358 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='24361 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='66646 E2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='78539 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='17053 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='57079 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='91229 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='70862 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='10368 E3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='78539 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='41055 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='57253 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='83745 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='46474 E4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='19911 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='84837 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='57079 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='18999 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='23475 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='88755 IMMERSED BOUNDARY METHOD FOR CEM IN EIT 27 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Conductivity reconstruction – Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4 with P = 1, that is with a single current pattern I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The extension to several current patterns is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We therefore fix I ∈ RM ⋄ and Umeas ∈ RM such that Umeas,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Let (u(σ, I), U(σ, I)) be the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) for a given conductivity σ ∈ L∞ ∩ H1(Ω) satisfying σ ≥ c > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' It is known that the mapping M : σ −→ (u(σ, I), U(σ, I)), is Frechet differentiable [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' More precisely, for all δσ in L∞ ∩ H1(Ω) such that σ + δσ ≥ ˜c > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' in Ω, one has M(σ + δσ) = M(σ) + (δu, δU) + o(∥δσ∥L∞(Ω)), where δu ∈ H1(Ω) and δU ∈ RM, δU1 = 0, are the only solution of the following variational problem: find (u, U) ∈ H1(Ω) × RM such that for all (v, V ) ∈ H1(Ω) × RM, one has (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1) � Ω σ∇u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='∇v dx + � m � Em ξm(u − Um)(v − Vm) ds(x) + εU1 V1 = − � Ω δσ∇u(σ, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='∇v dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Equivalently, (δu, δU) satisfies � � � � � � � � � � � −∇ · (σ∇δu) = −∇ · (δσ∇u(σ, I)) in Ω σ∂ν(δu) = ξm(δUm − δu) − (δσ)∂νu(σ, I) on Em σ∂ν(δu) = −(δσ)∂νu(σ, I) on ∂Ω\\E, � Em σ∂ν(δu) ds(x) + εδU1 = − � Em (δσ)∂ν(u(σ, I)) ds(x) for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As a consequence, we have (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2) F(σ + δσ) = F(σ) + (δU, U(σ, Iinput) − Umeas)RM + ϵ(δσ, σ − σ∗)H1(Ω) + o(∥δσ∥L∞∩H1(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We now define (w, W) as the unique solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) with f = g = 0 and input current I = U(σ, Iinput) − Umeas, that is w = u(σ, U(σ, Iinput) − Umeas), W = U(σ, U(σ, Iinput) − Umeas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1) and the variational formulation of problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5), we obtain that (δU, U(σ, Iinput) − Umeas)RM = − � Ω δσ∇u(σ, I) · ∇wdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As a consequence, for δσ in L∞ ∩ H1(Ω) such that σ + δσ ≥ ˜c > 0 in Ω, and t ∈ (0, 1), we have F(σ + δσ) = F(σ) + t � Ω � δσ(ϵ(σ − σ∗) − ∇u(σ, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='∇w) + ϵ∇(δσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='∇(σ − σ∗) � dx + o � t∥δσ∥L∞∩H1(Ω) � Our aim is now to choose δσ such that F(σ + tδσ) ≤ F(σ) for t > 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The previous formula shows that it is sufficient to choose δσ such that � Ω � δσ(ϵ(σ − σ∗) − ∇u(σ, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='∇w) + ϵ∇(δσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='∇(σ − σ∗) � dx < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Let us define v as the unique function of H1 0(Ω) such that for all ˜v in H1 0(Ω), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3) � Ω (∇v · ∇˜v + v ˜v) dx = − � Ω (G · ∇˜v + f ˜v) dx, where G = ϵ∇(σ − σ∗), f = ϵ(σ − σ∗) − ∇u(σ, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='∇w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' With this definition, we have � Ω � v(ϵ(σ − σ∗) − ∇u(σ, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='∇w) + ϵ∇v · ∇(σ − σ∗) � dx = −∥v∥2 H1(Ω) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' 28 J´ER´EMI DARD´E, NIAMI NASR, AND LISL WEYNANS Therefore, assuming that v belongs to L∞(Ω), we can choose δσ = v, which ends the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Clearly, v solution of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='3) might fail to be in L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' However, from a numerical point of view, we will obtain an approximation of v which will, by construction, be in L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Such approximation turns out to be a good direction descent, as shown in the presented reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Electrode position reconstruction – Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6 is a consequence of a more general result, that is the Frechet derivability of the measurement map with respect to small perturbation of the boundary of Ω, obtained in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We start by very briefly recalling the results proved in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' For h ∈ C1(∂Ω, Rd), we set F[h] : x ∈ ∂Ω �→ x + h(x) and ∂Ωh = F[h](∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' For h small enough, ∂Ωh is the boundary of a smooth domain Ωh, which of course is a perturbation of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' For h still small enough, Ωh is covered by M well separated electrodes Em,h defined by Em,h = {x + h(x), x ∈ Em} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As a consequence, for any I ∈ RM ⋄ , we can denote (u(h), U(h)) the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5) with (Ω, Em) replaced by (Ωh, Em,h), and f = g = 0, and define the measurement map R : (h, I) ∈ Bd × RM ⋄ �→ U(h), with Bd = {h ∈ C1(∂Ω, Rn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' ∥h∥C1(∂Ω,Rd) < d}, d being a small enough fixed constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The main result of [15] is the Frechet derivability of R with respect to h ([15, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' A sampling formula is also provided ([15, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='4]), allowing to easily compute this derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' These results are gathered in the following Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The operator R is Fr´echet differentiable at the origin with respect to the first variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In other words, there exists a bounded bilinear operator R′ : C1(∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Rd) × RM ⋄ �→ RM such that lim h→0 1 ∥h∥C1 ∥R(h, ·) − R(0, ·) − R′h∥L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Furthermore, one can compute R′h component by component using the following sampling formula: for I and ˜I in RM ⋄ , if one denotes (u, U) and (˜u, ˜U) the corresponding solutions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5), then � (R′h)I � ˜I = − M � m=1 � ∂Em (h · ν∂Em)(Um − u)( ˜Um − ˜u)ds(x) − M � m=1 1 zm � Em hν ((d − 1)(Um − u)H − ∂νu) ( ˜Um − ˜u)ds(x) − � ∂Ω hν(σ∇u)τ(∇˜u)τds(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Here, H is the mean curvature, ν (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' ν∂Em) is the outward normal vector to Ω (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' to Em), and hν = h · ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Now that we have all the necessary tools, we can prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' In the following, we focus on Θ2, the computations with Θ1 being similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We also do the computation for a single input current I ∈ RM ⋄ , the extension to P input currents being straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Finally, for any x ∈ ∂Ω, we define θ(x) as the unique element of [−π, π[ such that x = r(θ(x))u(θ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Let m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M} be fixed, and η ∈ R sufficiently small such that Θ1 m < Θ2 m + η < Θ1 m+1, IMMERSED BOUNDARY METHOD FOR CEM IN EIT 29 where, to simplify notations, we set Θ1 M+1 = Θ1 1 + 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We define Θ2 η by Θ2 η,k = Θ2 k + δkm η, ∀k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' We define, for a fixed constant c > 0, Hη = � � �h ∈ C1(∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' R2), ������ h(Θk 1) = 0 ∀k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M} , h(Θk 2) = δkmη ∀k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M} , ∥h∥C1 ≤ cη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' � � � The set Hη is non empty: indeed, take χ ∈ C∞ c (R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' [0, 1]) such that supp(χ) ∈] − 1, 1[ and χ(0) = 1 and define ϕη(θ) = ηχ �θ − Θ2 m ∆ � where ∆ = max �Θ2 m − Θ1 m 2 , Θ1 m+1 − Θ2 m 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Then set ωη : R �→ R 2 π-periodic, defined by ωη : θ ∈ [Θ1 1, Θ1 1 + 2 π[−→ θ + ϕη(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' The, if the constant c is chosen sufficiently large, the vector field h : x ∈ ∂Ω −→ r(ωη(θ(x))u(ωη(θ(x)) − r(θ)u(θ(x)), belongs to Hη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As, for η small enough, for hη in Hη, one has ∂Ωhη = ∂Ω, Ek,hη = Ek for all k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' , M}, k ̸= m, and Em,hη = � r(θ)u(θ), θ ∈ [Θ1 m, Θ2 m + η] � , we have U(Θ1, Θ2 η, I) = R(hη, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Now, cumbersome computations show that, for all hη in Hη, for all x in ∂Ω, one has hη(x) = � θ(x + hη(x)) − θ(x) � ρ(θ(x))τ(x) + o(∥hη∥C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Applying Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2 with d = 2, immediately gives that for all ˜I ∈ RM ⋄ and (˜u, ˜U) corresponding solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='5), one has (R′(hη)I) · ˜I = η ρ(Θ2 m)(Um − u(x2 m))( ˜Um − ˜u(x2 m)) + o(∥hη∥C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' As � U(Θ1, Θ2 η, I) − U(Θ1, Θ2, I) � ˜I − η ρ(Θ2 m)(Um − u(x2 m))( ˜Um − ˜u(x2 m)) = � R(hη, I) − R(0, I) − R′(hη)I � ˜I + o(∥hη∥C1), Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='2 directly implies Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' This work has been partially supported by the ANR LabEx CIMI (under grant ANR-11-LABX-0040) within the French State Programme “Investissements d’Avenir”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='1007/978-0-387-70914-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Br¨uhl and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Hanke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' “Recent progress in electrical impedance tomography”.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Dard´e) Institut de Math´ematiques de Toulouse, UMR 5219, Universit´e de Toulouse, CNRS, UPS, 118 route de Narbonne, 31062 Toulouse C´edex 9, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Nasr) Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Bordeaux, CNRS, INRIA, Bordeaux INP, IHU-LIRYC, IMB, UMR 5251, F-33400 Talence, France (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Weynans) Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Bordeaux, CNRS, INRIA, Bordeaux INP, IHU-LIRYC, IMB, UMR 5251, F-33400 Talence, France Email address, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Dard´e: jeremi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='darde@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='univ-toulouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='fr Email address, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Nasr: niami.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='nasr@u-bordeaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='fr Email address, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content=' Weynans: lisl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='weynans@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='u-bordeaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} +page_content='fr' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE0T4oBgHgl3EQfswEr/content/2301.02582v1.pdf'} diff --git a/wdFPT4oBgHgl3EQf_DXv/content/tmp_files/2301.13218v1.pdf.txt b/wdFPT4oBgHgl3EQf_DXv/content/tmp_files/2301.13218v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ea684679b435f1ffaaef8b63dd4b4e87f80f4006 --- /dev/null +++ b/wdFPT4oBgHgl3EQf_DXv/content/tmp_files/2301.13218v1.pdf.txt @@ -0,0 +1,1318 @@ +Identifying quantum many-body integrability and chaos using eigenstates trace +distances +Reyhaneh Khasseh,1 Jiaju Zhang,2 Markus Heyl,1 and M. A. Rajabpour3 +1Theoretical Physics III, Center for Electronic Correlations and Magnetism, +Institute of Physics, University of Augsburg, D-86135 Augsburg, Germany +2Center for Joint Quantum Studies and Department of Physics, +School of Science, Tianjin University, 135 Yaguan Road, Tianjin 300350, China +3Instituto de Fisica, Universidade Federal Fluminense, +Av. Gal. Milton Tavares de Souza s/n, Gragoat´a, 24210-346, Niter´oi, RJ, Brazil +While the concepts of quantum many-body integrability and chaos are of fundamental importance +for the understanding of quantum matter, their precise definition has so far remained an open ques- +tion. In this work, we introduce an alternative indicator for quantum many-body integrability and +chaos, which is based on the statistics of eigenstates by means of nearest-neighbor subsystem trace +distances. We show that this provides us with a faithful classification through extensive numerical +simulations for a large variety of paradigmatic model systems including random matrix theories, +free fermions, Bethe-ansatz solvable systems, and models of many-body localization. While exist- +ing indicators, such as those obtained from level-spacing statistics, have already been utilized with +great success, they also face limitations. This concerns for instance the quantum many-body kicked +top, which is exactly solvable but classified as chaotic in certain regimes based on the level-spacing +statistics, while our introduced indicator signals the expected quantum many-body integrability. +We discuss the universal behaviors we observe for the nearest-neighbor trace distances and point +out that our indicator might be useful also in other contexts such as for the many-body localization +transition. +Quantum chaos and integrability have been a major +focus of research for decades due to their key relevance +for the foundations of statistical physics and fundamen- +tal concepts such as thermalization. In classical physics, +chaos manifests as a divergence of initially close-by phase- +space trajectories. Integrability as a counterpart of chaos +is defined by the existence of a maximal number of +Poisson-commuting invariants [1, 2]. However, establish- +ing a precise measure of quantum chaos and integrability +in the quantum many-body regime has remained an out- +standing challenge [3]. The most widely used indicator +is based on level spacing statistics [4, 5]. +However, it +predicts chaotic behavior for some systems, which are +expected to be considered integrable in the many-body +sense [6, 7]. +In this letter, we introduce an alternative indicator +of quantum integrability and many-body quantum chaos +based on the eigenstate properties instead of the spec- +trum. We show that our indicator correctly classifies a +wide range of systems as quantum integrable, including +Bethe-ansatz solvable models, quantum spin chains in a +fully many-body localized (MBL) regime, and quadratic +fermionic systems. +It is a central result of this work +that our indicator detects quantum integrability also in +cases where the level spacing fails, such as the quantum +many-body kicked-top model. Our indicator is based on +subsystem trace distances between nearest-neighboring +Hamiltonian eigenstates, which provides a bound on the +smoothness of operator expectation values as a function +of energy and, therefore, a natural connection to the +eigenstate thermalization hypothesis (ETH). This mea- +sure is much more robust to the symmetries of the sys- +tems as compared to the level spacing statistics. This +can be useful when the symmetries of a model are not +fully understood. +To investigate whether a general quantum Hamiltonian +exhibits chaos or integrability, we evaluate the trace norm +distance between two reduced density matrices defined as +DA +n = 1 +2||ρA +n+1 − ρA +n ||1 . +(1) +Here, ρA +n = Tr ¯ +Aρn denotes the reduced density matrix of +a subsystem A and ρn = |ψn⟩ ⟨ψn| the density matrix of +an eigenstate |ψn⟩ of a given Hamiltonian. We order the +eigenstates |ψn⟩ with respect to their eigenvalues ϵn in as- +cending order, i.e., ϵn+1 > ϵn. While distances between +density matrices can be defined in various ways [8–10], +the definition in Eq. (1) turns out to be practically suit- +able, as we will discuss in the remainder of this letter. In +particular, it has been found that DA +n provides a general +upper bound on the smoothness of operator expectation +values as a function of energy [10, 11]: +|∆On| = |Tr(ρA +n+1 − ρA +n )O| ≤ 2sDA +n . +(2) +Here, ∆On = ⟨ψn+1| O |ψn+1⟩ − ⟨ψn| O |ψn⟩ denotes the +difference of operator expectation values in neighboring +eigenstates with O an operator defined in subsystem A +and s is the largest singular value of the operator O [10]. +The expectation values of local observables for various +Hamiltonian eigenstates may fluctuate between neigh- +boring eigenstates and conform to thermal predictions +if ETH is valid. +This distinction can be used to dif- +ferentiate between integrable and chaotic models. The +ETH, which pertains to chaotic Hamiltonians, stipulates +that the diagonal matrix elements of observables within +arXiv:2301.13218v1 [cond-mat.stat-mech] 30 Jan 2023 + +2 +Hamiltonian eigenstates exhibit a smooth energy depen- +dence and a narrow distribution. Conversely, in the in- +tegrable regime, the expectation values across the spec- +trum tend to fluctuate significantly, as previously demon- +strated in various research studies (see references [12] +and [13] for a review). We propose to use the eigenstate +trace distances measure in Eq. (1), which expands upon +the definition of the ETH based on subsystem trace dis- +tances in Ref. [14] by providing quantitative criteria for +quantum many-body chaos and integrability. +Through an extensive numerical analysis of different +classes of one-dimensional quantum many-body systems, +we have gained several key insights, which we will now +briefly outline. For quantum chaotic many-body systems, +we find that the distribution of DA +n resembles that of +random matrix theory models and that the mean value +⟨DA +n ⟩ averaged over the spectrum decays exponentially +with system size L as long as x = LA/L < 1/2 with +LA denoting the subsystem size. +In integrable Hamil- +tonians, on the other hand, we observe that this expo- +nential decay is absent, which we identify as the signa- +ture of quantum integrability. Overall, we identify two +different behaviors of ⟨DA +n ⟩ in quantum many-body in- +tegrable systems suggesting the possibility of two differ- +ent classes. +For conventional integrable systems, such +as free fermions and Bethe ansatz solvable models, we +find a seemingly universal linear dependence in ⟨DA +n ⟩ as +a function of x for x <∼ 1/2. In systems with emergent +integrability, such as those exhibiting MBL, the behavior +of ⟨DA +n ⟩ is not following the linear dependence as for con- +ventional integrable systems. Interestingly, we find that +the exactly solvable many-body kick-top model presents +an intriguing scenario where certain regions of the tun- +ing parameter exhibit behavior consistent with that of a +conventional integrable system while other regions dis- +play characteristics evocative of MBL. In the following, +we will introduce the microscopic chaotic and integrable +models we use to illustrate our findings. +Many-body quantum chaotic systems: Let us start with +the analysis of many-body quantum chaotic systems. In +this context, we will use for our analysis the paradigmatic +quantum Ising chain with both transverse and longitudi- +nal fields +HIsing = +L +� +l=1 +(Jσz +l σz +l+1 + hzσz +l + hxσx +l ) , +(3) +where σα +l (α = x, y, z) denote the Pauli spin operators at +site l, J is the coupling constant, and hβ(β = x, z) rep- +resent the strengths of the two magnetic fields. In what +follows, we will set the interaction J = 1. We assume pe- +riodic boundary conditions (PBC), i.e., σα +L+1 = σα +1 (α = +x, y, z), which implies translational invariance. This sym- +metry enables us to partition the Hamiltonian into differ- +ent sectors with a conserved momentum of K = 2πj/L, +where j = 0, ..., L − 1. Each sector can be independently +diagonalized, reducing the computational complexity. It +is worth noting that the main features of the statistics +x = 4/18 +x = 1/2 +FIG. 1: Ising model: Distribution P(Dn(x)) of trace +distances Dn(x) for nearest-neighbor eigenstates with +x = LA/L the ratio between subsystem size LA and +system size L = 18. The main plot shows data for +P(Dn(x)) at different magnetic field strengths for +LA = L/2 and the inset for LA = 4. +of DA +n are identical for single symmetry sectors or for +the full spectrum. We, therefore, focus on a single sector +K = 2π/L without loss of generality. Let us mention, +however, that the K = 0, π sectors provide an exception +due to the presence of further symmetries. Examples of +different symmetry blocks and comparisons with results +from the full spectrum can be found in the Supplemental +Material [16]. In the following, it will be suitable to ana- +lyze DA +n as a function of x = LA/L, so that we introduce +the following notation: +Dn(x) = 1 +2||ρn+1(x) − ρn(x)||1, +x = LA/L , +(4) +in addition to Eq. (1). +The Ising chain, which is described in Eq. (3), is known +to exhibit chaotic behavior as long as J and hx,z are non- +zero [17]. In Fig. 1, we show the qualitatively different be- +haviors of the distribution of Dn(x) in the integrable and +quantum chaotic regimes for a system of size L = 18 and +for x = 1/2, 2/9, respectively. The distribution is narrow +and strongly peaked for the quantum chaotic case, and +we find that the width of the distribution is exponen- +tially suppressed with the system size in this limit, see +Fig. S1(a) in the Supplemental Material [16]. This result +is consistent with RMT, where the distribution is Gaus- +sian, with a vanishing standard deviation by increasing +the size of the system, see Fig. S1(b, c) in the Supplemen- +tal Material [16]. Turning off either the longitudinal or +transverse field gives an integrable model. We see that +these limits have a drastically broader distribution, as +shown by the brown and purple curves in Fig. 1. +In what follows, we will consider as a detector of quan- +tum many-body chaos and integrability the mean value + +35 +5+V5 +1+V5 +b +8 +4 +口 +30 +4 +口 +5+V5 +25 +8 +口 +6 +3 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Dn(c)35 +30 +25 +4 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.03 +(a) +Chaotic Ising +(b) +GOE +(c) +x = 1/2 +FIG. 2: Quantum many-body chaotic models : (a) Average ⟨Dn(x)⟩ for the Ising chain at hz = (1 + +√ +5)/4 and +hx = (5 + +√ +5)/8 in the symmetry-sector at momentum K = 2π/L. (b) ⟨Dn(x)⟩ for RMT from the Gaussian +orthogonal ensemble (GOE) taken over 4000 random eigenstates. The inset shows the exponential decay of ⟨Dn(x)⟩ +with system size L for the particular case x = 1/3. In panel (c), we compare the system-size dependence of +⟨Dn(x = 1/2)⟩ in the quantum many-body chaotic Ising model with RMT. +⟨Dn(x)⟩ taken over all eigenstates in a single momen- +tum sector. In the chaotic regime, we see that ⟨Dn(x)⟩ +decays exponentially upon increasing system size L as +long as x < 1/2, see Fig. 2. This behavior is a direct +consequence of ETH. In Section II of the Supplemental +Material [16], we show analytically that +⟨Dn(x)⟩ = C∆ , +(5) +where ∆ = ⟨ϵn − ϵn+1⟩ denotes the mean level spacing. +As ∆ decays exponentially with system size L, ETH con- +sequently predicts the analogous behavior also for Dn(x). +Furthermore, we can bound the proportionality constant +C ≤ 1/∆E with ∆E = +� +⟨H2⟩ − ⟨H⟩2 the total energy +fluctuations yielding a system-size dependence propor- +tional to +√ +L. +In Fig. 2, we show numerical data of ⟨Dn(x)⟩ for many- +body quantum chaotic models. Specifically, Fig. 2(a) and +Fig. 2(b) include results for the quantum chaotic Ising +chains and the Gaussian Orthogonal Ensemble (GOE) of +RMT, respectively. We find that x = 1/2 is a fixed point, +and the behavior of ⟨Dn(x)⟩ is different for values of x +less than and greater than 1/2. For x < 1/2, ⟨Dn(x)⟩ de- +cays upon increasing L, ultimately tending towards zero +as L approaches infinity. The results of our numerical +calculations using the RMT indicate that the decay of +⟨Dn(x)⟩ is exponential with the system size, i.e., ∼ e−αx +with an exponent α ∼ 0.12 for x = 1/3, see inset of +Fig. 2(b). For x > 1/2, ⟨Dn(x)⟩ tends towards 1 by in- +creasing system size. To emphasize the consistency of the +results of the chaotic Ising chain with RMT, we study in +more detail ⟨Dn(1/2)⟩ in Fig. 2(c). One can see that for +large system sizes ⟨Dn(1/2)⟩ in the chaotic Ising model +approaches the saturation value of RMT. More details +about RMT are discussed in Sec. I of the Supplemental +Material [16]. +Integrable systems: After discussing the behavior of +⟨Dn(x)⟩ for quantum many-body chaotic systems, we +now move to the case of integrable models. In particular, +we first study the transverse-field Ising chain at hz = 0 in +Eq. (3). Since the model is integrable in this regime, we +expect the results to be significantly different from those +predicted by RMT. We show the markedly different be- +havior in Fig. 3(a) where we provide numerical data for +⟨Dn(x)⟩ for different system sizes. First of all, we can +not identify a fixed point anymore. Upon increasing sys- +tem size, we find that ⟨Dn(x)⟩ appears to converge to a +single non-zero curve that does not match the prediction +of ETH in Eq. 5. Instead, the numerical data seems to +approach a linear function ⟨Dn(x)⟩ ∼ ax upon increas- +ing system size with a slope a ≈ 2. +The linear slope +remains roughly unchanged when considering other sym- +metry blocks except those with K = 0, π, which have +a ≈ 1 [18]. This is discussed further in Sec. III of the +Supplemental Material [16]. +In order to gain some analytical insights into this +linear behavior, we have also studied the XY chain, +which can be solved exactly through a mapping to a free +fermion Hamiltonian using the Jordan-Wigner transfor- +mation [19–21]. The subsystem trace distance between +two eigenstates can be obtained by explicit construction +of the RDMs [22–24]. We find evidence that ⟨Dn(x)⟩ ∝ x +in the large L limit, especially in the range x ∈ (0.1, 0.4), +see Sec. IV of the supplemental material [16]. +In the following, we will now explore to what extent +this observed behavior for the Ising chain generalizes also +to other models. For that purpose, we will conduct the +same analysis on several different paradigmatic examples +of integrable systems, including the Bethe-ansatz inte- +grable homogeneous XXZ chain, the XXZ model with a +random field in the MBL regime, and the quantum many- +body kicked top. + +1.0 +0.8 +0.6 +L=8 +(Dn(α)) + L = 10 +Ｌ= 12 +0.4 +L=14 +←L= 16 +0.2 +★ L= 18 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0L=8 +L= 10 +L= 12 +L=14 +←L= 16 +0.35 +★L= 18 +0.25 +0.15 +0.05 +9 13 17 21 +L +0.0 +0.2 +0.4 +0.6 +0.8 +1.00.68 +0.63 +AB7Xi +cbVBNSwMxEJ2tX7V+VT16CRbBU9mVUj0WvYinCvYD2qVk02wbm02WJCuUbf+DFw+KePX/ePfmLZ70NYHA4/3ZpiZF8ScaeO6305ubX1jcyu/XdjZ3ds/ +KB4eNbVMFKENIrlU7QBrypmgDcMp+1YURwFnLaC0c3Mbz1RpZkUD2YcUz/CA8FCRrCxUvOul04m016x5JbdOdAq8TJSgz1XvGr25ckiagwhGOtO54b +Gz/FyjDC6bTQTSNMRnhAe1YKnBEtZ/Or52iM6v0USiVLWHQXP09keJI63EU2M4Im6Fe9mbif14nMeGVnzIRJ4YKslgUJhwZiWavoz5TlBg+tgQTxeyti +AyxwsTYgAo2BG/5VXSvCh71XLlvlKqXWdx5OETuEcPLiEGtxCHRpA4BGe4RXeHOm8O/Ox6I152Qzx/AHzucP4cKPWA=J|| +ACG3i 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(a) A two-dimensional zig-zag lattice of 6 rungs and 4 +legs holding 6 × 4 spin- 1 +2 particles. The spin-exchange interaction +strength along the rungs and the legs are J⊥ and J||, respectively, +while the spins interact diagonally with the interaction strengths +J1 +d and J2 +d along the left and the right diagonals respectively. The +periodic boundary condition along the rungs, legs, and diagonals +are represented by the dashed links to the edge spins of the lat- +tice. Each spin in the lattice is subjected to a magnetic field of +strength h in the z-direction. (b) Using degenerate perturbation +theory in the low-energy subspace in the strong rung-coupling limit +(J⊥ ≫ J||, J1 +d, J2 +d), for specific points in the parameter space of +J⊥ and h, the model can be mapped to a effective 1D Hamiltonian +(see Eq. (12)), where each lattice site hosts a d-dimensional Hilbert +space corresponding to each rung of the system. +degeneracy of the model in the strong rung-coupling limit +is two-fold. We prove that the 1D effective Hamiltonian is +rotationally invariant about the z-axis to any order in pertur- +bation theory. We focus on the leading order in perturbation, +which turns out to be an 1D spin-1/2 XXZ model, which +can be dealt with relatively easily due to the computational +advantage obtained from the reduction in the Hilbert space, +and considerable information is available about the model in +literature [92–95]. We analytically derive the effective cou- +pling constants of the 1D XXZ model as functions of the +coupling constants in the original model for arbitrary values +of N and L and for different boundary conditions on the 2D +lattice in the horizontal and vertical directions. We also point +out a set of typical states including the thermal state and the +time-evolved state constituted from the low-energy sector of +the model for which the effective 1D theory works, up to +O(J−1 +⊥ ), in estimating the states. We give a systematic pre- +scription for mapping matrix-elements of operators among +these states to their 1d effective counterparts. We also show +that the entanglement in low-energy states of the quasi-1D +or 2D model is satisfactorily proxied by the entanglement +in the corresponding mapped state in the 1D model. We il- +lustrate these results using typical observables and entangle- +ment measures computed using partial trace-based methods. +We also investigate measurement-based entanglement in the +quasi-1D and 2D model, for which analytical treatment is +difficult. However, for certain choices of measured Hermi- +tian operators, our numerical analysis shows that the 1D ef- +fective XXZ model can mimic the results in the quasi-1D +or 2D model. +The rest of the paper is organized as follows. In Sec. II, +we discuss the 2D Heisenberg model, the low-energy 1D ef- +fective Hamiltonian, and its symmetry, and point out typical +states where the 1D effective theory performs satisfactorily. +The technical details of working out the 1D effective XXZ +model are included in Sec. III. Sec. IV deals with the per- +formance of the 1D effective theory in computing matrix el- +ements of Hermitian operators, while the application of the +1D effective theory in quantifying entanglement using partial +trace-based and measurement-based approaches are included +in Sec. V. Sec. VI contains the concluding remarks and out- +look. +II. +LOW-ENERGY EFFECTIVE HAMILTONIAN +In this section, we introduce the Isotropic Heisenberg +model on the rectangular zig-zag lattice, and discuss map- +ping it to an effective one-dimensional (1D) lattice model in +the low-energy subspace. +A. +Isotropic Heisenberg model on a 2D zig-zag lattice +Let us consider an N × L lattice with L (N) lattice +sites along the vertical (horizontal) direction, each lattice site +hosting a spin- 1 +2 particle (see Fig. 1(a)), such that a 2NL- +dimensional Hilbert space H is associated to the system. We +call the vertical (horizontal) lines in the lattice as the rungs +(legs), where N (L) represents the number of rungs (legs). +While N = L represents a zig-zag square lattice, N ≫ L +represents a zig-zag ladder that is intermediate between the +one- and two-dimensional lattices, and is therefore called a +quasi-1D lattice model [73]. Isotropic Heisenberg interac- +tions [92–95] are present between the spins along the legs, +rungs, as well as the diagonals, and an external magnetic +field applies to all spins along the z direction. The spin sys- +tem is represented by the Hamiltonian [73] +H/J⊥ = HR + HI, +(1) +with +HR = 1 +4 +N +� +i=1 +L +� +j=1 +⃗σi,j.⃗σi,j+1 − +h +2J⊥ +N +� +i=1 +L +� +j=1 +σz +i,j − NEg, +(2) + +3 +and +HI = J|| +4J⊥ +N +� +i=1 +L +� +j=1 +⃗σi,j.⃗σi+1,j + J1 +d +4J⊥ +N +� +i=1 +L +� +j=1 +⃗σi,j+1.⃗σi+1,j ++ J2 +d +4J⊥ +N +� +i=1 +L +� +j=1 +⃗σi,j.⃗σi+1,j+1. +(3) +Here, J⊥ (J||) is the strength of the spin-exchange interac- +tions along the rungs (along the legs), J1 +d and J2 +d are the +diagonal interaction strengths, h is the strength of the mag- +netic field, ⃗σi,j ≡ {σx +i,j, σy +i,j, σz +i,j} are the Pauli operators +on the lattice-site denoted by the subscripts i, j, with i (j) +being the rung (leg) index running from 1 to N (L), and +Eg is a constant we will choose in section II B. We use the +computational basis {|0⟩ , |1⟩} to write the Pauli matrices, +where σz |0⟩ = |0⟩, σz |1⟩ = − |1⟩, such that |0⟩ ≡ |↑⟩, and +|1⟩ ≡ |↓⟩. +B. +Effective Hamiltonian in low-energy subspace +We now discuss the construction of a low-energy ef- +fective 1D Hamiltonian (LEH) [98] corresponding to the +spin models described in Sec. II A. We start with the limit +J|| = J1 +d = J2 +d = 0, where the model consists of N +non-interacting rungs, each of which corresponds to a 2L- +dimensional Hilbert space. The antiferromagnetic isotropic +Heisenberg Hamiltonian of each rung is given by +HRi = 1 +4 +L +� +j=1 +⃗σi,j.⃗σi,j+1 − +h +2J⊥ +L +� +j=1 +σz +i,j − Eg, +(4) +with HR = �N +i=1 HRi. Let us assume that for a given value +of h, the spectrum of HRi is given by {E(i) +ki , |ψ(i) +ki ⟩}, where +ki ∈ {0, 1, · · · , 2L − 1} ∀i ∈ {1, 2, · · · , N}, such that +HRi |ψ(i) +ki ⟩ = E(i) +ki |ψ(i) +ki ⟩ , +(5) +with ki = 0 representing the ground state. A d-fold degen- +eracy in the ground states can be imposed via tuning h to +specific values, denoted by h′ where typically h′ ∼ J⊥. At +this point, we choose Eg such that ground state energy of +HR with h = h′ vanishes. For a fixed L, there may be multi- +ple values of h′ resulting in a d-fold degenerate ground state. +For each h′, we relabel the states {|ψ(i) +ki ⟩} such that +E(i) +ki = 0 ∀ ki = 0, 1, 2, . . . , d − 1, +E(i) +ki > 0 ∀ ki = d, d + 2, . . . , 2L − 1, +(6) +and +denote +the +d-fold +degenerate +ground +states +by +{|ψ(i) +0 ⟩ , · · · , |ψ(i) +d−1⟩}. For each h′, the ground-state mani- +fold {|Ψl⟩} of HR is dN-fold degenerate, constituting the +low-energy subspace S ⊂ H, with |Ψl⟩ having the form +|Ψl⟩ = +N +� +i=1 +|ψ(i) +ki ⟩ . +(7) +Here, ki = 0, 1, · · · , d − 1, and l = 0, 1, 2, · · · , dN − 1 +labels the ground state manifold1. We will find it useful to +split the Hilbert space into two subspaces, one is spanned +by |Ψl⟩ (which we will term as low energy sector) and the +other (which we will term as high energy sector) orthogonal +to this space. We can also define a projector onto the low +energy sector as +Pg = +� +l +|Ψl⟩ ⟨Ψl| +(8) +and the projector on the high energy sector is Pe ≡ I − Pg. +Let us now rewrite the system Hamiltonian H (Eq. (1)) as +1 +J⊥ +H = H0 + 1 +J⊥ +H′, +(9) +where +H0 = 1 +4 +N +� +i=1 +L +� +j=1 +⃗σi,j.⃗σi,j+1 − h′ +2J⊥ +N +� +i=1 +L +� +j=1 +σz +i,j − NEg, +(10) +which is HR at h = h′, and +H′ = J|| +4 +N +� +i=1 +L +� +j=1 +⃗σi,j.⃗σi+1,j + J1 +d +4 +N +� +i=1 +L +� +j=1 +⃗σi,j+1.⃗σi+1,j ++ J2 +d +4 +N +� +i=1 +L +� +j=1 +⃗σi,j.⃗σi+1,j+1 − ∆h +2 +N +� +i=1 +L +� +j=1 +σz +i,j, +(11) +with ∆h = h − h′. Clearly, ground states of H0 have a dN- +fold degeneracy (see Eq. (7)), which is lifted by the pertur- +bation H′. For ∆h, J||, J1,2 +d +≪ J⊥, this leads to an effective +Hamiltonian operating in the low-energy subspace S, where +energy-eigenvalues of the n-th order (n = 1, 2, ...) effective +Hamiltonian in S provides the n-th order energy corrections +to the unperturbed states of low energy. In this paper, we fo- +cus only on the first-order (n = 1) effective Hamiltonians, +which can be obtained as +˜H = +dN−1 +� +l,l′=0 +⟨Ψl| H′ |Ψl′⟩ |Ψl⟩ ⟨Ψl′| . +(12) +Note here that the first-order effective Hamiltonian (12) rep- +resents a system of one lattice dimension with N-sites, where +each lattice site has d degrees of freedom. +In this paper, we focus on a subset of cases with d = 2 by +choosing h′ appropriately. The effective degrees of freedom +in this case is like a spin half system. The effective Hamilto- +nian will be built out of operators acting on this low energy +spin-1/2 degree of freedom. While we can use the defini- +tion (12) to find the effective Hamiltonian, we find it more +instructive to work out the symmetries of the system which +will then constrain the structure of the effective Hamiltonian. +1 Note that l can be identified as the decimal equivalent of the string +k1k2k3 · · · kN in base d. + +4 +The detailed structure of the effective Hamiltonian can be +subsequently worked out. We denote the doubly-degenerate +ground states of HRi at h = h′ by |ψ(i) +0 ⟩ and |ψ(i) +1 ⟩. Since +[Zi, HRi] = 0, with Zi = �L +j=1 σz +i,j defined on the rung i, +Zi |ψ(i) +k ⟩ = mk |ψ(i) +k ⟩ , +(13) +k = 0, 1, where mk are the eigenvalues of Zi corresponding +to the eigenvectors |ψ(i) +k ⟩. Note here that Zi is the generator +of the z-rotation on the Hilbert space of the ith rung. We now +propose the following. +■ Proposition 1. If m0 ̸= m1, ˜H is z-rotationally invariant. +Proof. Consider the operator +Z = +N +� +i=1 +Jz +i , +(14) +with +Jz +i = a + bZi, +(15) +where Z is a symmetry of the original system-Hamiltonian +H, i.e., +[Z, H0] = [Z, H′] = 0, +(16) +and a and b are real constants that can be chosen according +to convenience. Since m0 ̸= m1, we choose a = (m1 + +m0)/(m1 − m0) and b = 2/(m0 − m1) such that +Jz +i |ψ(i) +0 ⟩ = |ψ(i) +0 ⟩ , +Jz +i |ψ(i) +1 ⟩ = − |ψ(i) +1 ⟩ . +(17) +Therefore, the action of Jz +i on the rung-subspace spanned +by |ψ(i) +0(1)⟩ is similar to that of an effective Pauli-z operator2, +which can be defined as +τ z +i = |ψ(i) +0 ⟩ ⟨ψ(i) +0 | − |ψ(i) +1 ⟩ ⟨ψ(i) +1 | . +(18) +Let us further define the operator ηz = �N +i=1 τ z +i , and note +that +ηz |Ψl⟩ = Z |Ψl⟩ = ml |Ψl⟩ , +(19) +where ml = �N +i=1(−1)ki (see Eq. (7)). Since [ηz, H′] = +0, ηz can’t change the eigenvalue under H′ and hence +⟨Ψl| H′ |Ψl′⟩ = 0 if ml ̸= ml′. Since this vanishes for +ml ̸= m′ +l, we have +[ηz, ˜H] = +2N−1 +� +l,l′=0 +(ml − ml′) |Ψl⟩ ⟨Ψl| H′ |Ψl′⟩ ⟨Ψl′| += 0, +(20) +implying that ηz is a symmetry of the effective 1D Hamilto- +nian ˜H. +2 Note that the assumption m0 ̸= m1 is crucial for the Jz +i s to mimic the +action of Pauli-z operators +Since the first order effective Hamiltonian is guaranteed +to be at most nearest-neighbour in the effective spins, the +following corollary can be written. +⋄ Corollary 1.1 The most general form of z-rotationally in- +variant ˜H is a nearest-neighbor XXZ model [93] in a mag- +netic field, given by +˜H = +N +� +i=1 +� +˜Jxy +� +τ x +i τ x +i+1 + τ y +i τ y +i+1 +� ++ ˜Jzzτ z +i τ z +i+1 +� ++˜h +N +� +i=1 +τ z +i + C, +(21) +where C is an irrelevant additive constant. +Here, we have defined +τ x +i = |ψ(i) +0 ⟩ ⟨ψ(i) +1 | + |ψ(i) +1 ⟩ ⟨ψ(i) +0 | , +τ y +i = −i +� +|ψ(i) +0 ⟩ ⟨ψ(i) +1 | − |ψ(i) +1 ⟩ ⟨ψ(i) +0 | +� +(22) +as the effective Pauli-x and y operators defined on the rung +subspace spanned by |ψ(i) +0,1⟩ (see Eq. (18)). The coefficients +˜Jxy, ˜Jz, and ˜h can be determined as functions of the cou- +plings in the perturbation Hamiltonian H′, namely, J||, J1,2 +d , +and ∆h. The exact forms of the functions depend on the +forms of the doubly-degenerate ground states |ψ(i) +0,1⟩, and +subsequently on the specific point h = h′ where the per- +turbation calculation is carried out. +Note 1. In the low-energy sector, it is sufficient to consider +the effective Hamiltonian (21) as a model for N spin-1/2 +particles, where the Pauli matrices on the Hilbert space of +the spin-1/2 particle are given by τ x,y,z. In this picture, we +relabel |ψ(i) +0,1⟩ as +|0i⟩ = |ψ(i) +0 ⟩ , |1i⟩ = |ψ(i) +1 ⟩ , +(23) +where {|0i⟩ , |1i⟩} form τ z eigenbasis of the spin-1/2 par- +ticle at the ith site of the 1D lattice of size N. +Eigen- +states of the 1D model, |˜Φl⟩, can be written in terms of the +{|0i⟩ , |1i⟩} basis of the spins in the system. +Note 2. A couple of comments about the notations used in +the rest of the paper is in order here. +(a) Any state |˜Φ⟩ satisfying Pg |˜Φ⟩ = |˜Φ⟩ is essentially a +state in the 1D XXZ model, which can be written in +terms of |0⟩ and |1⟩. We will typically put ˜ on such +states to remind us that it is an XXZ state. Note that +the projection of a state |Φ⟩ onto the low energy sector +is Pg |Φ⟩. +(b) Similarly, any operator ˜A satisfying Pg ˜APg = ˜A is +essentially an operator defined on the Hilbert space of +the 1D XXZ model, which can be written in terms of +τ x,y,z. We will typically put ˜ on such operator to re- +mind us that it is an XXZ operator. This also applies +to the state ρ, which is a Hermitian operator. Note that +the projection of an operator A onto the low energy +sector is PgAPg. + +5 +C. +Merit of the mapping +We now ask how well the low-energy spectrum of the 2D +model is captured by the 1D effective Hamiltonian ˜H. In +Sec. II B, we see that for h = h′ and ∆h = J|| = J1,2 +d += 0, +the Hilbert space of the system naturally splits into two sub- +spaces - (a) the low-energy subspace S, spanned by the or- +thonormal degenerate states {|Ψl⟩}, where each of the state +|Ψl⟩ has zero energy w.r.t. H0, and (b) the excited subspace +S′, spanned by the excited eigenstates {|Ψ′ +α⟩} of H0 with +energy E′ +α ≥ O(J⊥), where ⟨Ψ′ +α|Ψl⟩ = 0 ∀l, α. The to- +tal state-space of the system is given by S ⊕ S′. We are +now interested in the spectrum of the system slightly away +from the point h = h′ in the space of coupling constants i.e +∆h, J||, J1,2 +d +≪ J⊥. Perturbation theory in the parameter +1/J⊥ gives the following structure for the energy eigenval- +ues and eigenvectors of the 2D model: +|Φl⟩ = |Φ0 +l ⟩ + J−1 +⊥ |Φ1 +l ⟩ + J−2 +⊥ |Φ2 +l ⟩ + . . . +(24) +El = E(1) +l ++ J−1 +⊥ E(2) +l ++ . . . +(25) +where |Φ0 +l ⟩ and E(1) +l +are respectively the non-degenerate3 +eigenvectors and eigenvalues of ˜H. This makes it explicit +that the low lying spectrum of the 2D Hamiltonian are very +close (upto 1/J⊥ corrections) to the XXZ spectrum. To see +that the eigenstates also match, note that the first order per- +turbation theory gives the first-order state corrections |Φ1 +l ⟩ to +have overlaps with the excited state manifold {|Ψ′ +α⟩} having +energy of O(J⊥), implying that |Φ1 +l ⟩ is a small correction to +|Φl⟩ in Eq. (24) due to the suppression factor J−1 +⊥ . One may +quantify the small corrections via the distance between |Φl⟩ +and |Φ0 +l ⟩, which can be quantified by a distance metric, eg. +the trace-distance [2] (see Appendix C for definition). +In the following, we describe four instances where the +mapping described in Sec. II B is particularly useful. +1. +States made of non-degenerate eigenstates +A state in the Hilbert space of the 2D model, which can be +written in terms of the non-degenerate eigenstates of H as +ρ = +� +l,l′ +cl,l′ |Φl⟩ ⟨Φl′| , +(26) +with � +l |cl,l|2 = 1, can be approximated as +ρ ≈ +� +l,l′ +cl,l′ |Φ0 +l ⟩ ⟨Φ0 +l′| = ˜ρ, +(27) +as long as the overall accumulated error due to the first order +corrections in |Φl⟩ ⟨Φl′| is ≤ O(J−1 +⊥ ) 4. We remind our- +selves here that ˜ρ is a density matrix entirely in the low en- +ergy sector (see Note 2, Sec. II B). As an example, consider +3 Degenerate eigenstates of ˜H would result in degeneracy in the energy +levels of the 2D model, which can be lifted by second, or higher order +perturbation theory. +4 This can be ensured by keeping the number of terms in ρ small. +the state +ρ = 1 +2 +� +|Φ0⟩ ⟨Φ0| + |Φ2N−1⟩ ⟨Φ2N−1| ++ |Φ0⟩ ⟨Φ2N−1| + |Φ2N−1⟩ ⟨Φ0| +� +(28) +of the 2D model, where |Φ0⟩ and |Φ2N−1⟩ are non- +degenerate eigenstates of H5. +This state can be approxi- +mated as +ρ ≈ ˜ρ = 1 +2 +� +|Φ0 +0⟩ ⟨Φ0 +0| + |Φ0 +2N−1⟩ ⟨Φ0 +2N−1| ++ |Φ0 +0⟩ ⟨Φ0 +2N−1| + |Φ0 +2N−1⟩ ⟨Φ0 +0| +� +(29) +up to the correction O(J−1 +⊥ ). Since |Φ0 +0⟩ = ⊗i |ψ(i) +0 ⟩, and +|Φ0 +2N−1⟩ = ⊗i |ψ(i) +1 ⟩, ˜ρ can be written as ˜ρ = |˜ΦGHZ⟩ ⟨˜ΦGHZ| +in the space of the 1D effective model, where +|˜ΦGHZ⟩ = +1 +√ +2 +� +|0⊗N⟩ + |1⊗N⟩ +� +(30) +is the N-qubit GHZ state [99, 100]. Note that for ferro- +magnetic couplings ˜Jxy, ˜Jzz < 0 and for ˜h > 0, |0⟩⊗N +and |1⟩⊗N represent the ground state and the highest energy +eigenstate of ˜H respectively. +2. +Equal mixture of degenerate states +Let us now assume that the energy level E(1) +l=l′ of ˜H is n- +fold degenerate, where the eigenstates of ˜H corresponding to +E(1) +l′ +are |Φ0 +l′,m⟩, with the index m running from 1 to n. This +leads to an n-fold degeneracy of the energy level El′ of H +with degenerate eigenstates |Φl′,m⟩ in first order perturbation +theory. Although |Φ0 +l′,m⟩ does not approximate the states +|Φl′,m⟩ up to small corrections, the total contribution of the +degenerate manifold {|Φl′,m⟩} in the density matrix at the +low-energy sector is approximated by the contribution of the +{|Φ0 +l′,m⟩}. More precisely, the contribution is represented by +an equal mixture of the degenerate states {|Φl′,m⟩} in the +Hilbert space of the 2D model, given by +ρ = 1 +n +n +� +m=1 +|Φl′,m⟩ ⟨|Φl′,m⟩| , +(31) +which is approximated, up to corrections of O(J−1 +⊥ ), by +an equal mixture of the corresponding degenerate states +{|Φ0 +l′,m⟩}, i.e., +ρ ≈ ˜ρ = 1 +n +n +� +m=1 +|Φ0 +l′,m⟩ ⟨|Φ0 +l′,m⟩| . +(32) +5 Here we work in a subspace of the space of coupling constants in H +where |Φ0⟩ and |Φ2N −1⟩ are non-degenerate. + +6 +3. +Thermal states +Consider the thermal state, +ρT = +� +α e−βEα |Φα⟩ ⟨Φα| +� +α e−βEα +, +(33) +of the 2D system at an absolute temperature kBT, with kB +being the Boltzmann constant, and β = 1/kBT, where +{|Φα⟩} is the full spectrum of H. +For low temperatures +such that kBT ≪ O(J⊥) (or, 1 ≪ βJ⊥), the state is well- +approximated by the low-energy spectrum as +ρT ≈ +�2N−1 +l=0 +e−βEl |Φl⟩ ⟨Φl| +�2N−1 +l=0 +e−βEl +≈ +�2N−1 +l=0 +e−βE(1) +l +|Φ0 +l ⟩ ⟨Φ0 +l | +�2N−1 +l=0 +e−βE(1) +l += ˜ρ, +(34) +up to correction of O(1/J⊥). +In the first approximation +above, we have truncated the sum to states with E ≪ O(J⊥) +and in the second, we have approximated both the states and +the energies by their XXZ counterparts. +4. +Closed system evolution +The mapping of the 2D model to the effective 1D XXZ +model also allows one to investigate the time-evolution ρ(t) +of the closed system as long as the initial state ρ(0) is of the +type described in Eq. (26). Such a state in the Hilbert space +of the 2D model evolves as +ρ(t) = +� +l,l′ +clc⋆ +l′e−i(El−El′)t |Φl⟩ ⟨Φl′| +≈ +� +l,l′ +clc⋆ +l′e−i(E(1) +l +−E(1) +l′ )t |Φ0 +l ⟩ ⟨Φ0 +l′| += ˜ρ, +(35) +up to a correction of O(1/J⊥) as long as t ≤ J⊥. +Note that all the density matrices which we described +above, have the property that +ρ ≈ PgρPg, +(36) +implying that these density matrices are approximately the +same as their projection onto the low energy sector. +In this paper, we focus on a number of typical Hermitian +operators and entanglement in the 2D model, and ask to what +extent their quantitative as well as qualitative behaviour are +approximated by the 1D XXZ model. For this investiga- +tion, it is important to work out the effective couplings of +˜H in terms of the original couplings in H, which depends +explicitly on the boundary conditions of the 2D lattice. In +this respect, one may consider four possible cases – (1) PBC +along both rungs and legs, (2) PBC along the rugs and OBC +along the legs, (3) OBC along the rungs and PBC along the +legs, and (4) OBC along both rungs and legs, all of which we +discuss in detail in Sec. III. Readers interested in the proper- +ties of expectation values of relevant operators and entangle- +ment can skip Sec. III, and proceed directly to Secs. IV and +V, where these results are discussed. +III. +WORKING OUT THE EFFECTIVE HAMILTONIAN +In this section, we explicitly work out the low-energy ef- +fective Hamiltonian (21) for different boundary conditions +along the legs and rungs. +For brevity, we have assumed +J⊥ = 1 only for this section. +A. +Periodic boundary condition along the rungs +For the case of PBC along the rungs, given by ⃗σi,L+1 ≡ +⃗σi,1, we present below analytical forms of the ground states +|ψ(i) +k ⟩ , k = 0, 1 of the rung Hamiltonian HRi. This in turn +helps us to get analytic expressions for the couplings of ef- +fective Hamiltonian. +From the Hamiltonian HRi given in Eq. (4) (recall that we +have set J⊥ = 1), it is obvious that for large h, the minimum +energy state is given by all up spins +|ψ(i) +0 ⟩ = ⊗L +j=1 |0i,j⟩ , +(37) +having energy L +� 1 +4 − h +2 +� +− Eg. +The next higher energy +states at large h are given by one spin flip - since this can +occur at any of the L rung sites, there are L such states. +Since the Hamiltonian HRi is translationally invariant, we +can switch to momentum basis via +|q⟩ = +1 +√ +L +L +� +j=1 +e2iqjπ/L σx +i,j |ψ(i) +0 ⟩ , +(38) +where q = 0, 1, 2, . . . , L − 1. These states are energy eigen- +states for all values of h, i.e +HRi |q⟩ = +� +cos +�2πq +L +� ++ L +�1 +4 − h +2 +� ++ h − 1 − Eg +� +|q⟩ . +(39) +For even L6, the minimum energy eigenstate corresponds to +q = L/2, with energy eigenvalue L +� 1 +4 − h +2 +� ++ h − 2 − Eg. +This state becomes degenerate with the minimum energy +state at +h′ = 2, +(40) +while the state itself is given by +|ψ(i) +1 ⟩ = |q = L/2⟩ = +1 +√ +L +L +� +j=1 +(−1)j |j⟩ . +(41) +6 For odd L, the minimum energy states in this sector correspond to (L ± +1)/2. This leads to a three-fold degeneracy of ground states of the rung +at h = h′, which is beyond the scope of our consideration in this paper. + +7 +We must comment here that while it is reasonable to expect +that the first excited state at large h becomes degenerate with +the minimum energy state as we dial down h, it is not guar- +anteed to be. However, our numerical investigation verifies +this expectation to be correct. +In order to compute parameters of the effective 1d XXZ +˜Jxy, ˜Jzz, ˜h (see Eq. (21)), we will match the matrix elements +of H and ˜H in the following states: +|Ψ0⟩ = ⊗N +i=1 |ψ(i) +0 ⟩ , +|Ψj⟩ = |ψ(j) +1 ⟩ ⊗N +i=1 +i̸=j |ψ(i) +0 ⟩ , +|Ψj,j+1⟩ = |ψ(j) +1 ⟩ ⊗ |ψ(j+1) +1 +⟩ ⊗N +i=1 +i̸=j,j+1 |ψ(i) +0 ⟩ . +(42) +By matching the matrix elements, the expressions for ˜Jxy, +˜Jzz, and ˜h can be obtained in terms of the original couplings +of the model for PBC along the legs (⃗σN+1,j ≡ ⃗σ1,j) as +˜Jxy = (J|| − J1 +d − J2 +d)/4, +˜Jzz = (J|| + J1 +d + J2 +d)/4L, +˜h = +� +(L − 1)(J|| + J1 +d + J2 +d) − L∆h +� +/2L. +(43) +Note that ˜Jxy, ˜Jzz, and ˜h are dimensionless numbers, since +the J−1 +⊥ factor, having value 1, is implicit in all expressions. +Similar mapping and analysis can also be done when the +legs obey open boundary condition (OBC). In this case, the +effective 1D model is given by +˜HOBC = ˜H + ˜h′(τ z +1 + τ z +N), +(44) +with ˜H as in Eq. (21) with its effective couplings given in +Eq. (43), the edge-inhomogeneity in the field strength as +˜h′ = −(L − 1)(J|| + J1 +d + J2 +d)/4L. +(45) +Note that for both PBC and OBC along the legs, in the +limit L → ∞, ˜Jzz → 0, implying that the 2D model +is mapped to an effective 1D XX model in a transverse +magnetic field [101–104], which can be exactly solved via +fermionization using the Jordan-Wigner transformation, fol- +lowed by a Fourier transformation. +B. +Open boundary condition along the rungs +Open boundary condition (OBC) along the rungs repre- +sents a geometry of the lattice that is different from the one +discussed in Sec. III A. Exact calculation of the degenerate +ground states and the strength of the magnetic field at which +the degeneracy occurs is difficult for arbitrary L. +How- +ever, our numerical analysis suggests that for open (periodic) +boundary condition along the rungs (legs), and for arbitrary +L, |ψ(i) +0 ⟩ is still given by Eq. (37), while |ψ(i) +1 ⟩ is found in +the ⟨Zi⟩ = L − 2 sector, and has the form +|ψ(i) +1 ⟩ = +L +� +j=1 +aj |j⟩ , +(46) +with aj ∈ R, and �L +j=1 a2 +j = 1. Note further that due to +Z2 symmetry, a1 = ±aL. Using Eqs. (37) and (46), the +effective couplings in Eq. (21) can be determined as +˜Jxy = [J|| + A0(J1 +d + J2 +d)]/4, +˜Jzz = [(4 − L + A1)J|| ++(L − 1 + A3 − 2A2)(J1 +d + J2 +d)]/16, +˜h = [(L − A1)J|| + (L − 1 − A3)(J1 +d + J2 +d) − 2∆h]/8, +(47) +with +A0 = +L−1 +� +j=1 +ajaj+1, +(48) +A1 = L +L +� +j=1 +a4 +j + (2L − 8) +L +� +j=1 +� +kACJH +icbZDLSsNAFIYn9V5vVZdugkVwY0mkqOCm2I07K9hWaGKZTE/aoZNJmDkRS+jDuPFV3Ljwgs3PovTy0KrPwz8fOczpw/SATX6DifVm5ufmFxaXklv7 +q2vrFZ2Npu6DhVDOosFrG6CagGwSXUkaOAm0QBjQIBzaBfHdWbd6A0j+U1DhLwI9qVPOSMokHtwpknpCcgxFbv1ksUj6CdeQj3mNXOq8Ph4Sy9HFP8W +4P/Xah6JScsey/xp2aIpmq1i68eZ2YpRFIZIJq3XKdBP2MKuRMwDvpRoSyvq0Cy1jJY1A+9n4yKG9b0jHDmNlnkR7TH9OZDTSehAFpjOi2NOztRH8r9 +ZKMTz1My6TFEGyaIwFTbG9igxu8MVMBQDYyhT3PzVZj2qKEOTa96E4M6e/Nc0jkrucal8VS5WzqdxLJNdskcOiEtOSIVckBqpE0YeyBN5Ia/Wo/VsvV +sfk9acNZ3ZIb9kfX0Dm2SmpQ= +ln [h0 +PBC � h0 +OBC] +AB7H +icbVA9SwNBEJ2LXzF+RS1tFoNgFe4kqGXQxsIigpcEkiPsbfaSJXt7x+6cEJ+g42FIrb+IDv/jZvkCk18MPB4b4aZeWEqhUHX/XYKa+sbm1vF7dLO7t +7+QfnwqGmSTDPus0Qmuh1Sw6VQ3EeBkrdTzWkcSt4KR7czv/XEtRGJesRxyoOYDpSIBKNoJb8rFbnvlStu1Z2DrBIvJxXI0eiVv7r9hGUxV8gkNabjuS +kGE6pRMmnpW5meErZiA54x1JFY26CyfzYKTmzSp9EibalkMzV3xMTGhszjkPbGVMcmVvJv7ndTKMroOJUGmGXLHFoiTBMy+5z0heYM5dgSyrSwtx +I2pJoytPmUbAje8surpHlR9S6rtYdapX6Tx1GEziFc/DgCupwBw3wgYGAZ3iFN0c5L86787FoLTj5zDH8gfP5A0RIjlc=ln L +ACB3 +icbVBNS8NAEN3Ur1q/qh4FWSyCp5JIUY+levBYwX5AE8pmu2mXbjZhdyKUtDcv/hUvHhTx6l/w5r9x2+agrQ8GHu/NMDPjwXYNvfVm5ldW19I79Z2N +re2d0r7h80dZQoyho0EpFq+0QzwSVrAfB2rFiJPQFa/nD6nfemBK80jewyhmXkj6kgecEjBSt3jsColdwQLouIEiNL2ZpOPaeOIq3h+A1y2W7LI9A1 +4mTkZKEO9W/xyexFNQiaBCqJ1x7Fj8FKigFPBJgU30SwmdEj6rGOoJCHTXjr7Y4JPjdLDQaRMScAz9fdESkKtR6FvOkMCA73oTcX/vE4CwZWXchknwC +SdLwoSgSHC01BwjytGQYwMIVRxcyumA2LiABNdwYTgL68TJrnZeiXLmrlKq1LI48OkIn6Aw56BJV0S2qowai6BE9o1f0Zj1ZL9a79TFvzVnZzCH6A+ +vzB6bZmdU= +ln + D +|B| +� +AB7HicbVA9SwNBEJ2LXzF+RS1tFoNgFe4kqGXQxsIigpcEkiPsbfaSJXt7x+6c +EJ+g42FIrb+IDv/jZvkCk18MPB4b4aZeWEqhUHX/XYKa+sbm1vF7dLO7t7+QfnwqGmSTDPus0Qmuh1Sw6VQ3EeBkrdTzWkcSt4KR7czv/XEtRGJesRxyoOYDpSIBKNoJb8rFbnvlStu1Z2DrBIvJxXI0eiVv7r9hGUxV8gkNabjuSkGE6pRMmnpW5meErZiA54x1JFY26CyfzYKTmzSp9EibalkMzV3xMTGhsz +jkPbGVMcmVvJv7ndTKMroOJUGmGXLHFoiTBMy+5z0heYM5dgSyrSwtxI2pJoytPmUbAje8surpHlR9S6rtYdapX6Tx1GEziFc/DgCupwBw3wgYGAZ3iFN0c5L86787FoLTj5zDH8gfP5A0RIjlc=ln L +AB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9mVol6EUi8eK9gPaJeSTbNtbDZkqxQtv0PXjwo4tX/481/Y9ruQVsfDzem2FmXhBzpo3rfju5tfWNza38dmF +nd2/oHh41NQyUYQ2iORStQOsKWeCNgwznLZjRXEUcNoKRrczv/VElWZSPJhxTP0IDwQLGcHGSs1JbXLjub1iyS27c6BV4mWkBnqveJXty9JElFhCMdadzw3Nn6KlWGE02mhm2gaYzLCA9qxVOCIaj+dXztFZ1bpo1AqW8Kgufp7IsWR1uMosJ0RNkO97M3E/7xOYsJrP2UiTgwVZLEoTDgyEs1eR32mKDF8bAkmitlbERlihYmxARVsCN7y6ukeVH2LsuV+0qpWsviyMJnMI5eHAFVbiDOjSAwCM8wyu8OdJ5cd6dj0VrzslmjuEPnM8fvO+Olw=|B| = 10 +AB7XicbVDLSgNBEOyNrxhfUY9eBoPgKexKfFyEC8eI5gHJEuYncwmY2ZnlplZIWzyD148KOLV/Hm3zhJ9qDRgoaiqpvuriDmTBvX/XJyK6tr6xv5zcL +W9s7uXnH/oKloghtEMmlagdYU84EbRhmOG3HiuIo4LQVjG5mfuRKs2kuDfjmPoRHgWMoKNlZqT2uT63O0VS27ZnQP9JV5GSpCh3it+dvuSJBEVhnCsdcdzY+OnWBlGOJ0WuomMSYjPKAdSwWOqPbT+bVTdGKVPgqlsiUMmqs/J1IcaT2OAtsZYTPUy95M/M/rJCa8lMm4sRQRaLwoQjI9HsdRnihLDx5Zgopi9FZEhVpgYG1DBhuAtv/yXNM/K3kW5clcpVWtZHk4gmM4BQ8uoQq3UIcGEHiAJ3iBV0c6z86b875ozTnZzCH8gvPxDcMDjps=|B| = 50 +AB7nicbVDLSgNBEOyNrxhfUY9eBoPgKeyKqBchxIvHCOYByRJmJ5NkyOzsMtMrhE0+wosHRbz6Pd78GyfJHjSxoKGo6qa7K4ilMOi6305ubX1jcyu/Xdj +Z3ds/KB4eNUyUaMbrLJKRbgXUcCkUr6NAyVux5jQMJG8Go7uZ3zi2ohIPeI45n5IB0r0BaNopeakOrn1XLdbLldw6ySryMlCBDrVv86vQiloRcIZPUmLbnxuinVKNgk8LncTwmLIRHfC2pYqG3Pjp/NwpObNKj/QjbUshmau/J1IaGjMOA9sZUhyaZW8m/ue1E+zf+KlQcYJcscWifiIJRmT2O+kJzRnKsSWUaWFvJWxINWVoEyrYELzl1dJ46LsXZUvHy5LlWoWRx5O4BTOwYNrqMA91KAODEbwDK/w5sTOi/PufCxac042cwx/4Hz+ACyEjtE=|B| = 100 +AB7nicbVDLSgNBEOzxGeMr6tHLYBA8hd0QHxchxIvHCOYByRJmJ7PJkNnZWZWCJt8hBcPinj1e7z5N06SPWhiQUNR1U13lx8Lro3jfKO19Y3Nre3cTn5 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+cmS5idJIhs7PLTK8YlvyFw+KePVvPk3TpI9aGJBQ1HVTXdXEth0HW/nZXVtfWNzdxWfntnd2+/cHDYMFGiOdR5JCPdCpgBKRTUaCEVqyBhYGEZjC6mfrNR9BGROoexzH4IRso0RecoZUeOghPmJaCs0m3UHTL7gx0mXgZKZIMtW7hq9OLeBKCQi6ZMW3Pj +dFPmUbBJUzyncRAzPiIDaBtqWIhGD+dXTyhp1bp0X6kbSmkM/X3RMpCY8ZhYDtDhkOz6E3F/7x2gv0rPxUqThAUny/qJ5JiRKfv057QwFGOLWFcC3sr5UOmGUcbUt6G4C2+vEwa52Xvoly5qxSr1kcOXJMTkiJeOSVMktqZE64USRZ/JK3hzjvDjvzse8dcXJZ +o7IHzifP1MwkLM=(b) +AB9HicbVBNS8NAEN3Ur1q/qh69BIvgqSRS1GPRi8cK9gPaUDabSbt0s4m7k2IJ/R1ePCji1R/jzX/jts1BWx8MPN6bYWaen +wiu0XG+rcLa+sbmVnG7tLO7t39QPjxq6ThVDJosFrHq+FSD4BKayFAJ1FAI19A2x/dzvz2GJTmsXzASQJeRAeSh5xRNJLXQ3jCLOSIEz75YpTdeawV4mbkwrJ0eiXv3pBzNIJDJBte6ToJeRhVyJmBa6qUaEspGdABdQyWNQHvZ/OipfWaUwA5jZUqiPVd/T2Q0noS+aYzojUy95M/M/rphexmXSYog2WJRmAobY3uWgB1wBQzFxBDKFDe32mxIFWVociqZENzl1dJ6LqXlZr97VK/SaPo +0hOyCk5Jy65InVyRxqkSRh5JM/klbxZY+vFerc+Fq0FK585Jn9gf4AjDOSng=fitted +AB+XicbVDLSgMxFM3UV62vUZdugkVwVWZE1GXRjcsK9gHtUDLpbRuayQzJnWIZ+iduXCji1j9x59+YtrPQ1gOBwzn 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(a) Variation of ln (h′ +PBC − h′ +OBC) as a function of ln L. The numerical data is obtained by considering 2 ≤ L ≤ 102, and is fitted +to Eq. (53) with a = 4.88251, and the power of L obtained as ∼ −1.99684 from the fitted curve. (b) Variation of ln [D/|B|] as a function +of ln L with different values of |B|. The numerical data is obtained by considering 102 ≤ L ≤ 5 × 103, and is fitted to Eq. (54), where the +power of L is obtained as ∼ −1.004 from the fitted curve. All quantities plotted in all figures are dimensionless. +the lattice. This naturally includes the bulk B of the rungs at +the middle of the lattice, the size |B| of which is determined +by the lattice sites included in it. The reduced state ρB of +the bulk of a rung can be determined by tracing out all spins +from |ψ(i) +1 ⟩ except the ones in the bulk. To quantitatively +estimate how ρOBC +B +corresponding to OBC along rungs ap- +proaches ρPBC +B +when the rungs form closed chains, we focus +on the trace distance [2] (see Appendix C for the definition), +D, between ρPBC +B +and ρOBC +B +. Our numerical results indicate +that D varies with L as +D ∼ |B| +L , +(54) +and tends to 0 for L → ∞. Here, second and higher powers +of 1/L are ignored. See Fig. 2(b) for a demonstration. This +result indicate that for L → ∞, a rung with OBC can be +reliably mimicked by a rung with PBC as long as the inves- +tigation is confined in the bulk of the chain. +2. +Mapping for odd L with vanishing magnetic field +We now discuss a special case where L is odd, and no +magnetic field is present in the system, i.e., h = 0. In this +situation, each rung i of the system has a doubly degenerate +ground state having the form +|ψ(i) +0 ⟩ = +� +j +ajPj +� +|0⟩⊗(L+1)/2 |1⟩⊗(L−1)/2� +, +(55) +with � +j a2 +j = 1, and +|ψ(i) +1 ⟩ = ⊗L +j=1σx +j |ψ(i) +0 ⟩ . +(56) +Here, {Pj} is the set of all possible permutations of L′ +(L−L′) spins at ground (excited) state, with L′ = (L−1)/2 +(L′ = (L + 1)/2) for |ψ(i) +0 ⟩ (|ψ(i) +1 ⟩). +Proposition 1 (see +Sec. II B) is valid in this case also, and calculation (see Ap- +AB9XicbVDJS +gNBEO1xjXGLevTSGARPYUbcjkEvHiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmPih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjM +w1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicglRJgUav8uUGCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1ZBN +rLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3Uj5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kcki +NyQhxySerkljRIk3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4AruS3A=˜� +AB8HicbVDJSgNBEK2JW4xb1KOXxiB4Cj +PidgwK4kWIYBZJhtDT6Uma9PQM3TViGPIVXjwo4tXP8ebf2FkOGn1Q8Hiviqp6QSKFQdf9cnILi0vLK/nVwtr6xuZWcXunbuJUM15jsYx1M6CGS6F4DQVK3kw0p1EgeSMYXI79xgPXRsTqDocJ9yPaUyIUjKV7tvIHzG7uhl1 +iW37E5A/hJvRkowQ7VT/Gx3Y5ZGXCGT1JiW5yboZ1SjYJKPCu3U8ISyAe3xlqWKRtz42eTgETmwSpeEsbalkEzUnxMZjYwZRoHtjCj2zbw3Fv/zWimG534mVJIiV2y6KEwlwZiMvydoTlDObSEMi3srYT1qaYMbUYFG4I3/ +JfUj8qe6flk9vjUuViFkce9mAfDsGDM6jANVShBgwieIXeHW08+y8Oe/T1pwzm9mFX3A+vgH615CKFM +AB8HicbVDJSgNBEO2JW4xb1KOXwSB4Cj +Pidgx68SJEMIskQ+jp1CRNunuG7hoxDPkKLx4U8ernePNv7CwHTXxQ8Hiviqp6YSK4Qc/7dnJLyura/n1wsbm1vZOcXevbuJUM6ixWMS6GVIDgiuoIUcBzUQDlaGARji4HvuNR9CGx+oehwkEkvYUjzijaKWHNsITZtXbUadY +8sreBO4i8WekRGaodopf7W7MUgkKmaDGtHwvwSCjGjkTMCq0UwMJZQPag5alikowQTY5eOQeWaXrRrG2pdCdqL8nMiqNGcrQdkqKfTPvjcX/vFaK0WQcZWkCIpNF0WpcDF2x9+7Xa6BoRhaQpnm9laX9amDG1GBRuCP/yIq +mflP3z8tndalyNYsjTw7ITkmPrkgFXJDqRGJHkmbySN0c7L8678zFtzTmzmX3yB87nDwoikJQ=PM +AB8XicbVBNS8NAEN3Ur1q/qh69BIvgqSTi17Eq +iBehgv3ANpTNdtIu3WzC7kQsof/CiwdFvPpvPlv3LY5aOuDgcd7M8zM82PBNTrOt5VbWFxaXsmvFtbWNza3its7dR0likGNRSJSTZ9qEFxCDTkKaMYKaOgLaPiDq7HfeASleSTvcRiDF9Ke5AFnFI30EZ4wvTi+nbUKZacsjOBPU/cjJRIhmqn ++NXuRiwJQSITVOuW68TopVQhZwJGhXaiIaZsQHvQMlTSELSXTi4e2QdG6dpBpExJtCfq74mUhloPQ90hT7etYbi/95rQSDcy/lMk4QJsuChJhY2SP37e7XAFDMTSEMsXNrTbrU0UZmpAKJgR39uV5Uj8qu6flk7vjUuUyiyNP9sg+OSQuOSMV +ckOqpEYkeSZvJI3S1sv1rv1MW3NWdnMLvkD6/MHhb2Q1Q=AFM +AB6HicbVDLSgNBEOz1GeMr6tHLYBA8hV3xdQx68ZiAeUCyhNlJbzJmdnaZmRXCk +i/w4kERr36SN/GSbIHTSxoKq6e4KEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6m/qtJ1Sax/LBjBP0IzqQPOSMGivVvV6p7FbcGcgy8XJShy1Xumr249ZGqE0TFCtO56bGD+jynAmcFLsphoTykZ0gB1LJY1Q+9ns0Ak5tUqfhLGyJQ2Zqb8nMhpPY4C2xlRM9S +L3lT8z+ukJrzxMy6T1KBk80VhKoiJyfRr0ucKmRFjSyhT3N5K2JAqyozNpmhD8BZfXibN84p3VbmsX5Srt3kcBTiGEzgD6hCvdQgwYwQHiGV3hzHp0X5935mLeuOPnMEfyB8/kDfd2Mvw=1 +AB6XicbVDLSgNBEOz1GeMr6tHLYBC8GHbF1zHoxWMU84BkCbOT3mTI7OwyMyuEJ +X/gxYMiXv0jb/6Nk2QPmljQUFR1090VJIJr47rfztLyuraemGjuLm1vbNb2tv6DhVDOsFrFqBVSj4BLrhuBrUQhjQKBzWB4O/GbT6g0j+WjGSXoR7QvecgZNVZ6OPW6pbJbcacgi8TLSRly1Lqlr04vZmE0jBtW57bmL8jCrDmcBxsZNqTCgb0j62LZU0Qu1n0vH5NgqPRLGypY0ZKr+nshopPUoCmxnRM1 +Az3sT8T+vnZrw2s+4TFKDks0WhakgJiaTt0mPK2RGjCyhTHF7K2EDqigzNpyiDcGbf3mRNM4q3mXl4v68XL3J4yjAIRzBCXhwBVW4gxrUgUEIz/AKb87QeXHenY9Z65KTzxzAHzifP+cijPY=�1 +AB63icbVDLSgNBEOyNrxhfUY9eBoPgxbgbfB2DXjxGMA9IljA7mU2GzMwuM7NCWPILXjwo4t +Uf8ubfOJvsQRMLGoqbrq7gpgzbVz32ymsrK6tbxQ3S1vbO7t75f2Dlo4SRWiTRDxSnQBrypmkTcMp51YUSwCTtvB+C7z209UaRbJRzOJqS/wULKQEWwy6cw7r/XLFbfqzoCWiZeTCuRo9MtfvUFEkGlIRxr3fXc2PgpVoYRTqelXqJpjMkYD2nXUokF1X46u3WKTqwyQGkbEmDZurviRQLrScisJ0Cm5Fe9DLxP6+bmPDGT5mME0MlmS +8KE45MhLH0YApSgyfWIKJYvZWREZYWJsPCUbgrf48jJp1areVfXy4aJSv83jKMIRHMpeHANdbiHBjSBwAie4RXeHOG8O/Ox7y14OQzh/AHzucPxO6Naw=�1/2 +AB6nicbVDLTgJBEOzF+IL9ehlIjHxhLvE15HoxSNGeSwIbNDAxNmZzczsyZkwyd48aAxXv +0ib/6NA+xBwUo6qVR1p7sriAXxnW/ndzK6tr6Rn6zsLW9s7tX3D9o6ChRDOsEpFqBVSj4BLrhuBrVghDQOBzWB0O/WbT6g0j+SjGcfoh3QgeZ8zaqz04J1VusWSW3ZnIMvEy0gJMtS6xa9OL2JiNIwQbVue25s/JQqw5nASaGTaIwpG9EBti2VNETtp7NTJ+TEKj3Sj5QtachM/T2R0lDrcRjYzpCaoV70puJ/Xjsx/Ws/5TJODEo2X9 +RPBDERmf5NelwhM2JsCWK21sJG1JFmbHpFGwI3uLy6RKXuX5Yv781L1JosjD0dwDKfgwRVU4Q5qUAcGA3iGV3hzhPivDsf89ack80cwh84nz9bS401/2 +AB8Hicb +VDLSgNBEOyNrxhfUY9eFoPgKeyKr2PQi8cI5iHJEmZnO8mQmdlZlYIS7CiwdFvPo53vwbJ8keNLGgoajqprsrTDjTxvO+ncLK6tr6RnGztLW9s7tX3j9o6j +hVFBs05rFqh0QjZxIbhmO7UQhESHVji6nfqtJ1SaxfLBjBMBlI1meUGCs9dg3jEWaDSa9c8areDO4y8XNSgRz1XvmrG8U0FSgN5UTrju8lJsiIMoxynJS +6qcaE0BEZYMdSQTqIJsdPHFPrBK5/VjZksadqb8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6J3en3bsQUsPHlhCqmL3VpUOiCDU2o5INwV98 +eZk0z6r+ZfXi/rxSu8njKMIRHMp+HAFNbiDOjSAgoBneIU3RzkvzrvzMW8tOPnMIfyB8/kDLMiQqw=˜g +FIG. 3. Phase diagram of the 1D XXZ model in a magnetic field +on the plane of (˜δ, ˜g) [95, 105], marking the antiferromagnetic +(AFM), ferromagnetic (FM), and paramagnetic (PM) phases and +the corresponding phase boundaries, where ˜δ = ˜Jzz/ ˜Jxy quanti- +fies the z-anisotropy, and ˜g = ˜h/ ˜Jxy. +pendix D) leads to an effective 1D model given by the Hamil- +tonian +˜H = +N +� +i=1 +� +˜Jxy +� +τ x +i τ x +i+1 + τ y +i τ y +i+1 +� ++ ˜Jzzτ z +i τ z +i+1 +� +. (57) +We demonstrate the important steps of the calculation in Ap- +pendix D, where we set J1 +d = J2 +d = 0 to keep the calculation +uncluttered. However, similar results can be obtained also +for non-zero perturbations J1 +d, J2 +d, where the overall form of +Eq. (57) remains unchanged. + +1.5 +1.0 +0.5 - +0.0 +-0.5 +-1.0 +-1.5 +-3 +-2 +-1 +0 +1 +2 +3 +49 +A +˜A +σα +i,j +(−1)j +√ +L τ α +i , for α = x, y +σz +i,j +L−1 +L Ii + 1 +Lτ z +i +�L +j=1 σz +i,j +(L − 1)Ii + τ z +i +�L +j=1 σx +i,j +0 +�L +j=1 σy +i,j +0 +⊗L +j=1σz +i,j +τ z +i +⊗L +j=1σx +i,j +1 +2(τ z +i − Ii) for L = 2 +0 for L > 2 +⊗L +j=1σy +i,j +1 +2(τ z +i − Ii) for L = 2 +0 for L > 2 +σx +i,1 ⊗ σz +i,2 +− 1 +√ +2τ x +i +σz +i,1 ⊗ σx +i,2 +1 +√ +2τ x +i +σy +i,1 ⊗ σz +i,2 +− 1 +√ +2τ y +i +σz +i,1 ⊗ σy +i,2 +1 +√ +2τ y +i +σx +i,1 ⊗ σy +i,2 +0 +σy +i,1 ⊗ σx +i,2 +0 +σα +i,j ⊗ σα +i+r,j′ +(−1)j+j′ +L +τ α +i ⊗ τ α +i+r for α = x, y +σz +i,j ⊗ σz +i+r,j′ +(L−1)2 +L2 +I + +1 +L2 τ z +i τ z +i+r + L−1 +L2 (τ z +i + τ z +i+r) +σx +i,j ⊗ σz +i+r,j′ +(−1)j +√ +L +� L−1 +L τ x +i + 1 +Lτ x +i τ z +i+r +� +σz +i,j ⊗ σx +i+r,j′ +(−1)j′ +√ +L +� L−1 +L τ x +i+r + 1 +Lτ z +i τ x +i+r +� +σy +i,j ⊗ σz +i+r,j′ +(−1)j +√ +L +� L−1 +L τ y +i + 1 +Lτ y +i τ z +i+r +� +σz +i,j ⊗ σz +i+r,j′ +(−1)j′ +√ +L +� L−1 +L τ y +i+r + 1 +Lτ z +i τ y +i+r +� +σx +i,j ⊗ σy +i+r,j′ +0 +σy +i,j ⊗ σx +i+r,j′ +0 +TABLE I. A list of typical observables on the ladder and the cor- +responding low-energy operators on the 1D effective XXZ model, +where PBC along both rungs and legs have been used for calcula- +tion. A subset of these operators for L = 2 can be found in [89]. +IV. +MAPPING OBSERVABLES +It has been seen in Sec. II that specific states in the low- +energy sector of the 2D model (1) is approximated, up to +corrections of O(1/J⊥), by the spectrum of the 1D effective +XXZ model. A natural question that arises at this point is +whether the low energy content of an observable (i.e., matrix +elements of Hermitian operators in the low energy subspace) +are captured by an observable defined entirely on the Hilbert +space of the 1D effective XXZ model?. We answer this +question affirmatively below. +To put the question in a more concrete mathematical foot- +ing, we focus on the matrix element ⟨Φl|A|Φl′⟩ of a Her- +mitian operator A on the space of the non-degenerate low +energy states of the 2D model, and ask whether there exists +a Hermitian operator ˜A on the space of the 1D XXZ model +such that +⟨Φl|A|Φl′⟩ ≈ ⟨Φ0 +l | ˜A|Φ0 +l′⟩, +(58) +Towards answering this question, we note that an alternative +definition for Pg would be Pg = � +l |Φ0 +l ⟩ ⟨Φ0 +l |, which is bet- +ter suited for subsequent discussion. Equipped with this, we +propose the following. +Proposition 2. Eq. (58) holds if ˜A = PgAPg. +Proof. To prove this, let us write +⟨Φl|A|Φl′⟩ = ⟨Φl|PgAPg|Φl′⟩ + ⟨Φl|PeAPg|Φl′⟩ ++ ⟨Φl|PgAPe|Φl′⟩ + ⟨Φl|PeAPe|Φl′⟩, (59) +and note that for the low-energy manifold of H, +Pg |Φl′⟩ = +� +l +|Φ0 +l ⟩ ⟨Φ0 +l |Φl′⟩ ≈ |Φ0 +l′⟩ , +(60) +since for l ̸= l′, ⟨Φ0 +l |Φl′⟩ → 0 due to Eq. (24) and orthonor- +mality of |Φ0 +l ⟩. Also note that Pe |Φl′⟩ ≈ Pe |Φ0 +l′⟩ = 0. +Therefore, +⟨Φl|A|Φl′⟩ ≈ ⟨Φl|PgAPg|Φl′⟩ = ⟨Φ0 +l | ˜A|Φ0 +l′⟩, +(61) +where ˜A = PgAPg. Hence proved. +We point out here that although the proof is presented for +non-degenerate eigenstates, it can be straightforwardly ex- +tended to the states constructed out of the non-degenerate +eigenstates discussed in Sec. II C. +We now focus on a number of relevant operators A on the +space of the 2D model, and determine the corresponding op- +erators ˜A for the 1D effective model. We assume PBC along +both rungs and legs for demonstration. However, the pre- +scription can also be applied to other boundary conditions +along legs and rungs. We consider mainly two types of op- +erators – (a) operators A having support only on one rung, +say, i, where the corresponding operator ˜Aα +i = PgAiPg is a +2 × 2 matrix in the {|0⟩ , |1⟩} basis, and has support on only +the ith lattice site of the 1D XXZ model, and (2) operators +of the form Ai,i′ = Ai ⊗ Ai′ having support on two dif- +ferent rungs, i, and i′, where the mapped operator ˜Ai,i′ is a +4×4 Hermitian matrix in the basis {|00⟩ , |01⟩ , |10⟩ , |11⟩} +on the spins (i, i′) in the 1D XXZ spin-chain. To determine +the low-energy component of operators of the latter type, we +note the following. +Proposition 3. For an operator on the space of the 2D model +having the form Ai,i′ = Ai⊗Ai′ with Ai and Ai′ having sup- +port only on rungs i and i′ respectively, ˜Ai,i′ = PgAi,i′Pg = +˜Ai ⊗ ˜Ai′, where ˜Ai = PgAiPg, and ˜Ai′ = PgAi′Pg. +Proof. To prove this, we write Pg = ⊗N +i=1P (i) +g , with P (i) +g += +�1 +ki=0 |ψ(i) +ki ⟩ ⟨ψ(i) +ki |, where we use notations introduced in +Eq. (7). This leads to +˜Ai,i′ = PgAi,i′Pg += [P (i) +g +⊗ P (i′) +g +](Ai ⊗ Ai′)[P (i) +g +⊗ P (i′) +g +] += [P (i) +g AiP (i) +g ] ⊗ [P (i′) +g +Ai′P (i′) +g +] += ˜Ai ⊗ ˜Ai′. +(62) + +10 +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjkEvHiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmP +ih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4 +SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicglRJgUav8uUGCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1 +ZBNrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3Uj +5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kckiNyQhxySerkljRIk3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4A +ruS3A=˜� +AB8HicbVDLSgNBE +OyNrxhfUY9eFoPgKeyKr2PQi8cI5iHJEmZnO8mQmdlZl +YIS7CiwdFvPo53vwbJ8keNLGgoajqprsrTDjTxvO+ncLK +6tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7 +UQhESHVji6nfqtJ1SaxfLBjBMBlI1meUGCs9dg3jEWa +DSa9c8areDO4y8XNSgRz1XvmrG8U0FSgN5UTrju8lJsiI +MoxynJS6qcaE0BEZYMdSQTqIJsdPHFPrBK5/VjZksadqb +8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6 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+K2n/ixb/ixYMi4s1/47aN4NfAwsP7zjA7b5BIYdB1P5yZ2bn5hcXCUnF5ZXVtvbSx2TRxqjk0eCxjfRMwA1IoaKBACTeJBhYFEq6D27Oxf30H2ohYXeEgXbEekqEgjO0UqdU9SWEOPQlUz0JtO7rCex/CT4K2YWsPsoNX4teH4edUtmtuJOif8HLoUzyuiU3v1uzNMIFHLJjGl5boLtjGkUXMKo6KcGEsZv +WQ9aFhWLwLSzyX0jumuVLg1jbZ9COlG/T2QsMmYQBbYzYtg3v72x+J/XSjE8aWdCJSmC4tNFYSopxnQcFu0KDRzlwALjWti/Ut5nmnG0kRZtCN7vk/9C86DiVStHl4fl2mkeR4Fskx2yRzxyTGrknFyQBuHknjySZ/LiPDhPzqvzNm2dcfKZLfKjnI9PfCWj5w=���hAi � h ˜Ai +��� +AFM +PM +FM +AFM +PM +FM +AFM +PM +FM +AB+XicbZDJSgNBEIZr4hbjNurRS2MQPIUZcTtGvXiMYBbIDKGnU0ma9PQM3T2BMORNvHhQxKtv4s23sbMcN +PpDw8dfVT1H6WCa+N5X05hZXVtfaO4Wdra3tndc/cPGjrJFM6S0SiWhHVKLjEuFGYCtVSONIYDMa3k3rzREqzRP5 +aMYphjHtS97jBprdVw3EFT2BZKbQM2g45a9ijcT+Qv+AsqwUK3jfgbdhGUxSsME1brte6kJc6oMZwInpSDTmFI2pH1 +sW5Q0Rh3ms8sn5MQ6XdJLlH3SkJn7cyKnsdbjOLKdMTUDvVybmv/V2pnpXYc5l2lmUL5ol4miEnINAbS5QqZEWMLlC +lubyVsQBVlxoZVsiH4y1/+C42zin9ZuXg4L1dvF3EU4QiO4R8uIq3EMN6sBgBE/wAq9O7jw7b87vLXgLGYO4Zecj +28ip5NahAi +ACAXicbVDLSsNAFJ34rPUVdSO4GSyCq5KIr2X +VjcsK9gFNKJPpTt0MgkzE6GEuvFX3LhQxK1/4c6/cZpmoa0HLhzOuZd7wkSzpR2nG9rYXFpeW1tFZe39jc2rZ3dpsqTiWFBo15LNsBUcCZgIZmkM7kUCigEMrGN5M/NYDSMVica9HCfgR6QsWMkq0kbr2vseJ6HPAnma8B9nV2JO50LUrT +tXJgeJW5AKlDv2l9eL6ZpBEJTpTquE6i/YxIzSiHcdlLFSEDkfOoYKEoHys/yDMT4ySg+HsTQlNM7V3xMZiZQaRYHpjIgeqFlvIv7ndVIdXvoZE0mqQdDpojDlWMd4EgfuMQlU85EhEpmbsV0QCSh2oRWNiG4sy/Pk+ZJ1T2vnt2dVmr +XRwldIAO0TFy0QWqoVtURw1E0SN6Rq/ozXqyXqx362PaumAVM3voD6zPH7CvlxA= +h ˜Ai +ACH3icbZBNS8NAEIY3ftb6VfXoZbEIXiyJaPXY6sWjgq1CU8pmO2kXN5uwOxF +K2n/ixb/ixYMi4s1/47aN4NfAwsP7zjA7b5BIYdB1P5yZ2bn5hcXCUnF5ZXVtvbSx2TRxqjk0eCxjfRMwA1IoaKBACTeJBhYFEq6D27Oxf30H2ohYXeEgXbEekqEgjO0UqdU9SWEOPQlUz0JtO7rCex/CT4K2YWsPsoNX4teH4edUtmtuJOif8HLoUzyuiU3v1uzNMIFHLJjGl5boLtjGkUXMKo6KcGEsZv +WQ9aFhWLwLSzyX0jumuVLg1jbZ9COlG/T2QsMmYQBbYzYtg3v72x+J/XSjE8aWdCJSmC4tNFYSopxnQcFu0KDRzlwALjWti/Ut5nmnG0kRZtCN7vk/9C86DiVStHl4fl2mkeR4Fskx2yRzxyTGrknFyQBuHknjySZ/LiPDhPzqvzNm2dcfKZLfKjnI9PfCWj5w=���hAi � h ˜Ai +��� +AB8HicbVDLSgNBE +OyNrxhfUY9eFoPgKeyKr2PQi8cI5iHJEmZnO8mQmdlZl +YIS7CiwdFvPo53vwbJ8keNLGgoajqprsrTDjTxvO+ncLK +6tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7 +UQhESHVji6nfqtJ1SaxfLBjBMBlI1meUGCs9dg3jEWa +DSa9c8areDO4y8XNSgRz1XvmrG8U0FSgN5UTrju8lJsiI +MoxynJS6qcaE0BEZYMdSQTqIJsdPHFPrBK5/VjZksadqb +8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6 +J3en3bsQUsPHlhCqmL3VpUOiCDU2o5INwV98eZk0z6r+Z +fXi/rxSu8njKMIRHMp+HAFNbiDOjSAgoBneIU3Rzkvzr +vzMW8tOPnMIfyB8/kDLMiQqw=˜g +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjkEvHiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmP +ih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4 +SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicglRJgUav8uUGCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1 +ZBNrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3Uj +5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kckiNyQhxySerkljRIk3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4A +ruS3A=˜� +AB8HicbVDLSgNBEOyNrxhfUY9eFoPgKeyKr2PQi8cI5iHJEmZnO8mQmdlZlYIS7CiwdFvPo53vwbJ8keN +LGgoajqprsrTDjTxvO+ncLK6tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7UQhESHVji6nfqtJ1SaxfLB +jBMBlI1meUGCs9dg3jEWaDSa9c8areDO4y8XNSgRz1XvmrG8U0FSgN5UTrju8lJsiIMoxynJS6qcaE0BEZYMdSQT +qIJsdPHFPrBK5/VjZksadqb8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6J3en3bsQUsPHlhCqmL3VpU +OiCDU2o5INwV98eZk0z6r+ZfXi/rxSu8njKMIRHMp+HAFNbiDOjSAgoBneIU3RzkvzrvzMW8tOPnMIfyB8/kDLMiQq +w=˜g +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjkE +vHiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmPih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicglRJg +Uav8uUGCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1ZBNrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3Uj5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kckiNyQhx +ySerkljRIk3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4AruS3A=˜� +AB8HicbVDLSgNBEOyNrxhfUY9eFoPgKeyKr2P +Qi8cI5iHJEmZnO8mQmdlZlYIS7CiwdFvPo53vwbJ8keNLGgoajqprsrTDjTxvO+ncLK6tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7UQhESHVji6nfqtJ1SaxfLBjBMBlI1meUGCs9dg3jEWaDSa9c8areDO4y8XNSgRz1X +vmrG8U0FSgN5UTrju8lJsiIMoxynJS6qcaE0BEZYMdSQTqIJsdPHFPrBK5/VjZksadqb8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6J3en3bsQUsPHlhCqmL3VpUOiCDU2o5INwV98eZk0z6r+ZfXi/rxSu8njKMIRHMp+HA +FNbiDOjSAgoBneIU3RzkvzrvzMW8tOPnMIfyB8/kDLMiQqw=˜g +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjkE +vHiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmPih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicglRJg +Uav8uUGCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1ZBNrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3Uj5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kckiNyQhx +ySerkljRIk3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4AruS3A=˜� +(a) L=2 +(b) L=2 +(c) L=2 +(d) L=3 +(e) L=3 +(f) L=3 +AFM +PM +FM +AFM +PM +FM +AFM +PM +FM +ACInicdVBLSwMxGMz6rPW16tFLsAgeZNmVvYgFL14rGAfsF1KNk3b0Gx2SbKFsvS3ePGvePGgqCfBH2O6bUFBwLDzHzJlw +liRqWy7Q9jZXVtfWMzt5Xf3tnd2zcPDpsySgQmDRyxSLQDJAmjnDQUVYy0Y0FQGDSCkbXM781JkLSiN+pSUz8EA047VOMlJa6pt2sks8MQj81LZc23FLlfM5qRYXpOxO2MkSCwpi/ilbdnOtGsWlnG4jMNlHDqWnaEAFqh3zbdOL8JSLjCDEnpOXas/BQJRTEj03wnkSRGeIQGxNOUo5BIP82m8JTrfRgPxL6cAUz9ftEikIpJ2GgkyFSQ/nbm4l/eV6i+lU/pTxOFOF4/lA/YVBFcNYX7FBsGIT +TRAWVO8K8RAJhJVuNa9LWP4U/k+aF5ZTtkq3xULtalFHDhyDE3AGHFABNXAD6qABMLgHj+AZvBgPxpPxarzPoyvGYuYI/IDx+QVJr6B6" = 0.01 +ACInicdVBLSwMxGMz6rPW16tFLsAgeZNmVvYgFL14rGAfsF1KNk3b0Gx2SbKFsvS3ePGvePGgqCfBH2O6bUFBwLDzHzJlwliRqWy7Q9jZXVtfWMzt5Xf3tnd2zcPDpsySgQmDR +yxSLQDJAmjnDQUVYy0Y0FQGDSCkbXM781JkLSiN+pSUz8EA047VOMlJa6pt2sks8MQj81LZc23FLlfM5qRYXpOxO2MkSCwpi/ilbdnOtGsWlnG4jMNlHDqWnaEAFqh3zbdOL8JSLjCDEnpOXas/BQJRTEj03wnkSRGeIQGxNOUo5BIP82m8JTrfRgPxL6cAUz9ftEikIpJ2GgkyFSQ/nbm4l/eV6i+lU/pTxOFOF4/lA/YVBFcNYX7FBsGITRAWVO8K8RAJhJVuNa9LWP4U/k+aF5ZTtkq3xULta +lFHDhyDE3AGHFABNXAD6qABMLgHj+AZvBgPxpPxarzPoyvGYuYI/IDx+QVJr6B6" = 0.01 +ACInicdVBLSwMxGMz6rPW16tFLsAgeZNmVvYgFL14rGAfsF1KNk3b0Gx2SbKFsvS3ePGvePGgqCfBH2O6bUFBwLDzHzJlw +liRqWy7Q9jZXVtfWMzt5Xf3tnd2zcPDpsySgQmDRyxSLQDJAmjnDQUVYy0Y0FQGDSCkbXM781JkLSiN+pSUz8EA047VOMlJa6pt2sks8MQj81LZc23FLlfM5qRYXpOxO2MkSCwpi/ilbdnOtGsWlnG4jMNlHDqWnaEAFqh3zbdOL8JSLjCDEnpOXas/BQJRTEj03wnkSRGeIQGxNOUo5BIP82m8JTrfRgPxL6cAUz9ftEikIpJ2GgkyFSQ/nbm4l/eV6i+lU/pTxOFOF4/lA/YVBFcNYX7FBsGIT +TRAWVO8K8RAJhJVuNa9LWP4U/k+aF5ZTtkq3xULtalFHDhyDE3AGHFABNXAD6qABMLgHj+AZvBgPxpPxarzPoyvGYuYI/IDx+QVJr6B6" = 0.01 +FIG. 4. Variations of ⟨A⟩, ⟨ ˜A⟩, and ε = |⟨A⟩−⟨ ˜A⟩| as functions of ˜δ and ˜g, where A = σz +i,1 ⊗σz +i+1,1, ˜A = (I +τ z +i +τ z +i+1 +τ z +i ⊗τ z +i+1)/4 +for L = 2, and ˜A = [25I + 5(τ z +i + τ z +i+1) + τ z +i ⊗ τ z +i+1]/36 for L = 3. All quantities plotted are dimensionless. +Here we have been explicit only about projectors in the rungs +i and i′. Hence the proof. +Note that this proposition can be extended to any operator on +the space of the 2D model, since it can always be written as a +linear combination of tensor products of operators localized +on rungs. +We now specifically consider four constructions of A, +namely, σα +i,j, �L +j=1 σα +i,j, ⊗L +j=1σαj +i,j +7, and σαi +i,j⊗σαi+r +i+r,j′, where +α, αi ∈ {x, y, z}. The expressions for the corresponding +low-energy components ˜A on the 1D XXZ model are given +in Table I. For ease of reference, we call the operators A hav- +ing vanishing low-energy component (i.e., ˜A = 0) as high- +energy operators. Note here that the low-energy component +of σα +i,j, α = x, y, z, is consistent with the operator identifi- +cation of Jz +i in terms of τ z +i , as given in Eq. (17) and Eq. (18). +To systematically test the performance of the first-order +approximation for the operators listed in Table I, we con- +sider the situation of PBC along the legs8. Note that the 1D +effective model can be described by two independent dimen- +sionless parameters, ˜g = ˜h/ ˜Jxy, and ˜δ = ˜Jzz/ ˜Jxy, where +the dimensionless parameter ˜Jxy ̸= 0, and the physics of +the model is expected to be invariant with a change in the +value of ˜Jxy. Using ˜g and ˜δ, the phase diagram of the 1D +XXZ model is given by Fig. 3, where the ferromagnetic +7 One might naively guess that if Ai, Bi maps to ˜ +Ai, ˜Bi in XXZ model, +then for Ci = AiBi the corresponding XXZ operator is ˜Ci = ˜ +Ai ˜Bi. +However this is not true - see Eq. (62) for the caveat. +8 OBC along the legs can also be considered in similar fashion. +(FM), antiferromagnetic (AFM), and the paramagnetic (PM) +phases and the corresponding phase boundaries are marked. +We use this phase diagram for reference during discussion of +results in subsequent sections. The effective coupling con- +stants are functions of J||/J⊥, J1,2 +d /J⊥, and ∆h/ ⊥ (see +Eqs. (43) and (47)). Let us define Jsum +d += |(J1 +d + J2 +d)/J⊥|, +Jdiff +d += |(J1 +d −J2 +d)/J⊥|, and observe that ˜Jxy, ˜Jzz, and ˜h de- +pend only on Jsum +d +, and are independent of Jdiff +d . Solving for +J||/J⊥, Jsum +d +/J⊥, and ∆h/J⊥ from Eq. (43), one can write +J|| +J⊥ += 2 ˜Jxy(L˜δ + 1), Jsum +d +J⊥ += 2 ˜Jxy(L˜δ − 1), , +∆h +J⊥ += 2 ˜Jxy[2(L − 1)˜δ − ˜g], +(63) +for PBC along the rungs. In the case of OBC along the rungs, +from Eq. (47), one obtains +J|| +J⊥ += 4 ˜Jxy(4A0˜δ + C1) +C1 + A0C2 +, Jsum +d +J⊥ += 4 ˜Jxy(C2 − 4˜δ) +C1 + A0C2 +, +∆h +J⊥ += 2 ˜Jxy +�(4 − C2)(4A0˜δ + C1) +C1 + A0C2 ++2(A2 − A3)(C2 − 4˜δ) +C1 + A0C2 +− 4˜g +� +, +(64) +where C1 = 2A2−A3−L+1, C2 = 4−L+A1, and A0, A1, +A2, and A3 are given in Eqs. (48)-(51). In the following, we +consider expectation values of specific Hermitian operators, +both in the 2D model as well as in the effective 1D model, +as functions of ˜g and ˜δ, under the constraints that the cor- +responding +� +J||/J⊥, Jsum +d +/J⊥, Jdiff +d /J⊥, ∆h/J⊥ +� +are small. + +1.5 +0.05 +1.0 +0.04 +0.5 +0.03 +0.0 +0.02 +-0.5 +0.01 +-1.0 +-1.5 +0.00 +-2 +0 +2 +41.5 +1.0 +0.8 +0.5 +0.6 +0.0 +0.4 +-0.5 +0.2 +-1.0 +0.0 +-1.5 +-2 +0 +2 +41.5 +1.0 +0.8 +0.5 +0.6 +0.0 +0.4 +-0.5 +0.2 +-1.0 +0.0 +-1.5 +-2 +0 +2 +41.5 +0.12 +1.0 +0.10 +0.5 +0.08 +0.0 +0.06 +-0.5 +0.04 +-1.0 +0.02 +-1.5 +0.00 +-2 +0 +2 +41.5 +1.0 +0.9 +0.5 +0.8 +0.0 +0.7 +-0.5 +0.6 +-1.0 +0.5 +-1.5 +-2 +0 +2 +41.5 +1.0 +0.9 +0.5 +0.8 +0.0 +0.7 +-0.5 +0.6 +-1.0 +0.5 +-1.5 +-2 +0 +2 +41.5 +0.14 +1.0 +0.12 +0.5 +0.10 +0.08 +0.0 +0.06 +-0.5 +0.04 +-1.0 +0.02 +-1.5 +0.00 +-2 +0 +2 +41.5 +0.14 +1.0 +0.12 +0.5 +0.10 +0.08 +0.0 +0.06 +-0.5 +0.04 +-1.0 +0.02 +-1.5 +0.00 +-2 +0 +2 +411 +Note that irrespective of the value of ˜δ and ˜g, this can be en- +sured by fixing ˜Jxy to a small value9. We fix ˜Jxy = 10−2 +and Jdiff +d += 0 for the range −3 ≤ ˜δ ≤ 4 and −1.5 ≤ ˜g ≤ 1.5 +considered in this work. +For demonstration, we choose A = σz +i,1 ⊗ σz +i+1,1 on a +3 × 2 and 3 × 3 lattice, for which the low-energy compo- +nent ˜A is given in Table I. We compute ⟨A⟩ = Tr(ρA) +and ⟨ ˜A⟩ = Tr(˜ρ ˜A), where ρ and ˜ρ are considered to be +a thermal state (see Eq. (34)). +To ensure that the state +lies in the low-energy sector of the 2D model, we consider +βJ⊥ = 102, where the strength of the rung-coupling is taken +to be J⊥ = 102. Fig. 4 depicts the variation of ⟨A⟩, ⟨ ˜A⟩, +and ε = |⟨A⟩ − ⟨ ˜A⟩| as functions of ˜δ and ˜g, where OBC is +assumed along the rungs of the lattices. In order to quanti- +tatively estimate the performance of the 1D effective XXZ +model to substitute the values of ⟨A⟩, on the terrain of ε, we +mark the regions in which ε ≤ J−1 +⊥ += 10−2. Note that in +the case of L = 2, major portions of the considered (˜δ, ˜g) +plane is included in this region, with relatively higher values +of ε occurring mostly in the FM phase with ˜g ≥ 0. In con- +trast, for L = 3, higher values of ε appear in the PM and the +AFM phases also. These observations remain qualitatively +unchanged for observables defined on a single rung instead +of two different rungs. For a discussion and demonstration +with A = σz +1,1 ⊗ σz +i,2 and A = σz +i,1, see Appendix F. We +want to emphasise here that that qualitative variations of ⟨A⟩ +in the 2D model is mimicked satisfactorily by the same for +⟨ ˜A⟩ in the effective 1D model, in the entire range of (˜δ, ˜g) +considered in this paper, even beyond the realm of perturba- +tion theory. +V. +ESTIMATING ENTANGLEMENT +Now that we have matched the spectrum and observables +from the 2D model to the 1D XXZ model, in this Sec- +tion, we focus on quantum correlation measures belonging +to the entanglement-separability paradigm [25, 26] that can +be computed in the system under consideration. +A. +Entanglement from reduced density matrices +While the distribution of entanglement in a quantum spin +model can be varied, such as bipartite and multipartite entan- +glement over different subsystems, it often adapts a partial +trace-based methodology for computation [25]. For exam- +ple, the bipartite entanglement between two subsystems of +the spin model in a pure state, is inferred from the reduced +density matrix of one of the subsystems. Therefore, the first +natural question to address here is if there exists a relation be- +tween the reduced density matrices of subsystems of the 2D +model, and the reduced density matrices of the correspond- +ing subsystems in the effective 1D XXZ model. In fact, it +9 This is the only constraint on ˜Jxy, which, in turn, implies that J||/J⊥ +and Jsum +d +/J⊥ have to be comparable. +is natural to expect that a single-rung reduced density matrix +in the 2D model will match to a single spin reduced density +matrix in the 1D XXZ model. We will now show that this +is indeed true. +Let us begin with a density matrix of the 2D system, which +has a description in the 1D effective XXZ model, i.e ρ ≈ +PgρPg = ˜ρ. For this, we propose the following. +■ Proposition 4. The reduced density matrix ρ(i) obtained +via tracing out the rung i from ρ and the reduced density +matrix ˜ρ(i) obtained from ˜ρ by tracing out the spin i obeys +ρ(i) ≈ ˜ρ(i). +Proof. Note that exploiting the basis-independence of partial +trace, and using notations introduced in Eq. (7) in Sec. II B, +ρ(i) = Triρ = +2L−1 +� +k=0 +⟨ψ(i) +k |ρ |ψ(i) +k ⟩ +(65) +whereas +˜ρ(i) = Tri˜ρ = +1 +� +k=0 +⟨ψ(i) +k |˜ρ |ψ(i) +k ⟩ +(66) +To prove that ρ(i) ≈ ˜ρ(i), we consider +⟨ψ(i) +k |ρ |ψ(i) +k ⟩ ≈ ⟨ψ(i) +k |˜ρ |ψ(i) +k ⟩ = ⟨ψ(i) +k |PgρPg |ψ(i) +k ⟩ .(67) +Since Pg = ⊗iP i +g and P i +g |ψ(i) +k ⟩ = 0 unless k = 0, 1, it +follows that the RHS of (67) is vanishing for k > 1. Thus +we have that the reduced density matrix on the space of the +2D model (Eq. (65)) becomes +ρ(i) ≈ +1 +� +k=0 +⟨ψ(i) +k |ρ |ψ(i) +k ⟩ = ˜ρ(i). +(68) +Hence proved. +The next corollary follows straightforwardly from Propo- +sition 4. +⋄ Corollary 4.1. The reduced density matrix ρS for a sub- +system composed of a set S of arbitrary number of rungs ob- +tained from the state ρ of the 2D model and the reduced den- +sity matrix ˜ρS of the corresponding set S of spins obtained +from the state ˜ρ of the 1D XXZ model obeys ρS ≈ ˜ρS. +Proposition 4 and Corollary 4.1 can be verified by com- +puting the trace distance between ρS and ˜ρS. This implies +that any function of the density matrices, for example, en- +tanglement quantified over subsystems using their reduced +density matrices, such as the bipartite entanglement between +two chosen parties [25, 26], entanglement spectrum [106], +distance-based multipartite entanglement [107–109], or mul- +tiparty quantum correlation quantified by monogamy-bases +approaches [110–117] should also match for ρS and ˜ρS. To +demonstrate this, we choose bipartite entanglement over a +subsystem S constituted of two nearest-neighbor rungs in +the 2D lattice, which corresponds to two nearest-neighbor +spins in the 1D effective model. We numerically compute +the bipartite entanglement Ei,i+1, between the two rungs i + +12 +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjkEvHiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmP +ih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4 +SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicglRJgUav8uUGCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1 +ZBNrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3U +j5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kckiNyQhxySerkljRIk3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4A +AruS3A=˜� +AB8HicbVDLSgNBE +OyNrxhfUY9eFoPgKeyKr2PQi8cI5iHJEmZnO8mQmdlZl +YIS7CiwdFvPo53vwbJ8keNLGgoajqprsrTDjTxvO+ncLK +6tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7 +UQhESHVji6nfqtJ1SaxfLBjBMBlI1meUGCs9dg3jEW +aDSa9c8areDO4y8XNSgRz1XvmrG8U0FSgN5UTrju8lJsiI +MoxynJS6qcaE0BEZYMdSQTqIJsdPHFPrBK5/VjZksadq +b8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6 +J3en3bsQUsPHlhCqmL3VpUOiCDU2o5INwV98eZk0z6r+ +ZfXi/rxSu8njKMIRHMp+HAFNbiDOjSAgoBneIU3Rzkvzr 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+AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjkE +vHiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmPih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicglRJg +Uav8uUGCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1ZBNrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3Uj5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kckiNyQhx +ySerkljRIk3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4AruS3A=˜� +AB8HicbVDLSgNBEOyNrxhfUY9eFoPgKeyKr2P +Qi8cI5iHJEmZnO8mQmdlZlYIS7CiwdFvPo53vwbJ8keNLGgoajqprsrTDjTxvO+ncLK6tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7UQhESHVji6nfqtJ1SaxfLBjBMBlI1meUGCs9dg3jEWaDSa9c8areDO4y8XNSgRz1X +vmrG8U0FSgN5UTrju8lJsiIMoxynJS6qcaE0BEZYMdSQTqIJsdPHFPrBK5/VjZksadqb8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6J3en3bsQUsPHlhCqmL3VpUOiCDU2o5INwV98eZk0z6r+ZfXi/rxSu8njKMIRHMp+HA +FNbiDOjSAgoBneIU3RzkvzrvzMW8tOPnMIfyB8/kDLMiQqw=˜g +AFM +PM +FM +(c) L=2 +ACIn 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Variations of Ei,i+1, ˜Ei,i+1, and ε = |Ei,i+1 − ˜Ei,i+1| as functions of ˜δ and ˜g. The top horizontal panel is for a 3×2 lattice, while +figures in the bottom horizontal panel are for a 3 × 3 lattice. All quantities plotted are dimensionless. +and i + 1 in the 2D model from their reduced density ma- +trix ρi,i+1, and the bipartite entanglement ˜Ei,i+1 between +the spins (i, i + 1) in the 1D effective XXZ model from +their reduced density matrix ˜ρi,i+1, where negativity [118– +124] (see Appendix E for a definition) is used for entangle- +ment quantification. These reduced density matrices ρi,i+1 +and ˜ρi,i+1 are obtained respectively by tracing out all other +rungs other than the rungs (i, i + 1) from the thermal state ρ +of the 2D model, and by tracing out all other spins except the +spins i and i + 1 from the thermal state ˜ρ of the 1D model +(see Sec. II C 3). To ensure low-energy of the states, we fix +βJ⊥ = 102, while J⊥ = 102. +The variations for Ei,i+1 and ˜Ei,i+1 as functions of (˜δ, ˜g) +are included in Fig. 5 in the case of a two-leg and a three-leg +system. Similar to the expectation values of observables dis- +cussed in Sec. IV, to quantitatively test the performance of +the 1D effective theory in estimating Ei,i+1, we look at the +variation of ε = |Ei,i+1− ˜Ei,i+1| as functions of ˜δ and ˜g, and +find that ε ≤ J−1 +⊥ += 10−2 for the whole of the AFM and the +PM phases, while relatively high values of ε occurs in the +FM phases for ˜g ≥ 0. These observations remain qualita- +tively unchanged with a change in the value of L from 2 to 3. +To check whether these observations change with a change +of the subsystem over which entanglement is computed, or +with a change from the bipartite entanglement to a multi- +partite quantum correlation, we further consider (a) bipar- +tite entanglement, as quantified by negativity, over a biparti- +tion of the entire system, and (b) a monogamy-based [110– +114, 117] multiparty quantum correlation, referred to as the +monogamy score [115–117], over the entire system, and find +the observations reported so far to be valid. See Appendix G +for details. Also, note that for the entirety of the (˜δ, ˜g) plane +considered in this paper, the behaviour of different quantum +correlations is qualitatively mimicked by the same for their +corresponding XXZ counterparts, although the perturbation +theory may not be valid everywhere on the plane. +B. +Measurement-based quantification of entanglement +Given a quantum state ρ of a composite quantum system, +one may also quantify entanglement over a subsystem us- +ing a measurement-based protocol [125–129]. For a biparti- +tion S : S of the system, a local projection measurement on +one of the subsystems, say, S, leads to an ensemble of post- +measured states ϱ(i) +SS = P (i) +S ρSSP (i)† +S +/Tr +� +P (i) +S ρSSP (i)† +S +� +with probabilities pi = Tr +� +P (i) +S ρSSP (i)† +S +� +. Here, P (i) +S += +|ψ(i) +S ⟩ ⟨ψ(i) +S |, i = 0, 2, · · · , dS − 1 are projection operators +corresponding to the eigenkets |ψ(i) +S ⟩ of a chosen Hermitian +operator MS on the Hilbert space of S with dimension dS. +For each choice of MS, one can define average entanglement +in the post-measured states of S, given by +⟨E⟩S = +dS−1 +� +i=0 +piE(ϱ(i) +S ), +(69) +where ϱ(i) +S += TrSϱ(i) +SS. Note that the subsystem S can be a +collection of smaller subsystems, and the entanglement mea- +sure E can be either a bipartite, or a multiparty measure. +In this work, we choose bipartite entanglement quantified + +1.5 +1.0 +0.05 +0.5 +0.04 +0.0 +0.03 +-0.5 +0.02 +-1.0 +0.01 +-1.5 +0.00 +-2 +0 +2 +41.5 +0.06 +1.0 +0.05 +0.5 +0.04 +0.0 +0.03 +-0.5 +0.02 +-1.0 +0.01 +-1.5 +0.00 +-2 +0 +2 +1 +41.5 +0.06 +1.0 +0.05 +0.5 +0.04 +0.0 +0.03 +-0.5 +0.02 +-1.0 +0.01 +-1.5 +0.00 +-2 +0 +2 +41.5 +0.06 +1.0 +0.05 +0.5 +0.04 +0.0 +0.03 +-0.5 +0.02 +-1.0 +0.01 +-1.5 +-2 +0 +2 +41.5 +0.06 +1.0 +0.05 +0.5 +0.04 +0.0 +0.03 +-0.5 +0.02 +-1.0 +0.01 +-1.5 +-2 +0 +2 +41.5 +0.025 +1.0 +0.020 +0.5 +0.015 +0.0 +0.010 +-0.5 +0.005 +-1.0 +-1.5 +0.000 +-2 +0 +2 +41.5 +1.0 +1.0 +0.8 +0.5 +0.6 +0.0 +0.4 +-0.5 +0.2 +-1.0 +-1.5 +0.0 +-2 +0 +2 +413 +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjkEvHiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmPih4 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+8cI5iHJEmZnO8mQmdlZlYIS7CiwdFvPo53vwbJ8keNLGgoajqprsrTDjTxvO+ncLK6tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7UQhESHVji6nfqtJ1SaxfLBjBMBlI1meUGCs9dg3jEWaDSa9c8areDO4y8XNSgRz1XvmrG8U +0FSgN5UTrju8lJsiIMoxynJS6qcaE0BEZYMdSQTqIJsdPHFPrBK5/VjZksadqb8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6J3en3bsQUsPHlhCqmL3VpUOiCDU2o5INwV98eZk0z6r+ZfXi/rxSu8njKMIRHMp+HAFNbiDOjSA +goBneIU3RzkvzrvzMW8tOPnMIfyB8/kDLMiQqw=˜g +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjkEvH +iOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmPih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicglRJgUav8uU +GCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1ZBNrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3Uj5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kckiNyQhxySerkljRI +k3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4AruS3A=˜� +(a) +(b) +(c) +(d) +(e) +(f) +AFM +PM +FM +AFM +PM +FM +AFM +PM +FM +AB8HicbVDLSgNBEO +yNrxhfUY9eFoPgKeyKr2PQi8cI5iHJEmZnO8mQmdlZlYI +S7CiwdFvPo53vwbJ8keNLGgoajqprsrTDjTxvO+ncLK6tr +6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7UQhE +SHVji6nfqtJ1SaxfLBjBMBlI1meUGCs9dg3jEWaDSa9c +8areDO4y8XNSgRz1XvmrG8U0FSgN5UTrju8lJsiIMoxynJ +S6qcaE0BEZYMdSQTqIJsdPHFPrBK5/VjZksadqb8nMiK0 +HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6J3en3bsQ +UsPHlhCqmL3VpUOiCDU2o5INwV98eZk0z6r+ZfXi/rxSu +8njKMIRHMp+HAFNbiDOjSAgoBneIU3RzkvzrvzMW8tOPnM +IfyB8/kDLMiQqw=˜g +AB8HicbVDLSgNBEOyNrxhfUY9eFoPgKeyKr2PQi8cI5iHJEmZnO8mQmdlZlYIS7CiwdFvPo53vwbJ8keNLGg +oajqprsrTDjTxvO+ncLK6tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7UQhESHVji6nfqtJ1SaxfLBjBM +BlI1meUGCs9dg3jEWaDSa9c8areDO4y8XNSgRz1XvmrG8U0FSgN5UTrju8lJsiIMoxynJS6qcaE0BEZYMdSQTqIJsdP +HFPrBK5/VjZksadqb8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6J3en3bsQUsPHlhCqmL3VpUOiCDU2o5 +INwV98eZk0z6r+ZfXi/rxSu8njKMIRHMp+HAFNbiDOjSAgoBneIU3RzkvzrvzMW8tOPnMIfyB8/kDLMiQqw=˜g +AB8HicbVDLSgNBEOyNrxhfUY9eFoPgKeyKr2PQi +8cI5iHJEmZnO8mQmdlZlYIS7CiwdFvPo53vwbJ8keNLGgoajqprsrTDjTxvO+ncLK6tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7UQhESHVji6nfqtJ1SaxfLBjBMBlI1meUGCs9dg3jEWaDSa9c8areDO4y8XNSgRz1XvmrG8U +0FSgN5UTrju8lJsiIMoxynJS6qcaE0BEZYMdSQTqIJsdPHFPrBK5/VjZksadqb8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6J3en3bsQUsPHlhCqmL3VpUOiCDU2o5INwV98eZk0z6r+ZfXi/rxSu8njKMIRHMp+HAFNbiDOjSA +goBneIU3RzkvzrvzMW8tOPnMIfyB8/kDLMiQqw=˜g +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjkEvHiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmPih4 +vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4SsGL +WD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicglRJgUav8uUGCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1ZBNrLp +1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3Uj5ginE0QZ +VNCM78y4ukdVpzLmrnd2fV+nUR4kckiNyQhxySerkljRIk3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4AruS3A=˜� +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjkEvH +iOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmPih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicglRJgUav8uU +GCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1ZBNrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3Uj5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kckiNyQhxySerkljRI +k3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4AruS3A=˜� +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjkEvH +iOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmPih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicglRJgUav8uU +GCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1ZBNrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3Uj5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kckiNyQhxySerkljRI +k3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4AruS3A=˜� +AFM +PM +FM +FM +PM +AFM +AFM +PM +FM +ACInicdVBLSwMxGMz6rPW16tFLsAgeZNmVvYgFL14rGAfsF1KNk3 +b0Gx2SbKFsvS3ePGvePGgqCfBH2O6bUFBwLDzHzJlwliRqWy7Q9jZXVtfWMzt5Xf3tnd2z +cPDpsySgQmDRyxSLQDJAmjnDQUVYy0Y0FQGDSCkbXM781JkLSiN+pSUz8EA047VOMlJa +6pt2sks8MQj81LZc23FLlfM5qRYXpOxO2MkSCwpi/ilbdnOtGsWlnG4jMNlHDqWnaEAF +qh3zbdOL8JSLjCDEnpOXas/BQJRTEj03wnkSRGeIQGxNOUo5BIP82m8JTrfRgPxL6cA +Uz9ftEikIpJ2GgkyFS +Q/nbm4l/eV6i+lU/pTxOFOF4/lA/YVBFcNYX7FBsGITRAWVO8K8 +RAJhJVuNa9LWP4U/k+aF5ZTtkq3xULtalFHDhyDE3AGHFABNXAD6qABMLgHj+AZvBgPxp +PxarzPoyvGYuYI/IDx+QVJr6B6" = 0.01 +ACInicdVBLSwMxGMz6r +PW16tFLsAgeZNmVvYgFL14rGAfsF1KNk3b0Gx2SbKFsvS3ePGvePGgqCfBH2O6bUFBwLDzHzJlwliRqWy7Q9jZXVtfWMzt5Xf3tnd2zcPDpsySgQmDRyxSLQDJAmjnDQUVYy0Y0FQ +GDSCkbXM781JkLSiN+pSUz8EA047VOMlJa6pt2sks8MQj81LZc23FLlfM5qRYXpOxO2MkSCwpi/ilbdnOtGsWlnG4jMNlHDqWnaEAFqh3zbdOL8JSLjCDEnpOXas/BQJRTEj03wn +kSRGeIQGxNOUo5BIP82m8JTrfRgPxL6cAUz9ftEikIpJ2GgkyFSQ/nbm4l/eV6i+lU +/pTxOFOF4/lA/YVBFcNYX7FBsGITRAWVO8K8RAJhJVuNa9LWP4U/k+aF5ZTtkq3xULtalF +HDhyDE3AGHFABNXAD6qABMLgHj+AZvBgPxpPxarzPoyvGYuYI/IDx+QVJr6B6" = 0.01 +ACInicdVBLSwMxGMz6rPW16tFLsAgeZNmVvYgFL14rGAfsF1KNk3b0Gx2SbKFsvS3ePGvePGgqCfBH2O +6bUFBwLDzHzJlwliRqWy7Q9jZXVtfWMzt5Xf3tnd2zcPDpsySgQmDRyxSLQDJAmjnDQUVYy0Y0FQGDSCkbXM781JkLSiN+pSUz8EA047VOMlJa6pt2sks8MQj81LZc23FLlfM5qRYXpOxO2MkSCwpi/ilbdnOtGsWlnG4jMNlHDqWnaEAFqh3zbdOL8JSLjCDEnpOXas/BQJRTEj03wnkSRGeIQGxNOUo5BIP82m8JTrfRgPxL6cAUz9ftEikIpJ2GgkyF +SQ/nbm4l/eV6i+lU/pTxOFOF4/lA/YVBFcNYX7FBsGITRAWVO8K8RAJhJVuNa9LWP4U/k+aF5ZTtkq3xULtalFHDhyDE3AGHFABNXAD6qABMLgHj+AZvBgPxpPxarzPoyvGYuYI/IDx+QVJr6B6" = 0.01 +ACInicdVBLSwMxGMz6rPW16tFLsAgeZNmVvYgFL14rGAfsF1KNk3b0Gx2SbKFsvS3ePGvePGgqCfBH2O +6bUFBwLDzHzJlwliRqWy7Q9jZXVtfWMzt5Xf3tnd2zcPDpsySgQmDRyxSLQDJAmjnDQUVYy0Y0FQGDSCkbXM781JkLSiN+pSUz8EA047VOMlJa6pt2sks8MQj81LZc23FLlfM5qRYXpOxO2MkSCwpi/ilbdnOtGsWlnG4jMNlHDqWnaEAFqh3zbdOL8JSLjCDEnpOXas/BQJRTEj03wnkSRGeIQGxNOUo5BIP82m8JTrfRgPxL6cAUz9ftEikIpJ2GgkyF +SQ/nbm4l/eV6i+lU/pTxOFOF4/lA/YVBFcNYX7FBsGITRAWVO8K8RAJhJVuNa9LWP4U/k+aF5ZTtkq3xULtalFHDhyDE3AGHFABNXAD6qABMLgHj+AZvBgPxpPxarzPoyvGYuYI/IDx+QVJr6B6" = 0.01 +(g) +(h) +(i) +ACInicdVBLSwMxGMz6rPW16tFLsAgeZNmVvYgFL14rGAfsF1KNk3b0Gx2SbKFsvS3ePGvePGgqCfBH2O +6bUFBwLDzHzJlwliRqWy7Q9jZXVtfWMzt5Xf3tnd2zcPDpsySgQmDRyxSLQDJAmjnDQUVYy0Y0FQGDSCkbXM781JkLSiN+pSUz8EA047VOMlJa6pt2sks8MQj81LZc23FLlfM5qRYXpOxO2MkSCwpi/ilbdnOtGsWlnG4jMNlHDqWnaEAFqh3zbdOL8JSLjCDEnpOXas/BQJRTEj03wnkSRGeIQGxNOUo5BIP82m8JTrfRgPxL6cAUz9ftEikIpJ2GgkyF +SQ/nbm4l/eV6i+lU/pTxOFOF4/lA/YVBFcNYX7FBsGITRAWVO8K8RAJhJVuNa9LWP4U/k+aF5ZTtkq3xULtalFHDhyDE3AGHFABNXAD6qABMLgHj+AZvBgPxpPxarzPoyvGYuYI/IDx+QVJr6B6" = 0.01 +ACInicdVBLSwMxGMz6rPW16tFLsAgeZNmVvYgFL14rGAfsF1KNk3b0Gx2SbKFsvS3ePGvePGgqCfBH2O +6bUFBwLDzHzJlwliRqWy7Q9jZXVtfWMzt5Xf3tnd2zcPDpsySgQmDRyxSLQDJAmjnDQUVYy0Y0FQGDSCkbXM781JkLSiN+pSUz8EA047VOMlJa6pt2sks8MQj81LZc23FLlfM5qRYXpOxO2MkSCwpi/ilbdnOtGsWlnG4jMNlHDqWnaEAFqh3zbdOL8JSLjCDEnpOXas/BQJRTEj03wnkSRGeIQGxNOUo5BIP82m8JTrfRgPxL6cAUz9ftEikIpJ2GgkyF +SQ/nbm4l/eV6i+lU/pTxOFOF4/lA/YVBFcNYX7FBsGITRAWVO8K8RAJhJVuNa9LWP4U/k+aF5ZTtkq3xULtalFHDhyDE3AGHFABNXAD6qABMLgHj+AZvBgPxpPxarzPoyvGYuYI/IDx+QVJr6B6" = 0.01 +ACInicdVBLSwMxGMz6rPW16tFLsAgeZNmVvYgFL14rGAfsF1KNk3b0Gx2SbKFsvS3ePGvePGgqCfBH2O6bUFBwLDzHzJlwliRqWy7Q9jZXVtfWMzt5Xf3tnd2zcPDpsySgQmDRyxSLQDJAmjnDQUVYy0Y0FQGDSCkbXM781JkLSiN+pSUz8EA047VOMlJa6pt2sks8MQj81LZc23FLlfM5qRYXpOxO +2MkSCwpi/ilbdnOtGsWlnG4jMNlHDqWnaEAFqh3zbdOL8JSLjCDEnpOXas/BQJRTEj03wnkSRGeIQGxNOUo5BIP82m8JTrfRgPxL6cAUz9ftEikIpJ2GgkyFSQ/nbm4l/eV6i+lU +/pTxOFOF4/lA/YVBFcNYX7FBsGITRAWVO8K8RAJhJVuNa9LWP4U/k+aF5ZTtkq3xULtalFHDhyDE3AGHFABNXAD6qABMLgHj+AZvBgPxpPxarzPoyvGYuYI/IDx+QVJr6B6" = 0.01 +ACHicbVDNS8MwHE3n15xfVY8eDA7B02jFr+NQBI8T3QespaRZuoWlaUlSYZQevfivePGgiFf/BG/+N6ZdD7r5 +IPB47/eS/J4fMyqVZX0blYXFpeWV6mptbX1jc8vc3unIKBGYtHEItHzkSMctJWVDHSiwVBoc9I1x9f5X73gQhJI36v +JjFxQzTkNKAYKS15r7DEB8yAq8dURAvdSIdyO9L7LM+tWwyoA54ldkjo0fLML2cQ4SQkXGpOzbVqzcFAlFMSNZz +UkiREeoyHpa8pRSKSbFotk8FArAxhEQh+uYKH+TqQolHIS+noyRGokZ71c/M/rJyq4cFPK40QRjqcPBQmDKoJ5K3BABc +GKTRBWFD9V4hHSCsdHc1XYI9u/I86Rw37LPG6e1JvXlZ1lEFe+AHAEbnIMmuAEt0AYPIJn8ArejCfjxXg3PqajFaP +M7I/MD5/AIpOmlY=hEiS +ACEHicbVDLSgMxFM3UV62vqks3wSK6KjPia1kUw +WVF+4BOKZnMbRuayQxJRijDfIbf8WNC0XcunTn35hpZ6GtBwKHc+5Jco8Xca0bX9bhYXFpeWV4mpbX1jc6u8vdNUYSwpNGjIQ9n2iALOBDQ0xzakQSeBxa3ugq81sPIBULxb0eR9ANyECwPqNEG6lXPnQ5EQMO2NWM+5Bcp6cCL3EDU0wuze +5S9NeuWJX7QnwPHFyUkE56r3yl+uHNA5AaMqJUh3HjnQ3IVIzyiEtubGCiNARGUDHUECUN1kslCKD4zi434ozREaT9TfiYQESo0Dz0wGRA/VrJeJ/3mdWPcvugkTUaxB0OlD/ZhjHeKsHewzCVTzsSGESmb+iumQSEK16bBkSnBmV54nzeOqc1Y9 +vT2p1C7zOopoD+2jI+Sgc1RDN6iOGoiR/SMXtGb9WS9WO/Wx3S0YOWZXfQH1ucPTFOeDA= +h ˜EiS +ACMnicbVDLSgMxFM3UV62vqks3wSK4scyIr2VRC +rqraB/QKSWT3rahmcyQZIQynW9y45cILnShiFs/wvQhaNsDgcM5ya5xws5U9q2X63UwuLS8kp6NbO2vrG5ld3eqagkhTKNOCBrHlEAWcCypDrVQAvE9DlWvdzX0qw8gFQvEve6H0PBJR7A2o0QbqZm9GbiciA4HXHTliDRjNzATwvjuyQ5+vV 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+JjFxQzTkNKAYKS15r7DEB8yAq8dURAvdSIdyO9L7LM+tWwyoA54ldkjo0fLML2cQ4SQkXGpOzbVqzcFAlFMSNZz +UkiREeoyHpa8pRSKSbFotk8FArAxhEQh+uYKH+TqQolHIS+noyRGokZ71c/M/rJyq4cFPK40QRjqcPBQmDKoJ5K3BABc +GKTRBWFD9V4hHSCsdHc1XYI9u/I86Rw37LPG6e1JvXlZ1lEFe+AHAEbnIMmuAEt0AYPIJn8ArejCfjxXg3PqajFaP +M7I/MD5/AIpOmlY=hEiS +ACMnicbVDLSgMxFM3UV62vqks3wSK4scyIr2VRC +rqraB/QKSWT3rahmcyQZIQynW9y45cILnShiFs/wvQhaNsDgcM5ya5xws5U9q2X63UwuLS8kp6NbO2vrG5ld3eqagkhTKNOCBrHlEAWcCypDrVQAvE9DlWvdzX0qw8gFQvEve6H0PBJR7A2o0QbqZm9GbiciA4HXHTliDRjNzATwvjuyQ5+vV +dzXgL4mIyPzdoZnN23h4BzxJnQnJoglIz+y2Ahr5IDTlRKm6Y4e6EROpGeWQZNxIQUhoj3SgbqgPqhGPFo5wQdGaeF2IM0RGo/UvxMx8ZXq+5J+kR31bQ3FOd59Ui3LxoxE2GkQdDxQ+2IYx3gYX+4xSRQzfuGECqZ+SumXSIJ1abljCnBmV5 +lSO85Z/vT2JFe4nNSRntoHx0iB52jArpGJVRGFD2iF/SOPqwn6836tL7G0ZQ1mdlF/2B9/wC/Uazw +|hEiS � h ˜EiS| +ACHicbVDNS8MwHE3n15xfVY8eDA7B02jFr+NQBI8T3QespaRZuoWlaUlSYZQevfivePGgiFf/BG/+N6ZdD7r5 +IPB47/eS/J4fMyqVZX0blYXFpeWV6mptbX1jc8vc3unIKBGYtHEItHzkSMctJWVDHSiwVBoc9I1x9f5X73gQhJI36v +JjFxQzTkNKAYKS15r7DEB8yAq8dURAvdSIdyO9L7LM+tWwyoA54ldkjo0fLML2cQ4SQkXGpOzbVqzcFAlFMSNZz +UkiREeoyHpa8pRSKSbFotk8FArAxhEQh+uYKH+TqQolHIS+noyRGokZ71c/M/rJyq4cFPK40QRjqcPBQmDKoJ5K3BABc +GKTRBWFD9V4hHSCsdHc1XYI9u/I86Rw37LPG6e1JvXlZ1lEFe+AHAEbnIMmuAEt0AYPIJn8ArejCfjxXg3PqajFaP +M7I/MD5/AIpOmlY=hEiS +ACEHicbVDLSgMxFM3UV62vqks3wSK6KjPia1kUw +WVF+4BOKZnMbRuayQxJRijDfIbf8WNC0XcunTn35hpZ6GtBwKHc+5Jco8Xca0bX9bhYXFpeWV4mpbX1jc6u8vdNUYSwpNGjIQ9n2iALOBDQ0xzakQSeBxa3ugq81sPIBULxb0eR9ANyECwPqNEG6lXPnQ5EQMO2NWM+5Bcp6cCL3EDU0wuze +5S9NeuWJX7QnwPHFyUkE56r3yl+uHNA5AaMqJUh3HjnQ3IVIzyiEtubGCiNARGUDHUECUN1kslCKD4zi434ozREaT9TfiYQESo0Dz0wGRA/VrJeJ/3mdWPcvugkTUaxB0OlD/ZhjHeKsHewzCVTzsSGESmb+iumQSEK16bBkSnBmV54nzeOqc1Y9 +vT2p1C7zOopoD+2jI+Sgc1RDN6iOGoiR/SMXtGb9WS9WO/Wx3S0YOWZXfQH1ucPTFOeDA= +h ˜EiS +ACMnicbVDLSgMxFM3UV62vqks3wSK4scyIr2VRC +rqraB/QKSWT3rahmcyQZIQynW9y45cILnShiFs/wvQhaNsDgcM5ya5xws5U9q2X63UwuLS8kp6NbO2vrG5ld3eqagkhTKNOCBrHlEAWcCypDrVQAvE9DlWvdzX0qw8gFQvEve6H0PBJR7A2o0QbqZm9GbiciA4HXHTliDRjNzATwvjuyQ5+vV +dzXgL4mIyPzdoZnN23h4BzxJnQnJoglIz+y2Ahr5IDTlRKm6Y4e6EROpGeWQZNxIQUhoj3SgbqgPqhGPFo5wQdGaeF2IM0RGo/UvxMx8ZXq+5J+kR31bQ3FOd59Ui3LxoxE2GkQdDxQ+2IYx3gYX+4xSRQzfuGECqZ+SumXSIJ1abljCnBmV5 +lSO85Z/vT2JFe4nNSRntoHx0iB52jArpGJVRGFD2iF/SOPqwn6836tL7G0ZQ1mdlF/2B9/wC/Uazw +|hEiS � h ˜EiS| +FIG. 6. Variations of ⟨E⟩S, ⟨ ˜E⟩S, and ε = |⟨E⟩S −⟨ ˜E⟩S| as functions of ˜δ and ˜g for a 3×2 lattice. The top, middle, and bottom horizontal +panels are respectively for the measurement of M zz +i , M zx +i , and M xx +i . All quantities plotted are dimensionless. +by negativity. For a given operator MS with a low-energy +description (i.e., if PgMSPg ̸= 0), we ask the question +as to whether the value ⟨E⟩S obtained from a low-energy +state ρSS of the 2D lattice is approximated by ⟨ ˜E⟩S via a +measurement of ˜ +MS in the corresponding state ˜ρSS of the +1D XXZ model? We ask this question for two classes of +operators - one keeps the low-energy sector invariant, i.e, +PgMSPg = MS, and the other which takes a low-energy +state out of the low-energy manifold, i.e., PgMSPg ̸= MS. +We now identify each S as the rung i, and for demon- +stration, we choose three examples of single-rung operators, +given by M zz +i += σz +i,1 ⊗ σz +i,2 belonging to the first category, +and M xx +i += σx +i,1 ⊗ σx +i,2, and M zx +i += σz +i,1 ⊗ σx +i,2 belonging to +the second category, with their respective low-energy com- +ponents given in Table I. For these operators, we compute +⟨E⟩S in a N × 2 ladder, ⟨ ˜E⟩S in the N-spin 1D effective +model, and ε = |⟨E⟩S − ⟨ ˜E⟩S| for the thermal state (see +Sec. II C 3), with βJ⊥ = 102, where J⊥ = 102. In all our +calculations, entanglement in the subsystem S is computed +over the bipartition 1 rung : rest of the subsystem (1 spin : +rest of the subsystem) in the case of the 2D model (effective +1D model). The behaviours of ⟨E⟩B, ⟨ ˜E⟩B, and ε, in the +case of a 3 × 2 lattice, as functions of ˜δ and ˜g, in the case of +these single-rung operators are depicted in Fig. 6. It is clear +from the figure that the performance of the 1D XXZ model +as a proxy for the 2D model worsens as one shifts from the +first category of operators to the second. We also point out +that in the case of M zz +i , the 1D XXZ model is a good sub- +stitute for the 2D model in the entire PM and AFM phases as +well as in the FM phase where ˜g < 0. +VI. +CONCLUSION AND OUTLOOK +In this paper, we consider a spin-1/2 isotropic Heisenberg +model in a magnetic field on a rectangular zig-zag lattice of +size N×L. We show that in some regimes of system parame- +ters, irrespective of the values of N and L, the 2D model can +be well-approximated by an 1D spin-1/2 XXZ model. In +particular, we show that for specific states in the low-energy +manifold of the 2D model, matrix elements of typical Hermi- + +1.5 +0.14 +1.0 +0.12 +0.5 +0.10 +0.08 +0.0 +0.06 +-0.5 +0.04 +-1.0 +0.02 +-1.5 +0.00 +-2 +0 +2 +41.5 +1.0 +1.0 +0.8 +0.5 +0.6 +0.0 +0.4 +-0.5 +0.2 +-1.0 +-1.5 +0.0 +-2 +0 +2 +41.5 +1.0 +1.0 +0.8 +0.5 +0.6 +0.0 +0.4 +-0.5 +0.2 +-1.0 +-1.5 +0.0 +-2 +0 +2 +41.5 +1.0 +1.0 +0.8 +0.5 +0.6 +0.0 +0.4 +-0.5 +0.2 +-1.0 +-1.5 +- +0.0 +-2 +0 +2 +41.5 +0.20 +1.0 +0.5 +0.15 +0.0 +0.10 +-0.5 +0.05 +-1.0 +-1.5 +0.00 +-2 +0 +2 +41.5 +0.20 +1.0 +0.5 +0.15 +0.0 +0.10 +-0.5 +0.05 +-1.0 +-1.5 +0.00 +-2 +0 +2 +41.5 +0.025 +1.0 +0.020 +0.5 +0.015 +0.0 +0.010 +-0.5 +-1.0 +0.005 +-1.5 +0.000 +-2 +0 +2 +41.5 +0.10 +1.0 +0.08 +0.5 +0.06 +0.0 +0.04 +-0.5 +-1.0 +0.02 +-1.5 +0.00 +-2 +0 +2 +41.5 +0.10 +1.0 +0.08 +0.5 +0.06 +0.0 +0.04 +-0.5 +-1.0 +0.02 +-1.5 +0.00 +-2 +0 +2 +41.5 +0.025 +1.0 +0.020 +0.5 +0.015 +0.0 +0.010 +-0.5 +-1.0 +0.005 +-1.5 +0.000 +-2 +0 +2 +41.5 +0.14 +1.0 +0.12 +0.5 +0.10 +0.08 +0.0 +0.06 +-0.5 +0.04 +-1.0 +0.02 +-1.5 +0.00 +-2 +0 +2 +41.5 +0.14 +1.0 +0.12 +0.5 +0.10 +0.08 +0.0 +0.06 +-0.5 +0.04 +-1.0 +0.02 +-1.5 +0.00 +-2 +0 +2 +414 +tian operators, and non-local quantum correlations such as +entanglement, the 1D XXZ model provides a satisfactory +proxy even when the perturbation parameters are not small. +We further consider the quantification of measurement-based +entanglement, where a measurement over a Hermitian opera- +tor is required, and find that the 1D model can mimic the 2D +model for certain choices of the Hermitian operators. While +we demonstrate these results for low values of N and L, +these positive findings open up an opportunity for investi- +gating observables as well as entanglement in the case of 2D +models with higher N and L using the 1D XXZ model as a +proxy. +We conclude with a discussion on possible future works. +An interesting direction would be to generalize the calcu- +lation where ground-states of individual rungs have higher +degeneracies (d +> +2), so that 2D models with d de- +grees of freedom per cite can be addressed. +It is also +important to ask whether expectation values of Hermitian +operators defined on the 2D model for specific purposes, +such as order parameters [130, 131], or entanglement wit- +ness operators [132, 133], can be estimated using the ef- +fective 1D model. +Moreover, motivated from the results +on measurement-based quantification of entanglement, other +quantum correlations belonging to the quantum information- +theoretic paradigms [7, 10], such as quantum discord [134, +135] and quantum work deficit [136, 137], requiring mea- +surements on subsystems can be investigated. +While our +mapping to 1D XXZ model works for specific sates (see +Sec. II C) in the low-energy manifold of the 2D model, the +calculation can be extended to other low-energy states by +working to higher orders in perturbation theory so that all +degeneracies are lifted. We expect that our formalism can +be applied to the strong leg, left-diagonal, and right-diagonal +limits also, providing a large subspace in the parameter space +of the coupling constants of the 2D model (see Fig. 7) where +the 1D effective theory works. It is also worthwhile to note +that the advantage in using the 1D effective model lies in +the drastic reduction in the degrees of freedom in certain pa- +rameter regimes. It would be interesting to look for other +quantum many-body models where this happens. +ACKNOWLEDGMENTS +We acknowledge the use of QIClib – a modern C++ li- +brary for general purpose quantum information processing +and quantum computing. +Appendix A: Rotational symmetry of the effective Hamiltonian +In this appendix, we take a slightly more abstract approach +and argue that the effective Hamiltonian preserves the rota- +tional symmetry in z direction to any order in perturbation +theory in 1/J⊥. More precisely, writing (note that in this +paper we work with only the leading order) +˜H = +∞ +� +n=0 +J−n +⊥ +˜H(n) +(A1) +ACNHicdVDLSgMxFM34rPVdekmWAQXUmbG2tZd0Y3UTQWrQmcMpl0D +M08SDJCmc5HufFD3IjgQhG3foOZPkBFL4Qczrn35Oa4MaNC6vqzNjM7N7+wWFgqLq+srq2XNjYvRZRwTDo4YhG/dpEgjIakI6lk5DrmBAUuI1du/yTXr+4IFzQKL+QgJnaA/JD2KEZSU7pLVGJl3u3aqV4aNfPQ3Ncrul43D2o5MOtV8yBrOamX3RjQ8n3YcqyY8HhfcNhl/ejZlT +qk8NYBTAzg1gIZi8iqDSbWd0qPlRTgJSCgxQ0J0DT2Wdoq4pJiRrGglgsQI95FPugqGKCDCTkf7ZnBXMR7sRVydUMIR+30iRYEQg8BVnQGSt+K3lpN/ad1E9hp2SsM4kSTE4d6CYMygnmC0KOcYMkGCiDMqdoV4lvEZYq56IKYfpT+D+4NCtGrVI9r5abx5M4CmAb7IA9YIA6aIJT0AYdg +ME9eAKv4E170F60d+1j3DqjTWa2wI/SPr8ADMinmg= +J1 +d � J?, J||, Jd2 +ACNHicdVDLSgMxFM34rPVdekmWAQXUmbG2tZd0Y3UTQWrQmcMpl0D +M08SDJCmc5HufFD3IjgQhG3foOZPkBFL4Qczrn35Oa4MaNC6vqzNjM7N7+wWFgqLq+srq2XNjYvRZRwTDo4YhG/dpEgjIakI6lk5DrmBAUuI1du/yTXr+4IFzQKL+QgJnaA/JD2KEZSU7pLVGJl3u3aqV4aNfPQ3Ncrul43D2o5MOtV8yBrOamX3ZjQ8n3YcqyY8HhfcNhl/ejZFlT +qk8NYBTAzg1gIZi8iqDSbWd0qPlRTgJSCgxQ0J0DT2Wdoq4pJiRrGglgsQI95FPugqGKCDCTkf7ZnBXMR7sRVydUMIR+30iRYEQg8BVnQGSt+K3lpN/ad1E9hp2SsM4kSTE4d6CYMygnmC0KOcYMkGCiDMqdoV4lvEZYq56IKYfpT+D+4NCtGrVI9r5abx5M4CmAb7IA9YIA6aIJT0AYdg +ME9eAKv4E170F60d+1j3DqjTWa2wI/SPr8ADOKnmg= +J2 +d � J?, J||, Jd1 +ACNHicdVDLSsNAFJ34rPVdelmsAguSknS2upOdCO6UbBVaGKYTCbp4 +OTBzEQoaT7KjR/iRgQXirj1G5zUFlT0wjCHc+49c+e4CaNC6vqTNjU9Mzs3X1oLy4tr6xW1ta7Ik45Jh0cs5hfuUgQRiPSkVQycpVwgkKXkUv35qjQL28JFzSOLuQgIXaIgoj6FCOpKdymlkjkx4PXDvT6/t7LXPXrOl1XW+bjVYBzHbTbOQnTjYc5tAKAnjiWAnhSU1RXn5tjG8zdyrVi +QGcGMCJATQU1QVjOvMqTxYXozTkEQSMyREz9ATaWeIS4oZyctWKkiC8A0KSE/BCIVE2Nlo3xuK8aDfszViSQcsd8nMhQKMQhd1Rki2Re/tYL8S+ul0t+zMxolqSQR/nrITxmUMSwShB7lBEs2UABhTtWuEPcR1iqnMsqhMlP4f+ga9aNVr153qweHI7jKIFNsAV2gAHa4AcgzPQARjcg +UfwAl61e+1Ze9Pev1qntPHMBvhR2scnFHynmg= +J|| � J?, J1 +d, J2 +d +ACNnicdVDLSsNAFJ3Ud31FXboZLIKLUpJYW92JbkQFOwDklgm02k6O +HkwMxFKzFe58TvcuXGhiFs/wUnbgIpeGO6Zc+89c+d4MaNCGsazVpqZnZtfWFwqL6+srq3rG5tESUckxaOWMS7HhKE0ZC0JWMdGNOUOAx0vFuT/N6545wQaPwWo5i4gbID+mAYiQV1dMvUmcsYnPfc1OjdnTYsA6sqlEzjKa138iB1axb+9l5L3ViwuMOr4P1e3+Pquq1M9uzGm2sp5eK +SRgIQELCWgqJo8KmMZlT39y+hFOAhJKzJAQtmnE0k0RlxQzkpWdRJAY4VvkE1vBEAVEuOl4wzuKqYPBxFXJ5RwzH6fSFEgxCjwVGeA5FD8ruXkXzU7kYNDN6VhnEgS4slDg4RBGcHcQ9inGDJRgogzKnaFeIh4ghL5XRZmVD8FP4P2lbNbNTqV/XK8cnUjkWwDXbAHjBExyDM3AJWgCDB +/AMXsGb9qi9aO/ax6S1pE1ntsCP0D6/Ax9qKY= +J? � J||, J1 +d, J2 +d +ACBXicbVDLSgMxFM3UV62vUZe6CBahgpSZUtRl0Y1 +0VcE+oB2HTCZtQzOZIckIZdqNG3/FjQtF3PoP7vwb0+kstHoguYdz7iW5x4sYlcqyvozc0vLK6lp+vbCxubW9Y+7utWQYC0yaOGSh6HhIEkY5aSqGOlEgqDAY6Ttja5mfvueCElDfqvGEXECNOC0TzFSWnLNw1Ld7UVERKd1N5lMpr4d3Z6V05cs2iV +rRTwL7EzUgQZGq752fNDHAeEK8yQlF3bipSTIKEoZmRa6MWSRAiP0IB0NeUoINJ0i2m8FgrPuyHQh+uYKr+nEhQIOU48HRngNRQLnoz8T+vG6v+hZNQHsWKcDx/qB8zqEI4iwT6VBCs2FgThAXVf4V4iATCSgdX0CHYiyv/Ja1K2T4rV2+qxdplFkceHI +AjUAI2OAc1cA0aoAkweABP4AW8Go/Gs/FmvM9bc0Y2sw9+wfj4BlKoly8= +(J?, J||, J1 +d, J2 +d) +A +B +C +D +FIG. 7. +Schematic representation of the subspaces denoted by +(A) J⊥ +≫ +J||, J1 +d, J2 +d, (B) J|| +≫ +J⊥, J1 +d, J2 +d, (C) J1 +d +≫ +J⊥, J||, J2 +d, and (D) J2 +d ≫ J⊥, J||, J1 +d in the space of the param- +eters J⊥, J||, J1 +d, and J2 +d. The thick lines in he lattices represent +the strong couplings. While this paper deals with only the sub- +space (A), the 1D effective model in each of these subspaces can be +worked out following the same methodology. +we show below that the [ ˜H(n), Z] = 0, ∀n. The general +formalism of perturbation theory provides [98] +˜H(n) = PgH0Pg + PgH′Pg + PgH′Pe(PeH0Pe)−1 +× +∞ +� +n=0 +� +(PeH′Pe − EI)(PeH0Pe)−1�n PeH′Pg. +(A2) +Note that the only operators on the RHS are H′ and Pg (re- +call that Pg are the projector to ground state manifold). The +proof that [ ˜H(n), Z] = 0 then follows simply from the fact +that [H′, Z] = 0 and [Pg, Z] = 0 – the former is simply +the consequence of rotational symmetry of the 2D system +(Eq. (1)) and the latter we prove below. Note that we can +construct Pg explicitly as +Pg = +� +l +|Ψl⟩ ⟨Ψl| . +(A3) +Since from Eq. (19), |Ψl⟩ is an eigenvector of the Hermitian +operator Z , it is obvious that [Pg, Z] = 0. +Appendix B: Low-energy effective Hamiltonian for OBC along +rungs +In this Section, we present the important steps of deter- +mining the effective coupling strengths ˜Jxy, ˜Jzz, and ˜h in +Eq. (21), with |ψ(i) +0 ⟩ and |ψ(i) +1 ⟩ given in Eqs. (37) and (46). +Similar to Sec. III, we have assumed J⊥ = 1 for brevity. +Also, For ease of reference, we write +H′ = HL + HD + HF . +(B1) + +15 +Here, HL is the leg-Hamiltonian given by +HL = J|| +4 +N +� +i=1 +L +� +j=1 +⃗σi,j.⃗σi+1,j, +(B2) +and HD = HDL + HDR represents the diagonal interactions +with +HDL = J1 +d +4 +N +� +i=1 +L +� +j=1 +⃗σi,j+1.⃗σi+1,j, +(B3) +and +HDR = J2 +d +4 +N +� +i=1 +L +� +j=1 +⃗σi,j.⃗σi+1,j+1 +(B4) +representing the left and right diagonals respectively (see +Fig. 1). The field-part HF of the perturbation Hamiltonian is +given by +HF = −∆h +2 +N +� +i=1 +L +� +j=1 +σz +i,j. +(B5) +We start with the field-contribution in H′, and write HF = +�N +i=1 HFi, HFi being the field-term corresponding to the ith +rung. The action of HFi on the states |ψ(i) +0,1⟩ along with the +definition of τ z +i leads to the effective Hamiltonian +˜HF = −∆h +2 +� +N(L − 1)I + +N +� +i=1 +τ z +i +� +, +(B6) +where I is the identity matrix on the ladder Hilbert space. +We next consider the leg-term HL, and treat the terms cor- +responding to the xy- and the zz-interactions separately, +i.e., HL = Hxy +L ++ Hzz +L . +In the basis |ψ(i) +ki ψ(i+1) +ki+1 ⟩, ac- +tion of the ith term of Hxy +L +provides non-zero values for +only ⟨ψ(i) +0 ψ(i+1) +1 +| . . . |ψ(i) +1 ψ(i+1) +0 +⟩ and its hermitian conju- +gate, leading to the effective Hamiltonian +˜Hxy +L = J|| +4 +N +� +i=1 +� +τ x +i τ x +i+1 + τ y +i τ y +i+1 +� +. +(B7) +On +the +other +hand, +ith +term +of +Hzz +L +additionally +has non-zero values for ⟨ψ(i) +0 ψ(i+1) +0 +| . . . |ψ(i) +0 ψ(i+1) +0 +⟩ and +⟨ψ(i) +1 ψ(i+1) +1 +| . . . |ψ(i) +1 ψ(i+1) +1 +⟩ also, which leads to +˜Hzz +L = NJ|| +16 [3L − 4 + A1] I ++J|| +8 [L − A1] +N +� +i=1 +τ z +i ++J|| +16 [4 − L + A1] +N +� +i=1 +τ z +i τ z +i+1, +(B8) +with the constant A1 given in Eq. (49). +Next, we pick up the diagonal terms HDL and HDR in the +perturbing Hamiltonian (see Eqs. (B3)-(B4)), and proceed +by writing Hxy +D = Hxy +DL + Hxy +DR and Hzz +D = Hzz +DL + Hzz +DR. +Calculation similar to the effective Hamiltonian of Hxy +L and +Hzz +L provides +˜Hxy +D = 1 +4A0(J1 +d + J2 +d) +N +� +i=1 +� +τ x +i τ x +i+1 + τ y +i τ y +i+1 +� +. +(B9) +and +˜Hzz +D = N +16 +� +(L − 1 + A3 + 2A2)(J1 +d + J2 +d) +� +I ++1 +8(L − 1 − A3)(J1 +d + J2 +d) +N +� +i=1 +τ z +i ++ 1 +16 +� +(L − 1 + A3 − 2A2)(J1 +d + J2 +d) +� N +� +i=1 +τ z +i τ z +i+1, +(B10) +with A0, A2, and A3 given by Eqs. (48), (50) and (51) +respectively. Therefore, combining Eqs. (B6), (B7), (B8), +(B9), and (B10), the effective Hamiltonian +˜H = ˜HF + ˜Hxy +L + ˜Hzz +L + ˜Hxy +D + ˜Hzz +D +(B11) +turns out to be as given in Eq. (21), with the effective cou- +plings and the additive constant are given in Eqs. (47). +In the following, we include the details of the above map- +ping in the case of L = 2 [71, 72, 87–89, 91], L = 3 [86, 91], +and L = 4, and provide the expressions for ˜Jxy, ˜Jzz, and ˜h, +where J⊥ is assumed to be 1. +1. +L = 2 +For L = 2 [71, 72, 87–89, 91], The ith rung has doubly +degenerate ground states (d = 2), given by +|ψ(i) +0 ⟩ = |00⟩ , +|ψ(i) +1 ⟩ = +1 +√ +2(|01⟩ − |10⟩), +(B12) +at h′ = 1, with the ground-state energy E0 = −3 ⊥ /4. The +coupling constants in ˜H are given by +˜Jxy = +� +2J|| − J1 +d − J2 +d +� +/8, +(B13) +˜Jzz = +� +2J|| + J1 +d + J2 +d +� +/16, +(B14) +˜h = +� +2J|| − 4∆h + J1 +d + J2 +d +� +/8. +(B15) +where we have used PBC along the legs. Using OBC along +the legs changes the 1D LEH to Eq. (44), with ˜Jxy, ˜Jzz, and +˜h given by Eqs. (B13)-(B15), and +˜h′ = − +� +2J|| + J1 +d + J2 +d +� +/16. +(B16) +2. +L = 3 +For L = 3 in Eq. (1) [86, 91], +|ψ(i) +0 ⟩ = |000⟩ , +|ψ(i) +1 ⟩ = (|001⟩ − 2 |010⟩ + |100⟩)/ +√ +6, +(B17) + +16 +which are degenerate at h′ = 3/2, with ground state energy +E0 = −7/4. The effective coupling constants in Eq. (21) are +given by +˜Jxy = +� +3J|| − 2(J1 +d + J2 +d) +� +/12, +(B18) +˜Jzz = +� +9J|| + 4(J1 +d + J2 +d) +� +/72, +(B19) +˜h = +� +9J|| + 11(J1 +d + J2 +d) − 18∆h +� +/36. +(B20) +In the case of OBC along the legs, the effective Hamiltonian +is modified to Eq. (44) with +˜h′ = − +� +9J|| + 11(J1 +d + J2 +d) +� +/72. +(B21) +3. +L = 4 +For L = 4, the doubly-degenerate ground states at h′ = +1 + 1/ +√ +2 are given by +|ψ0⟩ = |0000⟩ , +|ψ1⟩ = +� +− |0001⟩ + a |0010⟩ − a |0100⟩ ++ |1000⟩ +� +/ +� +2 + 2a2, +(B22) +with a = 1 + +√ +2. +The effective coupling constants in +Eq. (21) are given by +˜Jxy = +� +8J|| − (2 + 3 +√ +2)(J1 +d + J2 +d) +� +/32, +˜Jzz = +� +12J|| + (2 +√ +2 + 5)(J1 +d + J2 +d) +� +/128, +˜h = +� +20J|| − 32∆h + (2 +√ +2 + 19)(J1 +d + J2 +d) +� +/64. +(B23) +Appendix C: Trace distance +Here we briefly discuss the trace and Bures distance met- +rics used in this paper for quantifying the distance between +density matrices of subsystems of the quantum spin ladder. +Given two density operators ρ1 and ρ2 defined on a Hilbert +space of dimension d, the trace-distance between ρ1 and ρ2 +is defined as [2] +dT (ρ1, ρ2) = 1 +2 +� +(ρ1 − ρ2)†(ρ1 − ρ2) += 1 +2 +d +� +i=1 +|λi|, +(C1) +where λi are the singular values of the Hermitian matrix ρ1− +ρ2. Note that ρ1 − ρ2 is not necessarily positive. +Appendix D: Odd number of spins on a rung with vanishing +magnetic field +For ease of presenting the steps of the calculation, we set +j1 +d = J2 +d = 0, and redefine the states in Eqs. (55) and (56) as +|ψ(i) +0 ⟩ = +� +{j}∈S +a{j} |{j}⟩ , +|ψ(i) +1 ⟩ = +� +{j}∈S +a{j} |{j}⟩ . +(D1) +Here, {j} = {j1, j2, · · · , jl}, such that j1 < j2 < · · · < +j(l−1) < jl, with each j ∈ S where S ≡ {1, 2, . . . , L−1, L}. +In each of the terms in |ψ(i) +0 ⟩ (|ψ(i) +1 ⟩), the indices {j} in- +dicate the lattice sites of the spins that are in the state +|1⟩ (|0⟩), and |{j}⟩ is the complement of |{j}⟩, such that +|ψ(i) +1 ⟩ = (σx)⊗L |ψ(i) +0 ⟩ (see Eq. (56)). We follow the pre- +scription given in Sec. II B. First, we write the leg term as +HL = Hxy +L + Hzz +L (see Appendix B), where the action of +the ith term in Hxy +L , corresponding to the ith rung, on two +neighbouring rungs (i, i + 1) is given by +⟨ψ(i) +1 ψ(i+1) +0 +| Hxy +Li |ψ(i) +0 ψ(i+1) +1 +⟩ += +� +{j}∈S +� +{j′}∈S +� +k +a{j}a{j′}a{p}a{q}, +⟨ψ(i) +0 ψ(i+1) +1 +| Hxy +Li |ψ(i) +1 ψ(i+1) +0 +⟩ += +� +{j}∈S +� +{j′}∈S +� +k +a{j}a{j′}a{p}a{q}, +(D2) +with {p} = {p1, p2, · · · , pl} ∈ S1, where S1 = S\{k, {j}} +is the set of all elements in S except k and {j}. Similarly, +{q} = {q1, q2, ..ql} ∈ S2 with S2 = S\{k, {j′}}, and +k ∈ S3 with S3 = S\{{j}, {j′}}. The action of Hxy +Li on +|ψ(i) +0 ψ(i+1) +0 +⟩ and |ψ(i) +1 ψ(i+1) +1 +⟩ takes the system out of the +ground state manifold. +The effective Hamiltonian due to +Hxy +L therefore becomes +˜Hxy +L = AJ|| +4 +N +� +i=1 +� +τ x +i τ x +i+1 + τ y +i τ y +i+1 +� +, +(D3) +where +A = +� +{j}∈S +� +{j′}∈S +a{j}a{j′}a{p}a{q}. +(D4) + +17 +On the other hand, non-zero matrix elements of Hzz +Li corre- +sponding to the neighbouring rungs (i, i + 1) are +⟨ψ(i) +0 ψ(i+1) +0 +| Hzz +Li |ψ(i) +0 ψ(i+1) +0 +⟩ += +� +{j}∈S +� +{j′}∈S +[2(n1 + n2) − L] a2 +{j}a2 +{j′}, +⟨ψ(i) +0 ψ(i+1) +1 +| Hzz +Li |ψ(i) +0 ψ(i+1) +1 +⟩ += +� +{j}∈S +� +{j′}∈S +[L − 2(n1 + n2)] a2 +{j}a2 +{j′}, +⟨ψ(i) +1 ψ(i+1) +0 +| Hzz +Li |ψ(i) +1 ψ(i+1) +0 +⟩ += +� +{j}∈S +� +{j′}∈S +[L − 2(n1 + n2)] a2 +{j}a2 +{j′}, +⟨ψ(i) +1 ψ(i+1) +1 +| Hzz +Li |ψ(i) +1 ψ(i+1) +1 +⟩ += +� +{j}∈S +� +{j′}∈S +[2(n1 + n2) − L] a2 +{j}a2 +{j′}, +(D5) +where n1 is the cardinality of {j} ∩ {j′}, and n2 is the car- +dinality of (S\{j}) ∩ (S\j′). The effective Hamiltonian due +to Hzz +L , therefore, is +˜Hzz +L = B J|| +4 +N +� +i=1 +(τ z +i τ z +i+1) , +(D6) +where +B = +� +{j}∈S +� +{j′}∈S +[2(n1 + n2) − L] a2 +{j}a2 +{j′}. +(D7) +The effective Hamiltonian ˜H = ˜Hxy +L + ˜Hzz +L , therefore, takes +the form of an 1D XXZ Hamiltonian given by Eq. (57), +where the coupling constants ˜Jxy and ˜Jzz are given by +˜Jxy = (J||A4)/4, +˜Jzz = (J||B4)/4. +(D8) +We now demonstrate the crucial steps of the calculation in +the case of three-leg ladder at h = 0. We first write |ψ0⟩ and +|ψ1⟩ as (see Eq. (D1)) +|ψ0⟩ = (a1 |100⟩ + a2 |010⟩ + a3 |001⟩), +|ψ1⟩ = (a1 |011⟩ + a2 |101⟩ + a3 |110⟩), +(D9) +where a1 = a3 = 1/ +√ +6, and a2 = −2/ +√ +6. In this no- +tation, consider the action of Hxy +L +on one of the terms in +|ψ(i) +0 ψ(i+1) +1 +⟩, say, Hxy +L (a2 +1 |100011⟩). Here i1 = 1, j1 = 1, +S ≡ {1, 2, 3}, and S\{i1, j1} ≡ 2, 3, implying k ∈ {2, 3}. +For k = 2, p1 = S\{k, i1} = 3, q1 = S\{k, j1} = 3, and +for k = 3, p1 = S\{k, i1} = 2, q1 = S\{k, j1} = 2. This +leads to +⟨ψ(i) +0 ψ(i+1) +1 +| Hxy +L (a2 +1 |100011⟩) += +� +k +ai1aj1ap1aq1 = a2 +1a2 +2 + a2 +1a2 +3. +(D10) +Proceeding in the same way for other terms in |ψ(i) +0 ψ(i+1) +1 +⟩, +as well as the other elements of the basis +{|ψ(i) +0 ψ(i+1) +0 +⟩ , |ψ(i) +0 ψ(i+1) +1 +⟩ , |ψ(i) +1 ψ(i+1) +0 +⟩ , |ψ(i) +1 ψ(i+1) +1 +⟩}, +(D11) +and substituting values of a1, a2, a3, we obtain +⟨ψ(i) +1 ψ(i+1) +0 +| Hxy +L |ψ(i) +0 ψ(i+1) +1 +⟩ = 1, +⟨ψ(i) +0 ψ(i+1) +1 +| Hxy +L |ψ(i) +1 ψ(i+1) +0 +⟩ = 1, +(D12) +while the other matrix elements vanish. +To +see +the +action +of +Hzz +L +on +a +pair +of +neigh- +bouring rungs (i, i + 1), first consider the action of +Hzz +L +on +one +of +the +terms +of +in +|ψ(i) +0 ψ(i+1) +0 +⟩, +say +Hzz +L (a1a2 |100010⟩). To determine the contribution of this +term to ⟨ψ(i) +0 ψ(i+1) +0 +| Hzz +Li |ψ(i) +0 ψ(i+1) +0 +⟩, note that i1 = 1, +j1 = 2, I ≡ {1}, and J ≡ {2}, which leads to n1 = +0, and n2 = 1. +Therefore, for i1 = 1 and j1 = 2, +n1 + n2 − (L − n1 − n2) = −1. Using the same procedure +for other terms in |ψ(i) +0 ψ(i+1) +0 +⟩ as well as the other elements +of the basis, and substituting values for a1, a2 and a3, we +obtain non-zero matrix elements of Hzz +L as +⟨ψ(i) +0 ψ(i+1) +0 +| Hzz +Li |ψ(i) +0 ψ(i+1) +0 +⟩ = 1 +⟨ψ(i) +0 ψ(i+1) +1 +| Hzz +Li |ψ(i) +0 ψ(i+1) +1 +⟩ = −1 +⟨ψ(i) +1 ψ(i+1) +0 +| Hzz +Li |ψ(i) +1 ψ(i+1) +0 +⟩ = −1 +⟨ψ(i) +1 ψ(i+1) +1 +| Hzz +Li |ψ(i) +1 ψ(i+1) +1 +⟩ = 1. +(D13) +This leads to A4 = 1 = B4 = 1 (see Eqs. (D4) and (D7)). +Therefore, the coefficients of the 1D LEH (Eq. (57)), given +by Eq. (D8), are +˜Jxy = (J||)/4, +˜Jzz = (J||)/4. +(D14) +We point out here that the coupling constants can also be +determined in a similar fashion with perturbations J1 +d, J2 +d ̸= +0, where the overall form of Eq. (57) remains unchanged. +Appendix E: Entanglement and related measures of quantum +correlations +In this section, we provide brief definitions of the quan- +tum correlation measures investigated in the quantum spin +ladder. +More specifically, we focus on bipartite entan- +glement [25, 26] between different parts of the system, +as quantified by negativity [25, 118–123], and multipartite +quantum correlation over the complete system, or a cho- +sen subsystem, as quantified by the entanglement monogamy +score [112, 113, 115]. +1. +Negativity +Entanglement between two partitions A and B of a bipar- +tite quantum state ρAB is quantified by a bipartite entangle- +ment measure [25, 26], such as the negativity [118–124]. It +is defined as +ES:S = ||ρ +TS +SS|| − 1, +(E1) +with ||ϱ|| = Tr +� +ϱ†ϱ being the trace norm of ϱ (see also Ap- +pendix C), and ρ +TS +SS is obtained by performing partial trans- +position of the density matrix ρSS with respect to the sub- +system S. + +18 +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjkEvHiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmPi +h4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4Ss +GLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicglRJgUav8uUGCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1ZB +NrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3Uj5gi +nE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kckiNyQhxySerkljRIk3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4AruS3 +A=˜� +AB8HicbVDLSgNBE +OyNrxhfUY9eFoPgKeyKr2PQi8cI5iHJEmZnO8mQmdlZlY +IS7CiwdFvPo53vwbJ8keNLGgoajqprsrTDjTxvO+ncLK6 +tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7UQ +hESHVji6nfqtJ1SaxfLBjBMBlI1meUGCs9dg3jEWaDS +a9c8areDO4y8XNSgRz1XvmrG8U0FSgN5UTrju8lJsiIMo 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+tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7UQ +hESHVji6nfqtJ1SaxfLBjBMBlI1meUGCs9dg3jEWaDS +a9c8areDO4y8XNSgRz1XvmrG8U0FSgN5UTrju8lJsiIMo +xynJS6qcaE0BEZYMdSQTqIJsdPHFPrBK5/VjZksadqb8n +MiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6J3e +n3bsQUsPHlhCqmL3VpUOiCDU2o5INwV98eZk0z6r+ZfXi +/rxSu8njKMIRHMp+HAFNbiDOjSAgoBneIU3RzkvzrvzMW +8tOPnMIfyB8/kDLMiQqw=˜g +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjkEv +HiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmPih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicglRJgUa +v8uUGCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1ZBNrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3Uj5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kckiNyQhxySer +kljRIk3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4AruS3A=˜� +AB8HicbVDLSgNBEOyNrxhfUY9eFoPgKeyKr2PQi8cI5iHJEmZnO8mQmdlZlYIS7CiwdFvPo53vwbJ8keNL +GgoajqprsrTDjTxvO+ncLK6tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7UQhESHVji6nfqtJ1SaxfLBjB +MBlI1meUGCs9dg3jEWaDSa9c8areDO4y8XNSgRz1XvmrG8U0FSgN5UTrju8lJsiIMoxynJS6qcaE0BEZYMdSQTqI +JsdPHFPrBK5/VjZksadqb8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6J3en3bsQUsPHlhCqmL3VpUOiC +DU2o5INwV98eZk0z6r+ZfXi/rxSu8njKMIRHMp+HAFNbiDOjSAgoBneIU3RzkvzrvzMW8tOPnMIfyB8/kDLMiQqw=< +/latexit>˜g +AB8HicbVDLSgNBEOyNrxhfUY9eFoPgKeyKr2PQ +i8cI5iHJEmZnO8mQmdlZlYIS7CiwdFvPo53vwbJ8keNLGgoajqprsrTDjTxvO+ncLK6tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7UQhESHVji6nfqtJ1SaxfLBjBMBlI1meUGCs9dg3jEWaDSa9c8areDO4y8XNSgRz1Xvm +rG8U0FSgN5UTrju8lJsiIMoxynJS6qcaE0BEZYMdSQTqIJsdPHFPrBK5/VjZksadqb8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6J3en3bsQUsPHlhCqmL3VpUOiCDU2o5INwV98eZk0z6r+ZfXi/rxSu8njKMIRHMp+HAFNbi +DOjSAgoBneIU3RzkvzrvzMW8tOPnMIfyB8/kDLMiQqw=˜g +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjkEv +HiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmPih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicglRJgUa +v8uUGCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1ZBNrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3Uj5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kckiNyQhxySer +kljRIk3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4AruS3A=˜� +AFM +PM +FM +AFM +PM +FM +ACInicdVBLSwMxGMz6rPW16tFLsAgeZNmVvYgFL14rGAfsF1KNk3b0Gx2SbKFsvS3ePGvePGgqCfBH2O6bUFBwLDzHzJlwliRqWy7Q9jZXVtfWMzt5Xf3tnd2zcPDpsySgQ +mDRyxSLQDJAmjnDQUVYy0Y0FQGDSCkbXM781JkLSiN+pSUz8EA047VOMlJa6pt2sks8MQj81LZc23FLlfM5qRYXpOxO2MkSCwpi/ilbdnOtGsWlnG4jMNlHDqWnaEAFqh3zbdOL8JSLjCDEnp +OXas/BQJRTEj03wnkSRGeIQGxNOUo5BIP82m8JTrfRgPxL6cAUz9ftEikIpJ2GgkyFSQ/nbm4l/eV6i+lU/pTxOFOF4/lA/YVBFcNYX7FBsGITRAWVO8K8RAJhJVuNa9LWP4U/k+aF5ZTtkq3 +xULtalFHDhyDE3AGHFABNXAD6qABMLgHj+AZvBgPxpPxarzPoyvGYuYI/IDx+QVJr6B6" = 0.01 +ACInicdVBLSwMxGMz6rPW16tFLsAgeZNmVvYgFL14rGAfsF1KNk3b0Gx2SbKFsvS3ePGvePGgqCfBH2O6bUF +BwLDzHzJlwliRqWy7Q9jZXVtfWMzt5Xf3tnd2zcPDpsySgQmDRyxSLQDJAmjnDQUVYy0Y0FQGDSCkbXM781JkLSiN+pSUz8EA047VOMlJa6pt2sks8MQj81LZc23FLlfM5qRYXpOxO2MkSCwpi/ilbdnOtGsWlnG4jMNlHDqWnaEAFqh3zbdOL8JSLjCDEnpOXas/BQJRTEj03wnkSRGeIQGxNOUo5BIP82m8JTrfRgPxL6cAUz9ftEikIpJ2GgkyFSQ/nbm4l/eV6i+lU +/pTxOFOF4/lA/YVBFcNYX7FBsGITRAWVO8K8RAJhJVuNa9LWP4U/k+aF5ZTtkq3xULtalFHDhyDE3AGHFABNXAD6qABMLgHj+AZvBgPxpPxarzPoyvGYuYI/IDx+QVJr6B6" = 0.01 +(d) L=2 +(e) L=2 +(f) L=2 +(b) +(c) +(d) +(e) +(f) +FIG. 8. Variations of ⟨A⟩, ⟨ ˜A⟩, and ε = ⟨A⟩ − ⟨ ˜A⟩ as functions of ˜δ and ˜g for a 3 × 2 lattice, where A = σz +i,1 ⊗ σz +i,2 (plotted in (a)), +and A = σz +i,1 (plotted in (d)). The corresponding low-energy operators are ˜A = τ z +i (plotted in (b)) and ˜A = (Ii + τ z +i )/2 (plotted in (e)) +respectively (see Table I). All quantities plotted are dimensionless. +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjkEvHiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYM +mPih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRx +Pc4SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicglRJgUav8uUGCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj5 +0DY1ZBNrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDG +lTC3Uj5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kckiNyQhxySerkljRIk3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH +9gf4AruS3A=˜� +AB8HicbVDLSgNB +EOyNrxhfUY9eFoPgKeyKr2PQi8cI5iHJEmZnO8mQmdlZ +lYIS7CiwdFvPo53vwbJ8keNLGgoajqprsrTDjTxvO+nc +LK6tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbh +mO7UQhESHVji6nfqtJ1SaxfLBjBMBlI1meUGCs9dg3 +jEWaDSa9c8areDO4y8XNSgRz1XvmrG8U0FSgN5UTrju8l +JsiIMoxynJS6qcaE0BEZYMdSQTqIJsdPHFPrBK5/VjZk +sadqb8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfF +E/5a6J3en3bsQUsPHlhCqmL3VpUOiCDU2o5INwV98eZk +0z6r+ZfXi/rxSu8njKMIRHMp+HAFNbiDOjSAgoBneIU3 +RzkvzrvzMW8tOPnMIfyB8/kDLMiQqw=˜g +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjk +EvHiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmPih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicgl +RJgUav8uUGCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1ZBNrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3Uj5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kck +iNyQhxySerkljRIk3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4AruS3A=˜� +AB8HicbVDLSgNBEOyNrxhfUY9eFoPgKeyKr2PQi8cI5iHJEmZnO8mQmdlZlYIS7CiwdFvPo53vwbJ8k +eNLGgoajqprsrTDjTxvO+ncLK6tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7UQhESHVji6nfqtJ1Sax +fLBjBMBlI1meUGCs9dg3jEWaDSa9c8areDO4y8XNSgRz1XvmrG8U0FSgN5UTrju8lJsiIMoxynJS6qcaE0BEZYMd +SQTqIJsdPHFPrBK5/VjZksadqb8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6J3en3bsQUsPHlhCq +mL3VpUOiCDU2o5INwV98eZk0z6r+ZfXi/rxSu8njKMIRHMp+HAFNbiDOjSAgoBneIU3RzkvzrvzMW8tOPnMIfyB8/ +kDLMiQqw=˜g +AFM +PM +FM +AFM +PM +FM +(a) +(b) +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjk +EvHiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmPih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicgl +RJgUav8uUGCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1ZBNrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3Uj5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kck +iNyQhxySerkljRIk3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4AruS3A=˜� +AB8HicbVDLSgNBEOyNrxhfUY9eFoPgKeyKr2 +PQi8cI5iHJEmZnO8mQmdlZlYIS7CiwdFvPo53vwbJ8keNLGgoajqprsrTDjTxvO+ncLK6tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7UQhESHVji6nfqtJ1SaxfLBjBMBlI1meUGCs9dg3jEWaDSa9c8areDO4y8XNSgR +z1XvmrG8U0FSgN5UTrju8lJsiIMoxynJS6qcaE0BEZYMdSQTqIJsdPHFPrBK5/VjZksadqb8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6J3en3bsQUsPHlhCqmL3VpUOiCDU2o5INwV98eZk0z6r+ZfXi/rxSu8njKMIRH +Mp+HAFNbiDOjSAgoBneIU3RzkvzrvzMW8tOPnMIfyB8/kDLMiQqw=˜g +AFM +PM +FM +(c) +ACInicdVBLSwMxGMz6rPW16tFLsAgeZNmVvYgFL14rGAfsF1KNk3b0Gx2SbKFsvS3ePGvePGgqCfBH2O6bUFBwLDzHzJlwliRqWy7Q9jZXVtfW +Mzt5Xf3tnd2zcPDpsySgQmDRyxSLQDJAmjnDQUVYy0Y0FQGDSCkbXM781JkLSiN+pSUz8EA047VOMlJa6pt2sks8MQj81LZc23FLlfM5qRYXpOxO2MkSCwpi/ilbdnOtGsWlnG4jMNlHDqWnaEAFqh3zbdOL8JSLjCDEnpOXas/BQJRTEj03wnkSRGeIQGxNOUo5BIP82m8JTrfRgPxL6cAUz9ftEikIpJ2GgkyFSQ/nbm4l/eV6i+lU/pTxOFOF4/lA/ +YVBFcNYX7FBsGITRAWVO8K8RAJhJVuNa9LWP4U/k+aF5ZTtkq3xULtalFHDhyDE3AGHFABNXAD6qABMLgHj+AZvBgPxpPxarzPoyvGYuYI/IDx+QVJr6B6" = 0.01 +ACInicdVBLSwMxGMz6rPW16tFLsAgeZNmVvYg +FL14rGAfsF1KNk3b0Gx2SbKFsvS3ePGvePGgqCfBH2O6bUFBwLDzHzJlwliRqWy7Q9jZXVtfWMzt5Xf3tnd2zcPDpsySgQmDRyxSLQDJAmjnDQUVYy0Y0FQGDSCkbXM781JkLSiN+pSUz8EA047VOMlJa6pt2sks8MQj81LZc23FLlfM5qRYXpOxO2MkSCwpi/ilbdnOtGsWlnG4jMNlHDqWnaEAFqh3zbdOL8JSLjCDEnpOXas/BQJRTEj03wnkSRGeIQGxNOUo5B +IP82m8JTrfRgPxL6cAUz9ftEikIpJ2GgkyFSQ/nbm4l/eV6i+lU/pTxOFOF4/lA/YVBFcNYX7FBsGITRAWVO8K8RAJhJVuNa9LWP4U/k+aF5ZTtkq3xULtalFHDhyDE3AGHFABNXAD6qABMLgHj+AZvBgPxpPxarzPoyvGYuYI/IDx+QVJr6B6" = 0.01 +AB8HicbVDLSgNB +EOyNrxhfUY9eFoPgKeyKr2PQi8cI5iHJEmZnO8mQmdlZ +lYIS7CiwdFvPo53vwbJ8keNLGgoajqprsrTDjTxvO+nc +LK6tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbh +mO7UQhESHVji6nfqtJ1SaxfLBjBMBlI1meUGCs9dg3 +jEWaDSa9c8areDO4y8XNSgRz1XvmrG8U0FSgN5UTrju8l +JsiIMoxynJS6qcaE0BEZYMdSQTqIJsdPHFPrBK5/VjZk +sadqb8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfF +E/5a6J3en3bsQUsPHlhCqmL3VpUOiCDU2o5INwV98eZk +0z6r+ZfXi/rxSu8njKMIRHMp+HAFNbiDOjSAgoBneIU3 +RzkvzrvzMW8tOPnMIfyB8/kDLMiQqw=˜g +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjkEvHiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYM +mPih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRx +Pc4SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicglRJgUav8uUGCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj5 +0DY1ZBNrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDG +lTC3Uj5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kckiNyQhxySerkljRIk3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH +9gf4AruS3A=˜� +AB8HicbVDLSgNBEOyNrxhfUY9eFoPgKeyKr2PQi8cI5iHJEmZnO8mQmdlZlYIS7CiwdFvPo53vwbJ8k +eNLGgoajqprsrTDjTxvO+ncLK6tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7UQhESHVji6nfqtJ1Sax +fLBjBMBlI1meUGCs9dg3jEWaDSa9c8areDO4y8XNSgRz1XvmrG8U0FSgN5UTrju8lJsiIMoxynJS6qcaE0BEZYMd +SQTqIJsdPHFPrBK5/VjZksadqb8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6J3en3bsQUsPHlhCq +mL3VpUOiCDU2o5INwV98eZk0z6r+ZfXi/rxSu8njKMIRHMp+HAFNbiDOjSAgoBneIU3RzkvzrvzMW8tOPnMIfyB8/ +kDLMiQqw=˜g +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjk +EvHiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmPih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicgl +RJgUav8uUGCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1ZBNrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3Uj5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kck +iNyQhxySerkljRIk3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4AruS3A=˜� +AB9XicbVDJSgNBEO1xjXGLevTSGARPYUbcjk +EvHiOYBTJj6OmpSZr0LHTXKGHIf3jxoIhX/8Wbf2MnmYMmPih4vFdFVT0/lUKjbX9bS8srq2vrpY3y5tb2zm5lb7+lk0xaPJEJqrjMw1SxNBEgRI6qQIW+RLa/vBm4rcfQWmRxPc4SsGLWD8WoeAMjfTgopAB5G4AEtm4V6naNXsKukicgl +RJgUav8uUGCc8iJFLpnXsVP0cqZQcAnjsptpSBkfsj50DY1ZBNrLp1eP6bFRAhomylSMdKr+nshZpPUo8k1nxHCg572J+J/XzTC8nIRpxlCzGeLwkxSTOgkAhoIBRzlyBDGlTC3Uj5ginE0QZVNCM78y4ukdVpzLmrnd2fV+nUR4kck +iNyQhxySerkljRIk3CiyDN5JW/Wk/VivVsfs9Ylq5g5IH9gf4AruS3A=˜� +AFM +PM +FM +AFM +PM +FM +AFM +PM +FM +ACInicdVBLSwMxGMz6rPW16tFLsAgeZNmVvYgFL14rGAfsF1KNk3b0Gx2SbKFsvS3ePGvePGgqCfBH2O6bUFBwLDzHzJlwliRqWy7Q9jZXVtfWMzt5Xf3tnd2zcPDpsySgQmDRyxSLQDJAmjnDQUVYy0Y0FQGDSCkbXM781JkLSiN+pSUz8EA047VOMlJa6pt2sks8MQj81LZc23FLlfM5qRYXpOxO2MkSCwpi/ilbdnOtGs +WlnG4jMNlHDqWnaEAFqh3zbdOL8JSLjCDEnpOXas/BQJRTEj03wnkSRGeIQGxNOUo5BIP82m8JTrfRgPxL6cAUz9ftEikIpJ2GgkyFSQ/nbm4l/eV6i+lU/pTxOFOF4/lA/YVBFcNYX7FBsGITRAWVO8K8RAJhJVuNa9LWP4U/k+aF5ZTtkq3xULtalFHDhyDE3AGHFABNXAD6qABMLgHj+AZvBgPxpPxarzPoyvGYuYI/IDx+QVJr6B6" = 0.01 +(d) +(e) +(f) +AB8HicbVDLSgNBEOyNrxhfUY9eFoPgKeyKr2 +PQi8cI5iHJEmZnO8mQmdlZlYIS7CiwdFvPo53vwbJ8keNLGgoajqprsrTDjTxvO+ncLK6tr6RnGztLW9s7tX3j9o6jhVFBs05rFqh0QjZxIbhmO7UQhESHVji6nfqtJ1SaxfLBjBMBlI1meUGCs9dg3jEWaDSa9c8areDO4y8XNSgR +z1XvmrG8U0FSgN5UTrju8lJsiIMoxynJS6qcaE0BEZYMdSQTqIJsdPHFPrBK5/VjZksadqb8nMiK0HovQdgpihnrRm4r/eZ3U9K+DjMkNSjpfFE/5a6J3en3bsQUsPHlhCqmL3VpUOiCDU2o5INwV98eZk0z6r+ZfXi/rxSu8njKMIRH +Mp+HAFNbiDOjSAgoBneIU3RzkvzrvzMW8tOPnMIfyB8/kDLMiQqw=˜g +AB+nicbVDJSgNBEO2JW4zbRI9eBoPgKcyIG56CIniMYBZIhqGnU0ma9Cx016hnE/x4kERr36JN/GTjI +HjT4oeLxXRVU9PxZcoW1/GYWFxaXleJqaW19Y3PLG83VZRIBg0WiUi2fapA8BAayFAO5ZA19Ayx9dTvzWHUjFo +/AWxzG4AR2EvM8ZRS15ZvnKS53zLsIDphIUZplnVuyqPYX1lzg5qZAcdc/87PYilgQIhNUqY5jx+imVCJnArJSN1E +QUzaiA+hoGtIAlJtOT8+sfa30rH4kdYVoTdWfEykNlBoHvu4MKA7VvDcR/M6CfbP3JSHcYIQstmifiIsjKxJDlaP +S2AoxpQJrm+1WJDKilDnVZJh+DMv/yXNA+rzkn1+OaoUrvI4yiSXbJHDohDTkmNXJM6aRBG7skTeSGvxqPxbLwZ7 +PWgpHP7JBfMD6+AZYqlD0=E1:rest +ACAnicbVDJSgNBEO1xjXEb9SReBoPgKcyIG5 +6CIniMYBbIhNDTqSRNenqG7hoxDIMXf8WLB0W8+hXe/Bs7y0ETHxQ83quiql4QC67Rdb+tufmFxaXl3Ep+dW19Y9Pe2q7qKFEMKiwSkaoHVIPgEirIUA9VkDQEAt6F8N/do9KM0jeYeDGJoh7Ure4YyikVr2ro9ctCG9zlqpd+EjPGCqQG +OWteyCW3RHcGaJNyEFMkG5ZX/57YglIUhkgmrd8NwYmylVyJmALO8nGmLK+rQLDUMlDUE309ELmXNglLbTiZQpic5I/T2R0lDrQRiYzpBiT097Q/E/r5Fg57yZchknCJKNF3US4WDkDPNw2lwBQzEwhDLFza0O61FGZrU8iYEb/rlWVI9K +nqnxZPb40LpchJHjuyRfXJIPHJGSuSGlEmFMPJInskrebOerBfr3foYt85Zk5kd8gfW5w8rc5fz ˜E1:rest +ACFnicbVDJSgNBEO1xjXGLevQyGAQvCTPihq +egBDxGMAskIfT01CRNeha6a8Qwma/w4q948aCIV/Hm39hZDpr4oODxXhV9ZxIcIW9W0sLC4tr6xm1rLrG5tb27md3ZoKY8mgykIRyoZDFQgeQBU5CmhEqjvCKg7/euRX78HqXgY3OEgrZPuwH3OKOopU6uMCx3EvuyhfCAiQSFaVpoIR +cuJOV01hl2cnmraI1hzhN7SvJkikon9VyQxb7ECATVKmbUXYTqhEzgSk2VasIKsT7vQ1DSgPqh2Mn4rNQ+14peKHUFaI7V3xMJ9ZUa+I7u9Cn21Kw3Ev/zmjF6F+2EB1GMELDJIi8WJobmKCPT5RIYioEmlEmubzVZj0rKUCeZ1SHYs +y/Pk9px0T4rnt6e5EtX0zgyZJ8ckCNik3NSIjekQqEkUfyTF7Jm/FkvBjvxsekdcGYzuyRPzA+fwBKqC+ +|E1:rest � ˜E1:rest| +AB9HicbVDLSgMxFL2pr1pfVZdugkVwVWbE17Loxo1QwT6gHUsmzbShmcyYZApl6He4caGIWz/GnX9jp2 +FVg8EDufcyz05fiy4No7zhQpLyura8X10sbm1vZOeXevqaNEUdagkYhU2yeaCS5Zw3AjWDtWjIS+YC1/dJ35rTFTm +kfy3kxi5oVkIHnAKTFW8rohMUNKRHo7fXB75YpTdWbAf4mbkwrkqPfKn91+RJOQSUMF0brjOrHxUqIMp4JNS91Es5j +QERmwjqWShEx76Sz0FB9ZpY+DSNknDZ6pPzdSEmo9CX07mYXUi14m/ud1EhNceimXcWKYpPNDQSKwiXDWAO5zxagR +E0sIVdxmxXRIFKHG9lSyJbiLX/5LmidV97x6dndaqV3ldRThA7hGFy4gBrcQB0aQOERnuAFXtEYPaM39D4fLaB8Zx +9+AX18A7HbkhA= +M1 +AB/nicbVDLSsNAFJ34rPUVFVdugkVwVRLxtS +y6cSNUsA9oYplMbtqhk0mYmQglBPwVNy4Ucet3uPNvnLRZaOuBgcM593LPHD9hVCrb/jYWFpeWV1Yra9X1jc2tbXNnty3jVBokZjFoutjCYxyaCmqGHQTATjyGXT80Xhdx5BSBrzezVOwIvwgNOQEqy01Df3XUVZAJkbYTUkmGW3ef7g9M +2aXbcnsOaJU5IaKtHsm19uEJM0Aq4Iw1L2HDtRXoaFoRBXnVTCQkmIzyAnqYcRyC9bBI/t460ElhLPTjypqovzcyHEk5jnw9WaSUs14h/uf1UhVehnlSaqAk+mhMGWiq2iCyugAohiY0wEVRntcgQC0yUbqyqS3BmvzxP2id157x+d +nda1yVdVTQATpEx8hBF6iBblATtRBGXpGr+jNeDJejHfjYzq6YJQ7e+gPjM8fsBeV9w= ˜ +M1 +ACDnicbZC7TsMwFIadcivlFmBksagqsVAliN +tYwcKCVCR6kZpQOY7TWnWcyHaQqrRPwMKrsDCAECszG2+D02YoLb9k6d3zpHP+b2YUaks68coLC2vrK4V10sbm1vbO+buXlNGicCkgSMWibaHJGUk4aipF2LAgKPUZa3uA6q7ceiZA04vdqGBM3RD1OA4qR0qhrVkZOiFQfI5bejh/sY0 +dR5pN0Bmo6plq2pNBeNnZsyFXvmt+OH+EkJFxhqTs2Fas3BQJRTEj45KTSBIjPEA90tGWo5BIN52cM4YVTXwYREI/ruCEzk6kKJRyGHq6M1tTztcy+F+tk6jg0k0pjxNFOJ5+FCQMqghm2UCfCoIVG2qDsKB6V4j7SCsdIlHYI9f +/KiaZ5U7fPq2d1puXaVx1EB+AQHAEbXIAauAF10AYPIEX8AbejWfj1fgwPqetBSOf2Qd/ZHz9AhS5nMY= +|M1 � ˜ +M1| +FIG. 9. Variations of (a) E1:rest, (b) ˜E1:rest, (c) ε = |E1:rest − ˜E1:rest|, (d) M1, (e) ˜ +M1, and (f) ε = |M − ˜ +M1| as functions of ˜δ and ˜g for a +3 × 2 lattice. The continuous lines signify the value 10−2 for the respective quantities. All quantities plotted are dimensionless. + +1.5 +0.14 +1.0 +0.12 +0.5 +0.10 +0.08 +0.0 +0.06 +-0.5 +0.04 +-1.0 +0.02 +-1.5 +0.00 +-2 +0 +2 +41.5 +0.04 +1.0 +0.03 +0.5 +0.0 +0.02 +-0.5 +0.01 +-1.0 +-1.5 +0.00 +-2 +0 +2 +41.5 +1.0 +0.8 +0.5 +0.6 +0.0 +0.4 +-0.5 +0.2 +-1.0 +-1.5 +-2 +0 +2 +41.5 +1.0 +0.8 +0.5 +0.6 +0.0 +0.4 +-0.5 +-- +0.2 +-1.0 +-1.5 +-2 +0 +2 +41.5 +0.08 +1.0 +0.06 +0.5 +0.0 +0.04 +-0.5 +0.02 +-1.0 +-1.5 +0.00 +-2 +0 +2 +41.5 +0.75 +1.0 +0.50 +0.5 +0.25 +0.0 +0.00 +-0.25 +-0.5 +-0.50 +-1.0 +-0.75 +-1.5 +-2 +0 +2 +41.5 +0.75 +1.0 +0.50 +0.5 +0.25 +0.0 +0.00 +-0.25 +-0.5 +-0.50 +-1.0 +-0.75 +-1.5 +-2 +0 +2 +41.5 +0.14 +1.0 +0.12 +0.5 +0.10 +0.08 +0.0 +0.06 +-0.5 +0.04 +-1.0 +0.02 +-1.5 +0.00 +-2 +0 +2 +41.5 +0.150 +1.0 +0.125 +0.5 +0.100 +0.0 +0.075 +0.050 +-0.5 +0.025 +-1.0 +0.000 +-1.5 +-2 +0 +2 +41.5 +0.150 +1.0 +0.125 +0.5 +0.100 +0.0 +0.075 +0.050 +-0.5 +0.025 +-1.0 +0.000 +-1.5 +-2 +0 +2 +41.5 +0.035 +1.0 +0.030 +0.5 +0.025 +0.0 +0.020 +0.015 +-0.5 +0.010 +-1.0 +0.005 +-1.5 +0.000 +-2 +0 +2 +41.5 +1.0 +0.20 +0.5 +0.15 +0.0 +0.10 +-0.5 +0.05 +-1.0 +-1.5 +-2 +0 +2 +41.5 +1.0 +0.20 +0.5 +0.15 +0.0 +0.10 +-0.5 +0.05 +-1.0 +-1.5 +-2 +0 +2 +41.5 +1.0 +1.0 +0.8 +0.5 +0.6 +0.0 +0.4 +-0.5 +0.2 +-1.0 +-1.5 +0.0 +-2 +0 +2 +41.5 +0.014 +1.0 +0.012 +0.5 +0.010 +0.008 +0.0 +0.006 +-0.5 +0.004 +-1.0 +0.002 +-1.5 +0.000 +-2 +0 +2 +41.5 +1.0 +1.0 +0.8 +0.5 +0.6 +0.0 +0.4 +-0.5 +0.2 +-1.0 +-1.5 +0.0 +-2 +0 +2 +419 +2. +Monogamy of entanglement +For a given bipartite entanglement measure E and an N- +party quantum state ρS1S2···SN , the quantum state is said to +be monogamous [110–114, 117] for the entanglement mea- +sure E if +ESi:rest ≥ +� +i̸=j +ESiSj +(E2) +for i, j ∈ {1, 2, · · · , N} with Si as the nodal observer, +where ESi:rest = E(ρSi:rest) computed over the bipartition +Si : rest of the N-party system, and ESiSj = E(ρSiSj) +with ρSiSj = Tr{Sk;k̸=i,j}[ρS1S2···SN ]. The corresponding +monogamy score [115, 117] is defined as +MSi = ESi:rest − +� +i̸=j +ESiSj, +(E3) +where a positive (negative) value of MSi indicates that the +state ρS1S2···SN is monogamous (non-monogamous) for the +bipartite entanglement measure E with Si as the node. En- +tanglement monogamy score captures multiparty quantum +correlation present in the N-party system [116]. +Appendix F: Expectation values of observables on a rung +Here we briefly discuss the variations of the expectation +values of A = σz +i,1 and A = σz +i,1 ⊗ σz +i,2 on the ith rung of +a 3 × 2 lattice, and their corresponding low-energy compo- +nents ˜A, as functions of ˜δ and ˜g (see Fig. 8). Note that in +the strong rung-coupling limit, a rung is mapped to an ef- +fective two-level system constituting the 1D effective XXZ +model. Therefore, we expect A = σz +i,j and A = σz +i,1 ⊗ σz +i,2 +to qualitatively have similar behaviour as functions of sys- +tem parameters. This is confirmed by Figs. 8(a) and (d). The +low-energy components of A = σz +i,1 ⊗ σz +i,2 and A = σz +i,j +are given by ˜A = τ z +i and ˜A = (Ii + τ z +i )/2 respectively, and +⟨ ˜A⟩ in ˜ρ also match for these two observables over the entire +(˜δ, ˜g) plane. Moreover, ε = |⟨A⟩ − ⟨ ˜A⟩| also match for both +observables. +Appendix G: Bipartite entanglement and entanglement +monogamy score +We now discuss two specific types of quantum correlations +on a 3 × 2 lattice – (a) the bipartite entanglement, E1:rest, as +quantified by negativity (see Appendix E), between the rung +i = 1 and the rest of the system, and (b) the entanglement +monogamy score M1 on the lattice, taking one of the rungs, +say, rung i = 1, as the nodal observer. Correspondingly, in +the 1D effective XXZ model, we compute (a) the bipartite +entanglement ˜E1:rest between the spin i = 1 and the rest of +the system, and (b) the entanglement monogamy score ˜ +M1 +over the 3-spin effective model using the spin i = 1 as the +nodal observer. 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A 71, 062307 (2005). + diff --git a/zNE3T4oBgHgl3EQfmQrA/content/tmp_files/load_file.txt b/zNE3T4oBgHgl3EQfmQrA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a65500c207fcdfea2f0ff009c0e163062a6c6cd3 --- /dev/null +++ b/zNE3T4oBgHgl3EQfmQrA/content/tmp_files/load_file.txt @@ -0,0 +1,3293 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf,len=3292 +page_content='Estimating entanglement in 2D Heisenberg model in the strong rung-coupling limit Chandrima B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' Pushpan, Harikrishnan K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=', Prithvi Narayan, Amit Kumar Pal Department of Physics, Indian Institute of Technology Palakkad, Palakkad 678 623, India (Dated: January 12, 2023) In this paper, we calculate entanglement in the isotropic Heisenberg model in a magnetic field on a two- dimensional rectangular zig-zag lattice in the strong rung-coupling limit, using the one-dimensional XXZ model as a proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' Focusing on the leading order in perturbation, for arbitrary size of the lattice, we show how the one-dimensional effective description emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' We point out specific states in the low-energy sector of the two-dimensional model that are well-approximated by the one-dimensional spin-1/2 XXZ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' We propose a systematic approach for mapping matrix-elements of operators defined on the two-dimensional model to their low-energy counterparts on the one-dimensional XXZ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' We also show that partial trace-based description of entanglement in the two-dimensional model can be satisfactorily approximated using the one-dimensional XXZ model as a substitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' We further show numerically that the one-dimensional XXZ model performs well in estimating entanglement quantified using a measurement-based approach in the two-dimensional model for specific choices of measured Hermitian operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' INTRODUCTION The interface of quantum information theory [1, 2] and low-dimensional interacting quantum spin systems [3, 4], with small lattice dimension with each lattice site hosting a Hilbert space of a few levels, has grown into a rich area of interdisciplinary research [5–10] in the past two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' On one hand, these interacting quantum spin models have been identified as the natural candidates for testing and im- plementing quantum information and computation proto- cols, such as quantum state transfer [11–16], measurement- based quantum computation [17–20], and topological quan- tum error corrections [21–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' The motivation behind these studies has its origin in the natural occurrence of quan- tum states with rich quantum correlations belonging to both entanglement-separability [25, 26] and quantum information theoretic paradigms [7], which, alongside being fundamen- tally important can be used as resource in several quan- tum tasks [7, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' On the other hand, these quantum cor- relations have provided a refreshing perspective of charac- terizing quantum many-body systems [5], along with the development of tools and techniques like projected entan- gled pair states [27–31], and multiscale entanglement renor- malization ansatz [32–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' Current experimental advances allowing implementation and manipulation of these quan- tum spin models as well as quantum protocols designed on these models using trapped ions [39–43], superconducting qubits[44, 45], nuclear magnetic resonance [46, 47], solid- state systems[48, 49], and ultra-cold atoms [50–54] have also provided a major boost to these studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' Among a plethora of low-dimensional quantum spin mod- els, two-dimensional (2D) lattice models have always been specially challenging due to the faster growth of Hilbert space dimension with increasing number of spins in the sys- tem, compared to their one-dimensional (1D) counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' One such model is the Heisenberg model [55–63] in a mag- netic field on a rectangular lattice of NL sites, having re- spectively N and L lattice sites in the horizontal and verti- cal directions, where each lattice site hosts a spin-1/2 parti- cle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' A number of recent studies [64–67] have been carried out to understand the entanglement properties of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' Particular attention has been drawn towards quasi-1D mod- els [68, 69] like quantum spin ladders [70–73] with number of lattice sites in the horizontal direction being far greater than the same in the vertical direction (N ≫ L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' As nat- ural extensions of the 1D models while going towards 2D, quantum spin ladders with L = 2 has been investigated from the perspective of quantum state transfer [74, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' More- over, entanglement [75–83] and fidelity [79, 84] have been investigated in these ladder models from the perspective of phases characterizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' While most of these studies have concentrated on models with spin-1/2 particles, entangle- ment properties of quantum spin ladders with higher spin quantum numbers [85] have also been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' In the limit where the coupling strength J⊥ along the rungs of the ladder are much larger compared to other spin- exchange interactions present in the system, the rung behaves like a single degree of freedom, and the model becomes ef- fectively 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' By further tuning magnetic field strength, for low values of L (L = 2, 3), [86–91] show that the isotropic Heisenberg quantum spin-1/2 ladders map to an 1D XXZ model [92–95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' This mapping has been used to study quan- tum phase transitions in the case of antiferromagentic spin- 1/2 ladders with L = 2 [87–91, 96] and 3 [86, 91], us- ing magnetization properties [87, 88, 90, 91], and entangle- ment [96] (see also [97] for the mapping in the case of a mixed-spin variant of quantum spin ladders with L = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' While most of these studies have focused on local observ- ables, non-local correlations such as entanglement in the low-energy sector of the spin-1/2 Heisenberg model on a quasi-1D, or 2D lattice, using the effective 1D model as a proxy, is yet to be explored, to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' In this paper, we specifically ask whether entanglement properties in the 1D effective model can approximate the cor- responding entanglement in all states of the low-energy man- ifold of the isotropic Heisenberg model in a magnetic field, defined on a quasi-1D, or 2D rectangular zig-zag lattice (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' Towards approaching this question, we lay the ground by determining the 1D effective model in a system size-independent fashion, thereby bringing the specific mod- els, studied so far in a case-by-case basis [86–91, 96], under the same umbrella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content=' As mentioned above, we particularly fo- cus on tuning the magnetic field such that the ground-state arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE3T4oBgHgl3EQfmQrA/content/2301.04615v1.pdf'} +page_content='04615v1 ' metadata={'source': 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Under the same assump- +tions, we also give a quadratic lower bound for the ordered version of the central symmetric +polynomial. +1 +Introduction +Arithmetic Circuit Complexity aims to categorize polynomials according to how hard they are +to compute in algebraic models of computation. The most natural model is that of an arithmetic +circuit: starting from variables or constants, the circuit computes new polynomials by means of +addition and multiplication operations. The question is how many of these operations are needed. +The most challenging problem is to prove super-polynomial lower bounds against arithmetic cir- +cuits computing a low-degree polynomial. This is known as the VP vs VNP problem and is the +algebraic analogue of the famed P vs. NP question. The classical result of Baur and Strassen +[Str73, BS83] gives an Ω(n log d) lower bound for an n variate polynomial of degree d. A variety +of lower bounds has since been obtained by imposing various restrictions on the computational +model - e.g., arithmetic formulas or monotone circuits [Kal85, Val80]. But the result of Baur and +Strassen remains the strongest lower bound on unrestricted arithmetic circuits. +In this paper, we are interested in the non-commutative setting where multiplication does not +multiplicatively commute. Starting with the seminal works of Hyafil [Hya77] and Nisan [Nis91], +*Tel Aviv University, Israel. This work was done while the author was a postdoctoral researcher at the Institute of +Mathematics of the Czech Academy of Sciences, Prague and was supported by the Czech Science Foundation GA ˇCR +grant 19-27871X. Email: prerona.ch@gmail.com +†Institute of Mathematics of the Czech Academy of Sciences, Prague. This work was supported by Czech Science +Foundation GA ˇCR grant 19-27871X. Email: pahrubes@gmail.com. +1 +arXiv:2301.01676v1 [cs.CC] 4 Jan 2023 + +non-commutative circuits are a well-studied object. The lack of commutativity is a severe limita- +tion of the computational power which makes the task of proving circuit lower bounds apparently +easier. Nisan gave an exponential lower bound for non-commutative formulas whereas, commu- +tatively, the best bound is only quadratic [Kal85, CKSV22]. Since then, it seemed that exponential +non-commutative circuit lower bounds are just around the corner. Recently, Limaye, Srinivasan +and Tavenas [TLS22] proved such a lower bound in the homogeneous, constant depth setting. They +showed that any constant depth ∆ non-commutative homogeneous circuit for the iterated matrix +multiplication polynomial over n variables of degree d must have size nΩ(d +1 +∆ ). However for gen- +eral circuits, even in the non-commutative setting, the strongest lower bound remains Ω(n log d) +[Str73, BS83]. +We improve this lower bound to Ω(nd/ log d) under the additional assumption that the non- +commutative circuit is also homogeneous (see section 2 for definition). Non-commutatively, this +is already interesting if n = 2: we obtain a bivariate polynomial of degree d which requires circuit +size nearly linear in d. It is well-known that a (commutative or not) circuit computing a homo- +geneous polynomial of degree d can be converted to an equivalent homogeneous circuit with at +most a d2 increase in size (see, e.g., [HWY11]). Hence, homogeneity is not a serious restriction if +either d is small or if one is after a super-polynomial lower bound – as in the VP vs VNP problem. +However, our results fall in neither category and we do not know how to remove the homogeneity +restriction. Nevertheless, we strongly believe that it can be removed and non-commutative circuit +lower bounds are just around the corner. +2 +Notation and preliminaries +Let F be a field. A non-commutative polynomial over F is a formal sum of products of variables and +field elements. We assume that the variables do not multiplicatively commute, whereas they com- +mute additively, and with elements of F. The ring of non-commutative polynomials in variables +x1, . . . , xn is denoted F ⟨x1, . . . , xn⟩. A polynomial is said to be homogeneous if all monomials with +a non-zero coefficient in f have the same degree. +A non-commutative arithmetic circuit C is a directed acyclic graph as follows. Nodes (or gates) +of in-degree zero are labelled by either a variable or a field element in F. All the other nodes have +in-degree two and they are labelled by either + or ×. The two edges going into a gate labelled by +× are labelled by left and right to indicate the order of multiplication. Gates of in-degree zero will +be called input gates; gates of out-degree zero will be called output gates. +Every node in C computes a non-commutative polynomial in the obvious way. We say that C +computes a polynomial f if there is a gate in C computing f (not necessarily an output gate). C +will be called homogeneous if every gate in C computes a homogeneous polynomial. Given a circuit +C, let �C := { f : f is computed by some gate in C}. +2 + +A product gate will be called non-scalar, if both of its inputs compute a non-constant polyno- +mial. We define the size of C to be the number of non-input gates in it, and the non-scalar size of C +to be the number of non-scalar product gates in it. +Given integers n1, n2, [n1, n2] is the interval {n1, n1 + 1, . . . , n2} and [n] := [1, n]. +Note: Unless stated otherwise, circuits and polynomials are assumed to be non-commutative and +the underlying field F is fixed but arbitrary. +3 +Main results +For univariate polynomials there is no difference between commutative and non-commutative +computations. Already with two variables, non-commutative polynomials display much richer +structure. There are 2d monomials in variables x0, x1 of degree d (as opposed to d + 1 in the com- +mutative world); so a generic bivariate polynomial requires a circuit of size exponential in d. +Our first result is a lower bound that is almost linear in d. The hard polynomial is a bivariate +monomial (a specific product of variables x0, x1). +Theorem 1. For every d > 1, there exists an explicit bivariate monomial of degree d such that any homo- +geneous non-commutative circuit computing it has non-scalar size Ω(d/ log d). +In Remark 10, we point out a complementary O(d/ log d) upper bound for every bivariate mono- +mial. Note that commutatively every such monomial can be computed in size O(log d). +For n-variate polynomials, we obtain a stronger result (the hard polynomial is no longer a +monomial). +Theorem 2. For every n, d > 1 there exists an explicit n-variate homogeneous polynomial of degree d +which requires a homogenous non-commutative circuit of non-scalar size Ω(nd), if d ≤ n, or Ω(nd log n +log d), +if d ≥ n. +Theorem 1 and Theorem 2 are proved in section 4.1 and section 4.2 respectively. +Given 0 ≤ d, n, the ordered symmetric polynomial, OSd +n, is the polynomial1 +OSd +n(x1, . . . , xn) = +∑ +1≤i1<··· n. +3 + +For the central ordered symmetric polynomial OS⌊n/2⌋ +n +, the lower bound becomes Ω(n2). We +also observe that the known commutative upper bounds on elementary symmetric polynomials +work non-commutatively as well. +Proposition 4. OS1 +n, . . . , OSn +n can be simultaneously computed by a non-commutative circuit of size +O(n log2 n log log n), and by a homogeneous non-commutative circuit of size O(n2). +The polylog factor in the proposition depends on the underlying field and can be improved for +some Fs. Moreover, when measuring non-scalar size, one can obtain an O(n log n) upper bound +if F is infinite – this is tight by [BS83]. +The ordered symmetric polynomial can be contrasted with the truly symmetric polynomial +Sk +n = +∑ +i1,...,ik∈[n] distinct +xi1 · · · xik , +Non-commutatively, already Sn +n is as hard as the permanent [HWY11] and is expected to require +exponential circuits. +Remark 5. A polynomial of degree d can be uniquely written as f = ∑d +k=0 f (k) where f (k) is homogeneous +of degree k. It is well-known that if f has a circuit of size s, the homogeneous parts f (0), . . . , f (d) can be +simultaneously computed by a homogeneous circuit of size O(sd2) (this holds non-commutatively as well +[HWY11]). Note that OS0 +n, . . . , OSn +n are the homogeneous parts of ∏n +i=1(1 + xi) which has a circuit of a +linear size. Theorem 3 shows that in this case, homogenization provably costs a factor of the degree. +♦ +4 +Lower bounds against homogeneous non-commutative circuits +Let us define the measure we use to prove our lower bounds. Suppose f ∈ F ⟨x1, . . . , xn⟩ is a +homogeneous polynomial of degree d. Given an interval J = [a, b] ⊆ [d], the polynomial f J is +obtained be setting variables in position outside of J to one. More precisely, if α = ∏d +i=1 xji is a +monomial then αJ := ∏b +i=a xji, and the map is extended linearly so that f J = ∑k ckαJ +k whenever +f = ∑k ckαk. Given a non-negative integer ℓ, let +F ℓ( f ) = +� +f J : J ⊆ [d] is an interval of length ℓ +� +. +Given homogeneous polynomials f1, . . . , fm, our hardness measure is defined as +µℓ( f1, . . . , fm) := dim(span( +m +� +i=1 +F ℓ( fi))) . +Here, span(F) denotes the vector space of F-linear combinations of polynomials in F and dim is +its dimension. +4 + +The following lemma bounds the measure in terms of circuit size. +Lemma 6. Let C be a homogeneous circuit with s non-scalar multiplication gates. Then for every ℓ ≥ 2, +µℓ(�C) ≤ (ℓ − 1)s. +Proof. This is by induction on the size of C. If C consists of input gates only then F ℓ(�C) = ∅, as +we assumed ℓ ≥ 2 and �C consists of linear polynomials. +Otherwise, assume that u is some output gate of C and let C′ be the circuit obtained by remov- +ing that gate. If u is a sum gate or a scalar product gate then +µℓ(�C) ≤ µℓ( �C′) . +For if u computes f then f = a1 f1 + a2 f2 for some constants a1, a2 and f1, f2 ∈ �C′. If f has degree d +then for every interval J ⊆ [d] of length ℓ, f J = (a1 f1 + a2 f2)J = a1 f J +1 + a2 f J +2 ∈ span(F ℓ( �C′)). +If u is a non-scalar product gate computing f = f1 · f2 then +µℓ(�C) ≤ µℓ( �C′) + (ℓ − 1) . +To see this assume f1, f2 have degrees d1 and d2 respectively, and let J ⊆ [d1 + d2] be an interval +of length ℓ. If J is contained in [d1], f J = ( f1 f2)J = f J +1 f ∅ +2 is a scalar multiple of f J +1 and hence f J is +contained in span(F ℓ( �C′)); similarly if J is contained in [d1 + 1, d2]. Otherwise, both d1 and d1 + 1 +are contained in J. But there are only ℓ − 1 such intervals. Hence F ℓ(�C) contains at most ℓ − 1 +polynomials outside of span(F ℓ( �C′)). +This means that µℓ increases only at product gates, and that it increases only by ℓ − 1 at such +gates. Hence µℓ(�C) ≤ (ℓ − 1)s. +Remark 7. If f has n variables and degree d, the measure µℓ( f ) can be at most the minimum of d − (ℓ − 1) +and nℓ. Hence, Lemma 6 can by itself give a lower of at most the order of d log n/ log d. +♦ +4.1 +Lower bounds for a single monomial +Interestingly, Lemma 6 gives non-trivial lower bounds for f being merely a product of variables. +The simplest example is an n-variate product of a quadratic degree. +Proposition 8. Every homogeneous circuit computing f = ∏n +i=1 ∏n +j=1(xixj) contains at least n2 non- +scalar product gates. +Proof. This is an application of Lemma 6 with ℓ = 2. The family F 2( f ) consists of all monomials +xixj. Hence, µ2( f ) = n2. If C computes f, we have µ2(�C) ≥ µ2( f ) and hence C contains at least n2 +product gates. +5 + +Another case of interest is a monomial in two variables, x0, x1, of degree d. Suppose f = +∏d +i=1 xσi where σ = (σ1, . . . , σd) ∈ {0, 1}d. Then µℓ( f ) equals the number of distinct substrings +of σ of length ℓ. Hence we want to find a σ which contains as many substrings as possible. One +construction of such an object is provided by the de Bruijn sequence [dB46]. +de Bruijn sequences +For a given k, a de Bruijn sequence of order k over alphabet A is a cyclic +sequence σ in which every k-length string from Ak occurs exactly once as a substring. Note that +σ must have length |A|k. Furthermore, precisely k − 1 of the substrings overlap the beginning +and the end of the sequence and σ contains |A|k − (k − 1) substrings when viewed as an ordinary +sequence. de Bruijn sequences are widely studied and, in particular, they exist. Moreover, efficient +algorithms are known for constructing de Bruijn sequences (see, for example, [SWW16] and its +references). In the case of binary alphabet A = {0, 1}, this is especially so. We can start with a +string of k zeros. At each stage, extend the sequence by 1, unless this results in a k-string already +encounters, otherwise extend by 0. +Given d ≥ 2, let σ be a binary de Bruijn sequence of order ⌈log2 d⌉. It has length 2⌈log2 d⌉ ≥ d. +Define the polynomial +Bd(x0, x1) := +d +∏ +i=1 +xσi . +The following implies the result of Theorem 1. +Proposition 9. Every homogeneous circuit computing Bd contains Ω(d/ log d) non-scalar product gates. +Proof. This is an application of Lemma 6 with ℓ = ⌈log2 d⌉. [d] contains d − ℓ − 1 intervals of +length ℓ, all of which give rise to different substrings of σ. The family F ℓ(Bd) consists of d − +(ℓ − 1) different monomials and hence µℓ(Bd) = d − (ℓ − 1). By the lemma, assuming ℓ > 1, a +homogenous circuit for Bd must contain (d − (ℓ − 1))/(ℓ − 1) = Ω(d/ log d) product gates. +Remark 10. Using de Bruijn sequences over alphabet of size n, one can give an explicit monomial in n > 1 +variables and degree d ≥ n which requires homogeneous circuit of non-scalar size Ω(d log n/ log d). This +can also be deduced from Proposition 9 by viewing degree k bivariate monomials as a single variable. +Conversely, every such monomial α can be computed in size O(d log n/ log d) using multiplication +gates only (such a computation is automatically homogeneous). Indeed, we can first compute all monomials +of degree at most k by a circuit of size O(nk+1) and then compute α using ⌈d/k⌉ additional multiplication +gates. Choosing k around 0.5 log2 d log−1 +2 +n is sufficient. This also means the bound in Theorem 2 is +tight. +♦ +6 + +4.2 +Computing partial derivatives simultaneously +In order to obtain stronger lower bounds, we will translate the classical theorem of Baur and +Strassen [BS83] on computing partial derivatives to the non-commutative setting. +We define partial derivative with respect to first position only, as follows. Given a polynomial +f and a variable x, f can be uniquely written as f = x f0 + f1 where no monomial in f1 contains x +in the first position. We set ∂x f := f0. +The proof of the following lemma is almost the same as the one of Baur and Strassen. An ad- +ditional twist is added since we want the derivatives to be computed by a homogeneous circuit. +This requires the generalization of homogeneity to allow arbitrary variable weights. We empha- +size that taking derivatives with respect to the first position is essential in the non-commutative +setting. +Lemma 11. Assume that f ∈ F ⟨x1, . . . , xn⟩ can be computed by a homogeneous circuit of size s and non- +scalar size s×. Then ∂x1 f, . . . , ∂xn f can be simultaneously computed by a homogeneous circuit of size O(s) +and non-scalar size O(s×). +Proof. Given w = (w1, . . . , wn) ∈ Nn, let wi be the weight of xi and let the weight of a monomial +α = ∏d +j=1 xij be defined as wt(α) = ∑d +j=1 wij. A polynomial f ∈ F ⟨x1, . . . , xn⟩ is said to be w- +homogeneous if every monomial in it has the same weight. We call this the weight of f, denoted +by wt( f ). Furthermore we say that a circuit C is w-homogeneous if every gate in it computes a +w-homogeneous polynomial. The weight of any node, v, in a w-homogeneous circuit is defined +to be the weight of the polynomial being computed by it. +Note that if (w1, . . . , wn) = (1, . . . , 1), then w-homogeneity coincides with the usual notion of +homogeneity. Therefore Lemma 11 follows from the following claim. +Claim 12. For any w = (w1, . . . , wn) ∈ Nn, if there is a w-homogenous circuit that computes f ∈ +F ⟨x1, . . . , xn⟩ of size s and non-scalar size s×, then there is a w-homogeneous circuit that computes +D( f ) = {∂x1 f, . . . , ∂xn f } of size at most 5s and non-scalar size at most 2s×. +We prove this claim by induction on s. Recall that circuit size is measured by the number of +non-input gates. For the base case, s = 0, the circuit only consists of leaves. The derivatives are +then either 0 or 1 and can again be computed in zero size. +Assume s > 0. Let w = (w1, . . . , wn) ∈ Nn be arbitrarily fixed. Furthermore, suppose there +is a w-homogenous circuit C that computes f ∈ F ⟨x1, . . . , xn⟩ of size s. Choose a vertex v in C +such that both its children are leaves, and let �v be the polynomial it computes. �v is a homogeneous +polynomial in at most two variables and degree at most two; w.l.o.g., we can also assume that �v is +at least linear (otherwise v could be replaced by a leaf). +Let C′ be the circuit obtained from C by removing the incoming edges to v and labelling the +vertex v with a new variable, say x0. Let us assign it weight w0 := wt(�v). +7 + +Let f ′ be the polynomial computed by C′. Then, D( f ) = {∂x1 f, . . . , ∂xn f } can be recovered +from D( f ′) = {∂x0 f ′, ∂x1 f ′, . . . , ∂xn f ′} using the following version of chain rule: +∂xk f = (∂xk f ′ + ∂xk �v · ∂x0 f ′)|x0:=�v . +Note that ∂xk �v is a variable or a constant, and that it is zero except for at most two of the xk’s. +Let us set w′ = (w′ +0, w1, . . . , wn). Note that the weight of every vertex in C′ is the same as the +corresponding vertex in C. Therefore, since C is w-homogeneous, C′ is w′-homogeneous. Further- +more, C′ has s − 1 non-input gates and, by the inductive assumption, there is a w′-homogeneous +circuit D′ of size 5(s − 1) which computes D( f ′). Using D′ and the chain rule above, we can con- +struct a circuit with 5 additional gates which computes D( f ). The size of this circuit is at most +5(s − 1) + 5 = 5s and is easily seen to be w-homogeneous. +When counting non-scalar complexity, note that in the construction, only non-scalar product +gates introduce non-scalar gates, and we always introduce at most two such gates. +We can now prove Theorem 2. +Proof of Theorem 2. Let n, d be given with2 n > 1, d > 2. Let k be the smallest integer such that +nk ≥ n(d − 1). Take a de Bruijn sequence σ of order k in alphabet [n]. Take sequences σ1, . . . , σn ∈ +[n]d−1 so that their concatenation σ1 . . . σn is the initial segment of σ. Define the polynomial +f = x1α1 + · · · + xnαn , where αi = +d−1 +∏ +j=1 +xσi +j . +Assume f has a homogeneous circuit of non-scalar size s. Then, by Lemma 11, α1, . . . , αn +can be simultaneously computed by a homogeneous circuit of size s′ = O(s). We now apply +Lemma 6 with ℓ = k. By construction, µk(α1, . . . , αn) = n(d − 1 − (k − 1)) = n(d − k). This is +because αJ +i are distinct monomials for different i’s and intervals of length k. The lemma then gives +s′ ≥ n(d − k)/(k − 1). If d ≤ n, we have k = 2 and so s′ ≥ n(d − 2). If d > n, we have k ≤ +c1 log2 d/ log2 n and d − k ≥ c2d, for some constants c1, c2 > 0. Hence indeed s′ ≥ Ω(nd log n +log d). +4.3 +Lower bound for ordered symmetric polynomials +We now prove Theorem 3. We first note: +Remark 13. OS2 +n requires Ω(n) non-scalar product gates (even in the commutative setting). This can be +proved by a standard partial derivatives argument as in [NW97]. +♦ +Hence we can focus on degree d > 2, in which case we give the following strengthening of +Theorem 3: +2If d = 2, OS2 +n satisfies the theorem; see Remark 13. +8 + +Theorem 14. If 1 < k < n, any homogeneous circuit computing OSk+1 +n +(x1, . . . , xn) requires non-scalar +size Ω(k(n − k)). +Proof. Assume that a homogeneous circuit computes f = OSk+1 +n +(x1, . . . , xn) using s non-scalar +product gates. Then by Lemma 11 there is a homogeneous circuit of non-scalar size O(s) which +simultaneously computes {∂x1 f, . . . , ∂xn f }. Let this circuit be C. Then, by Lemma 6, µ2(�C) ≤ O(s). +Note that +∂xi f = OSk +n−i(xi+1, . . . , xn) . +Let fi,j := (∂xi f )[j,j+1]. We claim that the polynomials in F := +� +fi,j : i ∈ [n − k], j ∈ [k − 1] +� +are +linearly independent. +This implies that µ2(�C) ≥ (n − k)(k − 1) and gives a lower bound of +Ω(k(n − k)) as required. +We now prove that F is indeed linearly independent. Consider the lexicographic ordering on +S := [n − k] × [k − 1] defined by: +(i0, j0) < (i, j) iff (j0 > j) or (j0 = j and i0 < i) . +Let (i0, j0) ∈ S be given. Denote δi0,j0(g) the coefficient of the monomial xi0+j0xn+j0−k+1 in g. Then +for every (i, j) ∈ S, +δi0,j0( fi,j) = +� +� +� +1 +if (i0, j0) = (i, j) +0 +if (i0, j0) < (i, j) . +(15) +To see (15), assume that ∂xi f contains xn+j0−k+1 in position j + 1 in some monomial α with +a non-zero coefficient. The degree of α is k, and the positions j + 1, . . . , k need to be filled with +variables from xn+j0−k+1, . . . , xn in an ascending order. There are k − j such positions and k − +j0 such variables. Therefore j ≥ j0. Furthermore, if j = j0, the last k − j0 positions in α are +uniquely determined as the variables xn+j0−k+1, . . . , xn in that order. Similarly, if ∂xi f contains +xi0+j0 in position j0 in some α, the first j0 positions must be filled with variables from xi+1, . . . , xi0+j0. +Hence i ≤ i0, and in case of equality, the first j0 positions are uniquely determined. This means +that δi0,j0( fi,j) = 0 whenever (i0, j0) < (i, j). Furthermore, α := ∏ +i0+j0 +p=i0+1 xp ∏n +p=n+j0−k+1 xp is the +unique monomial in fi0,j0 with δi0,j0(α) = 1, concluding (15). +Finally, assume for the sake of contradiction that there exists a non-trivial linear combination +∑ +(i,j)∈S +γi,j fi,j = 0 . +9 + +Let (i0, j0) be the first pair in the lexicographic ordering with γi0,j0 ̸= 0. Then we have +0 = ∑ +(i,j)∈S +γi,jδi0,j0( fi,j) = γi0,j0δi0,j0( fi0,j0) + +∑ +(i,j)>(i0,j0) +γi,jδi0,j0( fi,j) . +Using (15), the last sum is zero and γi0,j0δi0,j0( fi0,j0) = γi0,j0 = 0, contrary to the assumption γi0,j0 ̸= +0. +5 +Upper bounds for ordered symmetric polynomials +In Proposition 4, we promised upper bounds on the complexity of elementary symmetric polyno- +mials. The promise we now fulfil. +A quadratic upper bound in the homogeneous setting +We want to show that for d ∈ {0, . . . , n}, +OSd +n can be simultaneously computed by a homogeneous circuit of size O(n2). +Note that +OSd +n(x1, . . . , xn) = OSd−1 +n−1(x1, . . . , xn−1) · xn + OSd +n−1(x1, . . . , xn−1). +Hence, once we have computed OSd +n−1, d ∈ {0, . . . , n − 1}, we can compute OSd +n, d ∈ {0, . . . , n} +using O(n) extra gates. The overall complexity is quadratic. +An almost linear upper bound in the non-homogeneous setting +We want to show that OSd +n, d ∈ +{0, . . . , n}, can be simultaneously computed by a non-commutative circuit of size n · poly(log n). +The proof is the same as its commutative analog for elementary symmetric polynomials, see +[BS83] or the monograph by Burgisser et al. [BCS, Chapters 2.1-2.3]. +The main observation is that polynomial multiplication can be done efficiently. Let +f = +n +∑ +i=0 +yiti, +g = +n +∑ +i=0 +ziti, +where f, g ∈ F ⟨y0, . . . , yn, z0, . . . , zn⟩ [t]. In other words, we assume that t commutes with other- +wise non-commuting variables y0, . . . , yn,z0, . . . , zn. We view f, g as univariate polynomials in the +variable t with non-commutative coefficients. Then f g = ∑2n +i=0 citi with ci = ∑i +j=0 yjzi−j. Commu- +tatively, the polynomials c0, . . . , c2n can be simultaneously computed by a small circuit. Indeed, if +F contains sufficiently many roots of unity, one can obtain an O(n log n) circuit using Fast Fourier +Transform; in other fields there are modification giving a circuit of size O(n log n log log n) see +[SS71, BCS]. When counting only non-scalar product gates, this can be improved to O(n) if F is +sufficiently large. We observe that the same holds if the coefficients of f, g do not commute. This +10 + +is because the polynomials ck are bilinear in y0, . . . , yn,z0, . . . , zn. Commutativity does not make a +difference in this case (an exercise). +Now consider the polynomial hn(t) = ∏n +i=1(xi + t) ∈ F ⟨x1, . . . , xn⟩ [t]. Then one can see that +OSd +n(x1, . . . , xn) is the coefficient of tn−d in h(t). The coefficients can be be recursively computed +by first computing ∏⌈n/2⌉ +i=1 (xi + t), ∏n +i=⌈n/2⌉+1(xi + t), and then combining the two by means of the +fast polynomial multiplication above. 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Sci., 12:303– +314, 1980. +12 + diff --git a/zdAzT4oBgHgl3EQftf1m/content/tmp_files/load_file.txt b/zdAzT4oBgHgl3EQftf1m/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd6de4b1b3a64707471789052183e0f37f2f72d1 --- /dev/null +++ b/zdAzT4oBgHgl3EQftf1m/content/tmp_files/load_file.txt @@ -0,0 +1,533 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf,len=532 +page_content='New Lower Bounds against Homogeneous Non-Commutative Circuits Prerona Chatterjee * Pavel Hrubeˇs† January 5, 2023 Abstract We give several new lower bounds on size of homogeneous non-commutative circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' We present an explicit homogeneous bivariate polynomial of degree d which requires homoge- neous non-commutative circuit of size Ω(d/ log d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' For an n-variate polynomial with n > 1, the result can be improved to Ω(nd), if d ≤ n, or Ω(nd log n log d ), if d ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Under the same assump- tions, we also give a quadratic lower bound for the ordered version of the central symmetric polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' 1 Introduction Arithmetic Circuit Complexity aims to categorize polynomials according to how hard they are to compute in algebraic models of computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' The most natural model is that of an arithmetic circuit: starting from variables or constants, the circuit computes new polynomials by means of addition and multiplication operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' The question is how many of these operations are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' The most challenging problem is to prove super-polynomial lower bounds against arithmetic cir- cuits computing a low-degree polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' This is known as the VP vs VNP problem and is the algebraic analogue of the famed P vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' NP question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' The classical result of Baur and Strassen [Str73, BS83] gives an Ω(n log d) lower bound for an n variate polynomial of degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' A variety of lower bounds has since been obtained by imposing various restrictions on the computational model - e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=', arithmetic formulas or monotone circuits [Kal85, Val80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' But the result of Baur and Strassen remains the strongest lower bound on unrestricted arithmetic circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' In this paper, we are interested in the non-commutative setting where multiplication does not multiplicatively commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Starting with the seminal works of Hyafil [Hya77] and Nisan [Nis91], Tel Aviv University, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' This work was done while the author was a postdoctoral researcher at the Institute of Mathematics of the Czech Academy of Sciences, Prague and was supported by the Czech Science Foundation GA ˇCR grant 19-27871X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Email: prerona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content='ch@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content='com †Institute of Mathematics of the Czech Academy of Sciences, Prague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' This work was supported by Czech Science Foundation GA ˇCR grant 19-27871X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Email: pahrubes@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content='01676v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content='CC] 4 Jan 2023 non-commutative circuits are a well-studied object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' The lack of commutativity is a severe limita- tion of the computational power which makes the task of proving circuit lower bounds apparently easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Nisan gave an exponential lower bound for non-commutative formulas whereas, commu- tatively, the best bound is only quadratic [Kal85, CKSV22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Since then, it seemed that exponential non-commutative circuit lower bounds are just around the corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Recently, Limaye, Srinivasan and Tavenas [TLS22] proved such a lower bound in the homogeneous, constant depth setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' They showed that any constant depth ∆ non-commutative homogeneous circuit for the iterated matrix multiplication polynomial over n variables of degree d must have size nΩ(d 1 ∆ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' However for gen- eral circuits, even in the non-commutative setting, the strongest lower bound remains Ω(n log d) [Str73, BS83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' We improve this lower bound to Ω(nd/ log d) under the additional assumption that the non- commutative circuit is also homogeneous (see section 2 for definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Non-commutatively, this is already interesting if n = 2: we obtain a bivariate polynomial of degree d which requires circuit size nearly linear in d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' It is well-known that a (commutative or not) circuit computing a homo- geneous polynomial of degree d can be converted to an equivalent homogeneous circuit with at most a d2 increase in size (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=', [HWY11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Hence, homogeneity is not a serious restriction if either d is small or if one is after a super-polynomial lower bound – as in the VP vs VNP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' However, our results fall in neither category and we do not know how to remove the homogeneity restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Nevertheless, we strongly believe that it can be removed and non-commutative circuit lower bounds are just around the corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' 2 Notation and preliminaries Let F be a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' A non-commutative polynomial over F is a formal sum of products of variables and field elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' We assume that the variables do not multiplicatively commute, whereas they com- mute additively, and with elements of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' The ring of non-commutative polynomials in variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' , xn is denoted F ⟨x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' , xn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' A polynomial is said to be homogeneous if all monomials with a non-zero coefficient in f have the same degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' A non-commutative arithmetic circuit C is a directed acyclic graph as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Nodes (or gates) of in-degree zero are labelled by either a variable or a field element in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' All the other nodes have in-degree two and they are labelled by either + or ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' The two edges going into a gate labelled by × are labelled by left and right to indicate the order of multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Gates of in-degree zero will be called input gates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' gates of out-degree zero will be called output gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Every node in C computes a non-commutative polynomial in the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' We say that C computes a polynomial f if there is a gate in C computing f (not necessarily an output gate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' C will be called homogeneous if every gate in C computes a homogeneous polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Given a circuit C, let �C := { f : f is computed by some gate in C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' 2 A product gate will be called non-scalar, if both of its inputs compute a non-constant polyno- mial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' We define the size of C to be the number of non-input gates in it, and the non-scalar size of C to be the number of non-scalar product gates in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Given integers n1, n2, [n1, n2] is the interval {n1, n1 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' , n2} and [n] := [1, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Note: Unless stated otherwise, circuits and polynomials are assumed to be non-commutative and the underlying field F is fixed but arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' 3 Main results For univariate polynomials there is no difference between commutative and non-commutative computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Already with two variables, non-commutative polynomials display much richer structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' There are 2d monomials in variables x0, x1 of degree d (as opposed to d + 1 in the com- mutative world);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' so a generic bivariate polynomial requires a circuit of size exponential in d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Our first result is a lower bound that is almost linear in d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' The hard polynomial is a bivariate monomial (a specific product of variables x0, x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' For every d > 1, there exists an explicit bivariate monomial of degree d such that any homo- geneous non-commutative circuit computing it has non-scalar size Ω(d/ log d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' In Remark 10, we point out a complementary O(d/ log d) upper bound for every bivariate mono- mial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Note that commutatively every such monomial can be computed in size O(log d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' For n-variate polynomials, we obtain a stronger result (the hard polynomial is no longer a monomial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' For every n, d > 1 there exists an explicit n-variate homogeneous polynomial of degree d which requires a homogenous non-commutative circuit of non-scalar size Ω(nd), if d ≤ n, or Ω(nd log n log d), if d ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Theorem 1 and Theorem 2 are proved in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content='1 and section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content='2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' Given 0 ≤ d, n, the ordered symmetric polynomial, OSd n, is the polynomial1 OSd n(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQftf1m/content/2301.01676v1.pdf'} +page_content=' , xn) = ∑ 1≤i1<··· 100M⊕) have al- +ready lost ≥ 80% of their initial particles. Over the next hundreds +of Myr, with a smaller number of total particles as well as a smaller +number of total perturbers, the ejection rate slows down. Overall the +most efficient ejection continues to occur in systems with simultane- +ously the most massive discs and the most massive GPs. +At the end of the simulations, the higher ejection efficiency occurs +for initial discs masses ≳10 M⊕, with the highest ejection efficiency +occurring when the GP mass is ∼ 100 M⊕ and the disc mass is ≳ +20 M⊕. Many systems exhibit the ejection of a substantial fraction +of the test particles in our simulations. The average ejection rate is +64.5% across all simulations in the grid, with an ejection rate of up +to 96.3% for the most extreme case. +The orbital characteristics (eccentricity, inclination) of particles +in the discs were calculated by averaging the elements of surviving +particles at each time step. In the animated version of Figure 1, we +show their evolution in 10 Myr time steps illustrating the change in +the remaining particles, their eccentricity, and inclination over 1 Gyr. +The color of each circle represents the mean values of the eccentricity +and inclination for each model at the timestep in question. +From the evolution seen in the middle and bottom panel animations +of fig. 1, we find that the disc response is monotonic for the lower GP +masses (𝑀GP ≤ 30 𝑀⊕). We find increasing excitation for decreasing +GP mass, increasing disc mass, and longer integration times. The +evolution of the disc excitation with time is more clearly visible in +eccentricity than inclination. +The middle panel of animated Figure 1 shows that after a few tens +of Myr of evolution, a more efficient stirring has been produced for +the middle rows of the grid, i.e. for 𝑀GP in the range ∼30 to ∼110 M⊕. +At this time, the stirring grows in proportion to the debris disc mass, +while for a given debris disc mass, the stirring increases with GP +mass, reaches a maximum around 70 to 100 M⊕, and decreases for +larger GP masses. This behaviour does not resemble the quadratic +behaviour presented in Muñoz-Gutiérrez et al. (2017), however that +study was for discs 2 to 4 orders of magnitude lighter than what we +are studying here. +Over time, during the first 400 Myr, we see less massive GPs +becoming progressively more efficient at exciting test particles; while +the more massive GPs models stop evolving. After 300 Myr the sweet +spot for efficient stirring becomes less evident, in part due to the +ejection rate of the most excited particles from these systems; after +600 Myr even the models with the least massive GP have stopped +evolving. By the end of the simulations, the largest mean eccentricity +occurs in the lower right corner of the grid, where the disc masses +are comparable to, or even greater than, the GPs masses in these +systems. +In the bottom panel of animated fig. 1 we observe a slower and +more linear trend for the evolution of the mean inclination; up to +100 Myr, the increment in mean inclination is small and its value +remains almost homogeneous across the grid. With time a small +tendency of larger excitation with larger disc masses and smaller GP +masses starts to develop; after 200 Myr the lower right corner of the +grid, where 𝑀GP ≤ 𝑀DD, starts to show clear signs of a stronger +stirring. By the end of the simulations, the final stirring is shown to +be a function of both GP mass and debris disc mass, with the greater +stirring observed in systems with lower GP masses and larger disc +masses. +When the mass of the disc is comparable to that of the GP, the +planet-disc interactions are warranted to be complex. The angular +momentum that can be transferred from the GP to the DPs is large +enough to produce a significant migration of the GP due to the +ejection of massive objects. Also, the reference plane (or “invariable +plane”) within such a massive debris disc is not well defined, as +the GP orbit no longer plays such an important role in determining +the total angular momentum of the system. These conditions are +satisfied for models in the lower right corner of our grid; in that +region, particles are excited but they are not efficiently ejected, so the +system effectively heats up and there is no way of cooling it down. +Complementary to the animated grids in fig. 1, we also present +the time evolution of each model as a curve on the three panels +of animated fig. 2. There we can see the evolution of each model +across the animation, with the survival fraction on the top panel, +mean eccentricity in the middle panel, and mean inclination in the +bottom panel; the last images (as well as the still frames) highlight +the average of all the models with the same GP mass. +In the top panel of animated fig. 2, we see the decline in particle +numbers as a function of time. As expected, the more massive planets +are more efficient at ejecting test particles from the system. For a +given planet mass, the ejection is more efficient with a more massive +disc. In the middle panel of animated fig. 2 we present the eccentricity +evolution for each of our 286 models; as in fig. 1, we are presenting +the evolution of the mean eccentricity of all particles remaining in the +simulations. For models with GP masses less than ≲ 80 M⊕ we can +see that the eccentricities keep increasing over the whole duration +of most of the simulations; all models slow down with time, but for +models with GP masses between ∼ 30 M⊕ and ∼ 80 M⊕ there seem +to be two phases: first, a fast increase and then they reach a plateau +with very little increase in eccentricity thereafter; the change between +these two phases occurs sooner and at a lower average eccentricity for +the more massive GPs, and will likely occur even at GP masses less +than 30 M⊕, but it probably requires more than 1 Gyr for the same +to happen, while for 80 M⊕ it only requires approximately 100 Myr. +For the most massive GPs (≳ 100 M⊕) a third phase appears, after +the fast increase, and before the plateau, a moderately fast decrease +occurs due to the rapid ejection of the most eccentric objects; again +the evolution is faster for more massive GPs, this new phase seems +to be most pronounced for our 223 M⊕ models, but perhaps with +smaller time steps it might be even more important for the 316 M⊕ +GP. Finally, models with GPs more massive than 220 M⊕ seem to +reach saturation, perhaps even a small decline, near the end of the +simulations. +Regarding the effect of the disc mass on the overall eccentricity, we +find that, for a given time and GP mass, larger disc masses produce +larger mean eccentricities. +We present the inclination evolution in the bottom panel of ani- +mated fig. 2; as for eccentricity, we are presenting the evolution of +the mean inclination of all particles remaining in the simulations. +Here we show that the evolution of the inclinations is much slower +than for the eccentricities, in fact, the inclination for all models con- +tinues to rise until the end of the simulations. As with eccentricities, +simulations with more massive discs tend to evolve faster and have +larger mean inclinations. +MNRAS 000, 1–11 (2023) + +Mixed Stirring of Debris Discs +5 +Figure 2. Evolution of the survival fraction (top), mean eccentricity (middle), +and inclination (bottom) of test particles in the discs. The different colors of +the lines in the three panels indicate the mass of the GP in the models. The +initial debris disc mass in the models is represented by the thickness of each +line, with thicker lines corresponding to more massive discs (pale lines in the +still frames, all but the last frame in the animated figures). The thickest lines +in the still frames (and those of the last animated frames) correspond to the +average of all disc masses for any given GP mass (An animated version of +this figure can be found online at Figshare). +In general, for very large GP masses, both eccentricity and inclina- +tion show a mostly smooth evolution. This is related to the dominance +of the GP mass on the overall dynamics, as well as to the number of +particles quickly ejected from the system. This shows a dependence +in GP mass on the degree of stirring of the disc. Very massive GPs +become less efficient with time at heating the discs, and in fact, those +discs cool off at later times, whereas less massive GPs continually stir +their discs throughout the timescale of the simulations. This effect +can be explained through the ejection efficiency of the GPs at differ- +ent masses. High-mass GPs (top rows) quickly excite and eject disc +particles and DPs that stray into regions of strong interaction with the +GP, leaving a depleted but dynamically cold system in their wake. In +contrast, low-mass GPs do very little to stir the discs, but also very +little to suppress stirring by the DPs or to eject particles excited by +DPs, leaving a well-populated but dynamically hot system. +3.2 Evolution of DPs in the Discs +We find that the evolution of massive DPs in the discs follows a +similar trend to that of massless particles, but their self-stirring is +slightly less efficient, as shown in fig. 3 (cf. fig. 2). There we present +animations showing the evolution of the survival fraction (top panel), +mean eccentricities (middle panel), and mean inclinations (bottom +panel) for surviving DPs in the simulations as a function of time, in +the same scheme as for the test particles in the previous sub-section. +In the top panel of fig. 3 we see the surviving fraction of DPs +in each model system as a function of time. Again, consistent with +the analogous plot for test particles in fig. 2, we see that the DPs +are more efficiently removed from the system with a more massive +GP and a more massive disc. In the middle panel of fig. 3 we can +see trends in the behaviour of the DPs can be delineated for models +with different GP masses, following the same general behaviour as +for the test particles. The models with the lowest GP masses, below +15 M⊕, exhibit a rising mean eccentricity for the DPs up until the +end point of our simulations. Models with GPs above that, but below +60 M⊕, again reach a plateau, and have a slow increase, but this time +they have an obvious maximum before having a slow decrease in +eccentricity at some point between 400 Myr and 1 Gyr. The time at +which the highest value occurs, and its value are both dependent on +the GP mass; more massive GPs have their maxima at earlier times +and with lower mean eccentricity values. This is again a result of the +increasing strength of interaction for DPs that more closely approach +the GP. Furthermore, we see that overall the mean eccentricity of +the DPs is lower than that of test particles. For models with GPs +> 60 M⊕ we again observe a third phase of evolution, in the first a +rapid increase in mean eccentricity occurs, quickly reaching a peak +within the first 200 Myr which is faster for more massive GPs; after +this follows a decline, also faster the more massive the GP; finally, +after the decline, a slow increment begins again until reaching an +approximate steady state by the end of the simulations. +The maximum values of the average mean eccentricity for the +models remain below ≃0.35 for DPs (cf. 0.55 for test particles, which +continue growing for the lowest mass GP models), with an apparent +saturation limit at this value independently of GP mass. We can also +see that the behaviour of the lines in fig. 3 is noisier than in the case +of test particles (fig. 2), this is because the DP population is 10 times +less numerous than the particles. We would expect this to also be true +for any real disc since the number of DPs containing a substantial +amount of the disc mass will always be a minority compared to the +total population (starting with the largest bodies, which are the most +dynamically relevant). +For any given GP mass there is a trend of larger eccentricities for +MNRAS 000, 1–11 (2023) + +1.0 +0.8 +Survival Fraction +0.6 +0.4 +0.2 +0.0 +GP Mass (M) +316.22 +112.20 +39.81 +14.12 +223.87 +79.43 +28.18 +10.0 +158.49 +56.23 +19.95 +0 +200 +400 +600 +800 +1000 +Time [Myr]0.6 +0.5 +Eccentricity +0.4 +0.3 +Mean +0.2 +0.1 +GP Mass (M) +316.22 +112.20 +39.81 +14.12 +223.87 +79.43 +28.18 +10.0 +158.49 +56.23 +19.95 +0.0 +0 +200 +400 +600 +800 +1000 +Time [Myr]GP Mass (M) +316.22 +112.20 +39.81 +14.12 +223.87 +79.43 +28.18 +10.0 +158.49 +56.23 +19.95 +40 +Mean Inclination [°] +30 +20 +10 +0 +0 +200 +400 +600 +800 +1000 +Time [Myr]6 +Muñoz-Gutiérrez et al. +Figure 3. Same as fig. 2 but for the evolution of the survival fraction (top), +mean eccentricity (middle), and inclination (bottom) of DPs in the discs. +The color of the lines in both panels indicates the mass of the GP in the +models, while line thickness represents the mass of the disc, with thicker +lines corresponding to more massive discs (pale lines in the still frame, all +but the last frame in the animated figure). The thickest lines in the still frame +(and those of the last animated frame), correspond to the average of all disc +masses for any given GP mass (An animated version of this figure can be +found online at Figshare). +larger disc masses. However, there is an overall dispersion for the +evolution of each suite of simulations, and some individual simula- +tions fall outside of the global trend e.g. the most massive discs for +the systems with 28 M⊕ and 112 M⊕ GPs lie well above the other +systems in their respective suites. These “outliers” may be attributed +to stochastic events involving DP interactions or ejections influencing +the overall evolution of that system. +The evolution of the mean inclination for DPs in our models is +shown in the bottom panel of fig. 3. Again, we observe a similar be- +haviour to the one described above for the test particles, finding lower +inclinations for larger GP masses, and also that the final inclination +values are consistently lower. In this case, almost all the systems +show a monotonic rise in mean inclination over the duration of the +simulations with no turnover. Only the most massive GP systems +(> 220 M⊕) seem to reach a peak in their respective mean inclina- +tion within the duration of the simulations. Systems with lower mass +GPs, MGP < 30 M⊕, are not yet slowing down at the end of the +simulations. We also find that, for a given GP mass, more massive +discs will produce larger mean inclinations. Overall, the greatest in- +clination values lie below 30◦ for the DPs, regardless of GP mass, +and take longer to undergo the same relative degree of excitation, +as compared to the test particles in the same systems that can reach +values close to 45◦. +3.3 Evolution of the Discs as a Whole +To better understand the evolution of discs as complete systems, +containing both massive and massless particles, as well as the rela- +tionship between the two, we begin by comparing the final values of +the mean orbital parameters and survival fractions of test particles +and DPs. In fig. 4 we show the final distribution of mean eccentricities +(left panel), mean inclinations (middle panel), and survival fractions +(right panel), of both populations, for all 286 systems; the different +colors indicate the mass of the GP in that system, while the size of +the dots represents the initial mass of the corresponding debris disc. +In the left and middle panels of fig. 4 we see that for both mean +eccentricity and inclination, the distribution of final values remains +above the identity line (indicated by the solid black line) except +for one outlier case in eccentricity which corresponds to one of the +models with the most massive GP. We can see that the final conditions +for all models closely follow a straight line. A comparison of the +corresponding panels in figs. 2 and 3, shows that massless particles +are more easily disturbed than DPs (as seen in fig. 4); it can also +be seen that the evolution of the eccentricity is much less mass +dependent than that of the inclination. We applied a linear fit in both +cases (dashed black lines in the left and middle panels of fig. 4) to +quantify how efficient the stirring of test particles is when compared +to that of massive DPs. The best fit for the models in the eccentricity +panel is given by +� +𝑒particles +� += 1.366 ⟨𝑒DPs⟩ +0.004 and for the final +inclination +� +𝑖particles +� += 1.916 ⟨𝑖DPs⟩ − 2.602◦; these fits show that +the stirring of test particles is more efficient than that of DPs by +factors of 1.366 for eccentricity and 1.916 for inclination. Both of +these fits lie close to the gray star representing the initial conditions +of all the distributions. +As in figs. 2 and 3, fig. 4 shows that the more massive discs (larger +dots) are more efficient at stirring their particles than less massive +ones (smaller dots) but that more massive GPs have a stabilising +effect on the discs after a quick removal of the initially unstable +minor bodies (both DPs and test particles); this comes about because +massive GPs will tend to eject particles that pass close to them, +MNRAS 000, 1–11 (2023) + +1.0 +0.8 +DPs Survival Fraction +0.6 +0.4 +0.2 +0.0 +GP Mass (M) +316.22 +112.20 +39.81 +14.12 +223.87 +79.43 +28.18 +10.0 +158.49 +56.23 +19.95 +0 +200 +400 +600 +800 +1000 +Time [Myr]0.6 +0.5 +Eccentricity +0.4 +DPs Mean E +0.3 +0.2 +0.1 +GP Mass (M) +316.22 +112.20 +39.81 +14.12 +223.87 +79.43 +28.18 +10.0 +158.49 +56.23 +19.95 +0.0 +200 +400 +600 +800 +1000 +Time [Myr]GP Mass (M) +316.22 +112.20 +39.81 +14.12 +223.87 +79.43 +28.18 +10.0 +158.49 +56.23 +19.95 +40 +DPs Mean Inclination [ +30 +20 +10 +200 +400 +600 +800 +1000 +Time [Myr]Mixed Stirring of Debris Discs +7 +Figure 4. Comparison of the final mean orbital parameters and survival fractions of DPs vs test particles. The left panel shows the distribution of mean +eccentricities, the middle panel for mean inclinations, and the right panel for survival fractions. The colors indicate the mass of the GP in the system, while +the size of the dot represents the initial mass of the debris disc. The identity is indicated by the solid black line, while best fits are indicated by dashed black +lines. Linear fits were done for both eccentricity and inclination, while a third-order polynomial was fitted to the survival fraction. The gray star in each panel +represents the initial conditions of our systems. +whereas lighter GPs will perturb their orbits without ejecting them +from the system. +The final distribution of survival fractions (right panel of fig. 4) +remains below the identity line, illustrating the greater difficulty for +a planet in ejecting massive objects (DPs) than massless ones (test +particles). A strong dependence on GP mass is observed in the final +survival rate for both populations of minor bodies, demonstrating the +efficiency of ejection. We find the relationship between the surviving +fractions (SFparticles and SFDPs) is best represented by a 3rd order +polynomial of the form: +SFparticles += +0.932 SFDPs3 − 0.703 SFDPs2 +(1) ++ 0.746 SFDPs − 0.031. +As with eccentricity and inclination, extrapolation of this trend to- +wards the less perturbed discs leads to the gray star representing +the initial conditions; for survival fractions, there is also an obvi- +ous extrapolation to more violent systems and we find that our trend +leads toward the (0,0) point where all particles would be ejected. +The scatter of simulations around this trend line is generally more +pronounced for the systems with higher GP masses (in the survival of +both test particles and DPs). This is to be expected as it is interactions +with the GP in each system that will dominate the removal of smaller +bodies (either by collision or ejection). We find that the number of +test particles removed by collisions remains approximately constant +over the simulation grid, comprising about 2% of the particles over +the duration of each model run. By contrast, the number of ejection +events is strongly correlated with the GP mass, with removals initially +about 5%, and swiftly becoming greater by an order of magnitude or +more with up to 95% during a model run. As the GP mass decreases +so too does the ejection efficiency, and they will only dynamically +heat their companion discs rather than deplete them. This leads to +a lower dispersion in the survival of DPs, but a comparable scatter +in test particle ejection. The most massive GPs exhibit the tightest +correlation with the observed trend. In these simulations, the GP +rapidly stirs and depletes the disc (cf. fig. 2) and if any minor body +subsequently migrates into the perturbation region of the GP it is +swiftly removed. +The most massive discs in the simulations for a given GP mass tend +to lie below the trend line identified by section 3.3. This indicates +segregation by disc mass within the distribution of surviving minor +bodies, where the more massive (initial) discs are more depleted +in both test particles and DPs for a given GP mass. This is the +natural consequence of greater dynamical stirring by more massive +individual DPs within the more massive disc for a given system, +leading to particles (and DPs) passing into close interaction with +the GPs. This tendency weakens and breaks down as the GP mass +decreases, representing the decreasing capacity of the GP to deplete +mass from the disc. +Most of the analysis of sections 3.1 and 3.2 is focused on the point +of view of the models, this is: we are classifying each model accord- +ing to its initial conditions. However, this is not directly applicable +to observations. From an observational point of view, it is more in- +teresting to characterise a model according to its current parameters, +and while the GP mass will not change, the disc mass will change +with the ejection of DPs. Therefore, similar to the animated fig. 2, +in the animated version of fig. 5 we show the time evolution of our +286 models by plotting the survival fractions, mean eccentricities, +and mean inclinations of surviving particles at each time step, as a +function of the evolving mass of the disc, instead of its initial mass. +In the top panel of fig. 5 we show the survival fraction of test +particles, where each snapshot corresponds to a 10 Myr evolution. +The color of each circle indicates the particle survival fraction present +in each disc at that time, with darker colours representing a lower +survival fraction. +We can see that the evolution of the survival rate is fastest for the +most massive GP and (initial) disc mass combinations, with more +than 50% depletion of those discs occurring within the first few tens +of Myr; in the same time frame, barely any ejections have occurred +amongst the lower mass systems. By 100 Myr, systems with GP +masses greater than 60 M⊕ have experienced substantial ejection, +losing up to half the particles (but not necessarily half their mass), +whereas systems below that have yet to experience any substantial +ejections. At the 500 Myr point, the most massive systems have lost +up to 90% of their initial particles and only the least massive GP/disc +systems are untouched by ejections. Beyond this time up to 1 Gyr the +overall picture remains constant and the systems’ evolution is more +gradual. +If we focus on a fixed small area of the grid, instead of following +MNRAS 000, 1–11 (2023) + +GP Mass (M@) +0.6 +316.22 +223.87 +158.49 +112.20 +0.5 +79.43 +Eccentricity +56.23 +39.81 +28.18 +0.4 +19.95 +14.12 +Mean l +10.0 +0.3 +Particles T +0.2 +0.1 +0.0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +DPs Mean EccentricityGP Mass (M) +316.22 +223.87 +40 +158.49 +112.20 +79.43 +Mean Inclination [ +56.23 +39.81 +28.18 +30 +19.95 +14.12 +10.0 +20 +Particles N +10 +5 +10 +15 +20 +25 +DPs Mean Inclination f1.0 +GP Mass (M) +316.22 +223.87 +158.49 +112.20 +0.8 +79.43 +Survival Fraction +56.23 +39.81 +28.18 +0.6 +19.95 +14.12 +10.0 +Particles S +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +DPs Survival Fraction8 +Muñoz-Gutiérrez et al. +Figure 5. Animated figures illustrating the surviving fraction of test particles +(top), mean eccentricity (middle) and mean inclination (bottom) as a function +of the evolving disc mass vs. GP mass. The time step is in increments of +10 Myr (An animated version of this figure can be found online at Figshare). +the evolution of individual coloured circles, the behaviour of the +survival fraction in that region becomes even more extreme, e.g., for +a disc mass of ∼ 7 M⊕ the difference in survival fraction goes from +≈ 90% (at low GP masses) to ≈ 10% (at high GM masses) during +the 1 Gyr simulation. +In the same sense, one should look at the animated middle panel +of fig. 5, following the eccentricity evolution, as we looked at the +animated top panel, i.e. we should focus on an area and not let our +eyes drift away from it; by looking at a column centered at around +∼ 10 M⊕, we see that the eccentricity slowly rises with time; after +the first few time steps the most eccentric models were those with a +GP mass of ∼ 200 M⊕, but with time this maximum went all the way +down to 10 M⊕ (although this took the best part of the 1 Gyr of our +simulations). Another thing to note is that, while many individual +dots reach saturation within our simulations, by looking at a fixed +area we see that it keeps on evolving, mostly because simulations with +more massive initial discs keep passing through our observation area +(akin to the difference between Eulerian vs. Lagrangian evolution). +By the end of our simulation, we find that there is a triangular region, +in the lower right of the plot, that is mostly saturated with mean +eccentricities ∼ 0.6. This is quite extreme for discs that were initially +dynamically cold with ⟨𝑒0⟩ = 0.025, a ∼25 fold increase. +Finally, for the bottom panel of the animated fig. 5, following the +inclination evolution, we observe that the evolution of inclination +is slower than for eccentricity. After ∼ 100 Myr the inclination is +mostly homogeneous with only the most massive disc models show- +ing signs of a significant stirring. During the next hundreds of Myr, a +differentiation in the level of stirring becomes evident for individual +columns, which seem to have uniform colors evolving in time, i.e., +the excitation level for the inclination is more clearly dependent on +the remaining debris disc mass than on the initial disc mass or the GP +mass. At the end of the simulations, the maximum stirring has oc- +curred for the most massive debris discs and the least massive GPs, +however, since the ejection fraction grows with GP mass, as time +passes, what would be an equally excited component in our most +massive GP models has already been depleted. +A skewed initial grid (rather than the rectangular one we consid- +ered here), with more massive debris discs for the more massive GP +systems, might fill in some of this depleted parameter space. How- +ever, as the disc evolution timescale decreases with increasing disc +mass, the observed regions of parameter space that are vacated in +our simulations are necessarily void given the duration of the simu- +lations. In this sense the structure we observe in our grid at 1 Gyr is +not fixed; longer integration would necessarily drive all the systems +to lower disc masses, leading to a more pronounced “gap” in the +top right of these plots. This diagram, therefore, provides some con- +straints on the evolutionary pathway undertaken by observed debris +discs with the constraints of the stellar age and inferred disc mass. +3.4 Evolution of GPs +Besides the evolution of the debris disc systems as a whole, the +GPs in our models experience modifications to their initial orbital +parameters; this is due to the interactions between the GP and massive +DPs, which results in the interchange of angular momentum that leads +to an overall increase in their eccentricity and ultimately to ejections +of some DPs. Although small in most cases, the orbital perturbations +experienced by some of the GPs in our models can be significant; +specifically: large inward orbital drifts, of up to 10 au, are observed +in systems with the less massive GPs and the most massive debris +discs, i.e. in those systems with the largest mass ratio, as given by +𝑀DD/𝑀GP. +In fig. 6 we only show the final distribution, in logarithmic values, +for the eccentricities (left panel), inclinations (middle panel), and +semimajor axis changes (right panel) of the GPs in our 286 models, +as a function of the logarithm of the mass ratio of the system. In log- +log space, those three distributions can be well described by linear +fits. +At the end of the simulations we found that most of our GPs would +be considered to have remained in cold orbits (only 3, out of 286, +have 𝑒 > 0.1, while only 5 have 𝑖 > 5◦); however, about 30% of the +GPs in our simulations have lost a significant fraction of their angular +momentum, having a noticeable decrease in their semimajor axis by +the 1 Gyr mark, 𝑎 < 0.9𝑎0. +At any point during the simulations, the three distributions (𝑒, 𝑖, +and |Δ𝑎| /𝑎0) can be well described by linear fits that slowly evolve +with time, with both the absolute value as well as the mass fraction +dependence slowly increasing. By fitting all simulations at 100 Myr +MNRAS 000, 1–11 (2023) + +t = 1000 Myr +1.0 +（ +300 +0.8 +Giant Planet Mass [M +（（ +Surviving Fraction +（（ +100 +（（ +0.6 +0.4 +30 +0.2 +10 +0.0 +3 +10 +30 +Disk Mass [M]t = 1000 Myr +1.0 +300 +0.8 +Giant Planet Mass [M +Eccentricity +～（ +100 +0.6 +0.4 +Mean +30 +0.2 +10 +0.0 +3 +10 +30 +Disk Mass [M@]t = 1000 Myr +50 +300 +40 +Giant Planet Mass [M +（（ +（（ +100 +30 +（（ +20 +30 +10 +10 +0 +3 +10 +30 +Disk Mass [M@]Mixed Stirring of Debris Discs +9 +Figure 6. Final orbital values for the GPs as a function of the mass ratio. The left panel shows the distribution of final eccentricities, the middle panel for final +inclinations, and the right panel for final semimajor axes changes. As in fig. 4, the colors indicate the mass of the GP in the system, while the size of the dot +indicates the initial mass of the debris disc. Linear fits in these Log-Log planes are indicated by the dashed lines (see text for details). +intervals, and subsequently fitting a time dependence to the linear +fits we obtain: +log10(𝑒GP) = 0.3383 +� 𝑇 +Myr +�0.1005 +log10(𝑀DD/𝑀GP) ++ 0.2061 log10 +� 𝑇 +Myr +� +− 2.0871, +(2) +log10(𝑖GP) = 0.6454 +� 𝑇 +Myr +�0.0751 +log10(𝑀DD/𝑀GP) ++ 0.3265 log10 +� 𝑇 +Myr +� +− 0.9025, +(3) +log10(|Δ𝑎| /𝑎0) = 0.3138 +� 𝑇 +Myr +�0.1281 +log10(𝑀DD/𝑀GP) ++ 0.4582 log10 +� 𝑇 +Myr +� +− 2.0646, +(4) +for eccentricity, inclination, and semimajor axis change, respectively. +4 DISCUSSION +In this work, we did not focus on the sculpting process of the edges +of the discs, nor on the disc shapes; this is why we choose 10 Hill +radii as the inner edge of the discs and not 5 Hill radii as been done +elsewhere (e.g. Pearce & Wyatt 2014). We further assumed a GP in +a circular and planar orbit to minimise the impact of the planet on +the disc. We instead focused on the stirring process produced as a +result of the interaction of the massive DPs embedded in the debris +discs, but such a process was somewhat dominated by the presence +of the GP. We focused on determining the stirring levels as functions +of both the GP and debris disc masses (assuming the shapes and +disc edges are imprinted on the debris discs by the giant planetary +companion). +Our model spans GP masses between 10 M⊕ (approximately +60% the mass of Neptune) and 316 M⊕ (approximately the mass +of Jupiter); Pearce et al. (2022) estimate that Neptune to Saturn-mass +planets are the minimum needed to stir most of their 178 modeled +discs (though some needing Jupiter mass planets, assuming maxi- +mum eccentricities of 0.3). Similarly, the range of disc masses in this +analysis, 3.16 to 31.6 M⊕, are consistent with expectations based +on both observations and theoretical considerations (Mulders et al. +2021; Krivov & Wyatt 2021). Several other studies predict larger +masses (> 100 M⊕) in order for debris discs to be self-stirred (e.g. +Krivov & Booth 2018; Krivov & Wyatt 2021). Nonetheless, in this +work, we found that small masses in debris discs can result in large +stirring values, up to a 25-fold increase in the most extreme cases. +Thus an efficient stirring is possible for small disc masses (< 10 M⊕), +if ever perhaps containing larger than expected perturbers, as some +of the DPs present in the most massive discs we considered have +assigned masses close to 1 M⊕. +Our massive objects are initially thought to be real ‘dwarf planets’ +(DPs), as long as we adopt the definition of DP as an object that +has not cleared its neighborhood from debris (yet). We could expect +massive debris discs (much more massive than our Kuiper belt) to +contain more massive objects, though this is not necessarily true, +depending on planetary formation mechanisms, disc mass density, +etc. Recent studies on dust formation and excitation place limits to +the most massive objects present in massive debris discs to be around +5 times the mass of Pluto. Based on spatially resolved observations +of the vertical scale heights of the debris discs around AU Mic and +𝛽 Pic, the most massive bodies present in those discs could be up +to 9×10−5 and 0.4 M⊕, respectively (Daley et al. 2019; Matrà et al. +2019). However, more massive objects might be present in debris +discs (without leaving a piece of observational evidence, such as +bumps or gaps), if we assume the mass range in planetesimals scales +linearly with the overall mass of the disc. +Limiting the mass of the DPs to be similar to the mass of Pluto +would diminish the stirring effect in both 𝑒 and𝑖. According to shorter +duration simulations (𝑡 = 50 Myr) with 41, 100, and 250 DPs (and +410, 1 000, and 2 500 test particles) we ran as consistency checks, +this correction should be approximately a factor of 1.5, and definitely +less than a factor of 2. +Notably, by using Mercury, the problem becomes computation- +ally intractable when considering more than a few hundred massive +DPs; new tools are required to expand the grid with a greater number +of DPs and particles, such as GPU-based simulations. We leave this +question open for future work. +The main stage for excitation evolution in our simulations occurs +in timescales of the order of a few and up to 100 Myr; while the +time required for our systems to acquire their final configurations, +i.e. reach their saturation levels, is of the order of 150 Myr to more +MNRAS 000, 1–11 (2023) + +GP Mass (M) +316.22 +223.87 +-1.0 +158.49 +112.20 +79.43 +56.23 +-1.5 +39.81 +28.18 +19.95 +Log10(eGp) +14.12 +2.0 +10.0 +-2.5 +-3.0 +-3.5 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +LOg10(MDD/MGP)1.0 +GP Mass (M) +316.22 +223.87 +0.5 +158.49 +112.20 +79.43 +0.0 +56.23 +39.81 +28.18 +19.95 +0.5 +Log1o(iGP) +14.12 +10.0 +1.0 +-1.5 +-2.0 +-2.5 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +LOg10(MDD/MGP)GP Mass (M) +316.22 +-0.50 +223.87 +158.49 +-0.75 +112.20 +79.43 +56.23 +-1.00 +39.81 +28.18 +(0e/levl)01607 +19.95 +1.25 +14.12 +10.0 +1.50 +-1.75 +-2.00 +-2.25 +-2.50 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +LOg10(MDD/MGP)10 +Muñoz-Gutiérrez et al. +than 1 Gyr scales. These timescales are similar to the ages of host +stars for many observed debris disc systems; thus we would expect +that many of the observed systems with similar physical parameters +as those covered in this work, would already be settled in their final +configuration, currently experiencing a quiet steady-state evolution. +The timescales derived here are a function of the chosen architec- +ture of the model, adopting a 1 M⊙ star with a planetary companion +at 30 au and a Kuiper belt-like disc beyond that. However, the evo- +lutionary timescales can be easily scaled for different stellar masses +or disc semi-major axes, as the dynamics should be self-similar, pro- +vided physical collisions are a negligible cause of removal of bodies. +For a different central star mass, all the masses should scale propor- +tional to the new stellar mass, and the timescales should be modified +as the inverse of the square root of the mass; for a different GP orbital +radius, the masses should not be modified, and the timescales should +be modified as the mass to the -3/2 power. For the Vega system, with +a stellar mass ∼2 M⊙ and a planetesimal belt around 100 au (Matrà +et al. 2020; Marshall et al. 2022), the equivalent timescale would be +nearly three times longer than the evolution of the models considered +here (depending on the exact location of the GP used in Vega). +A debris disc is the result of a collisional cascade within a plan- +etesimal belt triggered by dynamical excitation, either intrinsically +by the largest planetesimals within the belt (Krivov & Booth 2018) +or extrinsically by an external perturber (e.g. Mustill & Wyatt 2009). +The range of relative velocities among planetesimals, required to +trigger the onset of the collisional cascade, is typically estimated +as 100 to 300 m/s (e.g. Kenyon & Bromley 2001). On the other +hand, such relative velocities can be estimated from the average or- +bital parameters of the dust-producing small objects in the discs, as +𝑉rel = 𝑉𝐾 +√ +1.25𝑒2 + 𝐼2, where 𝑉𝐾 is the Keplerian velocity at the +distance 𝑎 from the star (Lissauer & Stewart 1993; Wyatt & Dent +2002). Krivov & Booth (2018) argued though, that the average in- +clinations are not terribly important when determining the relative +velocities among planetesimals, since eccentricities grow much faster +than inclinations in debris disc models. Thus, one can simply esti- +mate the relative velocities from the root mean square eccentricity of +the planetesimals as 𝑉rel = 𝑉𝐾 +√︃� +𝑒2� +. In any case, an estimation of +such relative velocities in all of our models shows that values close +to 1 km/s are quickly reached, in less than 10 Myr, regardless of the +initial debris disc mass or the GP mass of the model. Indeed, veloci- +ties of collisions within the belt modeled here, range from ∼250 m/s +to ∼2000 m/s, therefore locating themselves safely on the side of a +collisional cascade capable of producing dust. +A population of planetesimals on eccentric orbits within a debris +disc would produce a halo of millimetre dust grains. Such structures +have been identified in ALMA observations of several systems, in- +cluding HR 8799 (Geiler et al. 2019), HD 32997 and HD 61005 +(MacGregor et al. 2018). The typical eccentricity of dust grains +within debris discs inferred from their spatially resolved belts lies in +the range 0.1 to 0.3 (based on 11 discs, see figure 9 of Marino 2021); +this level of eccentricity is consistent with the mean eccentricity +induced by the dwarf planets in this set of simulations. +5 SUMMARY AND CONCLUSIONS +In this work, we performed a suite of 286 numerical simulations to +explore the stirring effects that a combination of giant and dwarf +planetary perturbations would have on the long-term evolution of +initially cold debris disc models. Our systems are formed by a solar +mass star, a giant planet initially located at 30 au in a circular and +planar orbit, and 100 massive dwarf planets embedded in a disc +described by 1000 test particles. The orbital distribution of the discs +was drawn randomly for small values of eccentricity (between 0 and +0.05) and inclination (between 0◦ and 5◦). We initially located the +inner edge of our discs at 10 Hill radii from the GP, with a total +width of 30 au. Our 1 Gyr long simulations take into account the +perturbations from the GP and the DPs over test particles and among +themselves. +The evolution timescale for the eccentricity and inclination de- +pends mostly on GP mass, where the simulations with more massive +GPs evolved faster than those with less massive GPs. On the other +hand, the limit to the heating depends on both GP mass and disc +mass, with large disc masses and small GP masses being able to heat +the disc more than simulations with light discs and/or heavy GPs. +Part of the reason why massive GPs are less efficient at heating the +disc is their tendency of ejecting “warm” particles before they can +get extreme values of either eccentricity or inclination. +In all models the mean inclination rises quickly (or at least rela- +tively quickly) before slowing down, only the most massive GPs seem +to be able to level off before the 1 Gyr mark. The eccentricity evolves +faster with many of the simulations reaching a plateau before the end +of the simulation. Very massive GPs heat their discs very quickly +which then slowly cool down by ejecting the more excited particles. +The effect on the eccentricity is larger than for the inclination. +Massless particles, which in real systems could be considered as +the less massive members of the discs, such as cometary nucleii, +are more mobile than massive objects (DPs), therefore they become +’hotter’, i.e. more eccentric, more inclined, and are easier to be ejected +(they have a poorer survival rate). Nonetheless, DPs reach significant +stirring levels as well and have only slightly better chances of survival +than test particles. +The values of both eccentricity and inclination for test particles at +a given time have a better correlation with the remaining mass of the +debris discs than with the GP mass or the initial debris disc masses; +this is particularly evident for the inclination. +GPs themselves are perturbed by their interactions with massive +DPs, the most significant perturbations occur when the mass of the +disc is comparable to the mass of the GP. In such cases, a significant +inward migration of the GP takes place, of up to ∼ 10 au, leaving a +stirred disc that is not able to cool off by ejecting “warm” particles, +with a far away GP closer to its star. +The masses in debris discs explored in this work, and specifically +their evolving remaining masses, are indeed very small when com- +pared to those expected to be able to stir the disc by the self-stirring +scenarios (Krivov & Wyatt 2021; Krivov & Booth 2018), but here we +highlight the fact that even with such small masses, which involve a +small number of massive perturbers (100 DPs initially), and perhaps +more importantly, not-so-massive objects, are capable of increas- +ing in an important percent, while acting together with the GP, the +eccentricities and inclinations of debris disc particles. This result is +similar to the enhancement of cometary production in the Kuiper belt +found by Muñoz-Gutiérrez et al. (2019) and could have additional +implications for the production of exocomets in extrasolar planetary +systems. +Taking everything into account, we have found that a combination +of perturbers, consisting of embedded dwarf and external giant plan- +etary masses, is in general more efficient in the stirring of cold debris +discs than one or the other mechanism acting independently. +MNRAS 000, 1–11 (2023) + +Mixed Stirring of Debris Discs +11 +DATA AVAILABILITY +The data underlying this article are available in the article and in its +online supplementary material. The animations, supplementary data, +and analysis scripts are provided for public access on Figshare. +ACKNOWLEDGEMENTS +The authors thank the referee, Alex Mustill, for his constructive and +helpful comments. JPM acknowledges research support by the Min- +istry of Science and Technology of Taiwan under grants MOST107- +2119-M-001-031-MY3 and MOST109-2112-M-001-036-MY3, and +Academia Sinica under grant AS-IA-106-M03. +Software: This work has made use of the symplectic integrator +package mercury (Chambers 1999), and the Python modules Mat- +plotlib (Hunter 2007), and NumPy (Harris et al. 2020). +REFERENCES +Andrews S. M., et al., 2018, ApJ, 869, L41 +Bai X.-N., Stone J. M., 2010a, ApJ, 722, 1437 +Bai X.-N., Stone J. M., 2010b, ApJ, 722, L220 +Birnstiel T., Dullemond C. P., Brauer F., 2010, A&A, 513, A79 +Blum J., Wurm G., 2008, ARA&A, 46, 21 +Brauer F., Henning T., Dullemond C. P., 2008, A&A, 487, L1 +Burns J. A., Lamy P. L., Soter S., 1979, Icarus, 40, 1 +Chambers J. E., 1999, MNRAS, 304, 793 +Daley C., et al., 2019, ApJ, 875, 87 +Dohnanyi J. S., 1969, J. Geophys. 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P., Tobin J., 2019, +ApJ, 872, 112 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–11 (2023) + diff --git a/zdE4T4oBgHgl3EQfyQ3C/content/tmp_files/load_file.txt b/zdE4T4oBgHgl3EQfyQ3C/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..782101a594b4bbbd5cffd1585b6e1789fa1a10b4 --- /dev/null +++ b/zdE4T4oBgHgl3EQfyQ3C/content/tmp_files/load_file.txt @@ -0,0 +1,1235 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf,len=1234 +page_content='MNRAS 000, 1–11 (2023) Preprint 16 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 A Mixed Stirring Mechanism for Debris Discs with Giant and Dwarf Planetary Perturbations Marco A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Muñoz-Gutiérrez,1★ Jonathan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Marshall,1,2 and Antonio Peimbert3 1Institute of Astronomy and Astrophysics, Academia Sinica, 11F of AS/NTU Astronomy-Mathematics Building, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='1, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 4, Roosevelt Rd, Taipei 10617, Taiwan 2Centre for Astrophysics, University of Southern Queensland, Toowoomba, QLD 4350, Australia 3Instituto de Astronomía, Universidad Nacional Autónoma de México, Apdo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' postal 70-264, Ciudad Universitaria, México Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' in original form ZZZ ABSTRACT Debris discs consist of belts of bodies ranging in size from dust grains to planetesimals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' these belts are visible markers of planetary systems around other stars that can reveal the influence of extrasolar planets through their shape and structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Two key stirring mechanisms — self-stirring by planetesimals and secular perturbation by an external giant planet — have been identified to explain the dynamics of planetesimal belts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' their relative importance has been studied independently, but are yet to be considered in combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In this work we perform a suite of 286 N-body simulations exploring the evolution of debris discs over 1 Gyr, combining the gravitational perturbations of both dwarf planets embedded in the discs, and an interior giant planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Our systems were somewhat modeled after the architecture of the outer Solar system: a Solar mass star, a single massive giant planet at 30 au (𝑀GP = 10 to 316 M⊕), and a debris disc formed by 100 massive dwarf planets and 1 000 massless particles (𝑀DD = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='16 to 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='6 M⊕).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We present the evolution of both the disc and the giant planet after 1 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The time evolution of the average eccentricity and inclination of the disc is strongly dependent on the giant planet mass as well as on the remaining disc mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We also found that efficient stirring is achieved even with small disc masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In general, we find that a mixed mechanism is more efficient in the stirring of cold debris discs than either mechanism acting in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Key words: circumstellar matter – planetary systems – planet–disc interactions – dynamical evolution and stability 1 INTRODUCTION Debris discs are massive structures observed around 20 to 30 per- cent of main sequence stars (for recent reviews see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=', Wyatt 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' their presence is signaled by the pres- ence of excess emission in thermal emission at infrared to millimetre wavelengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=', Eiroa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Thureau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Holland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Sibthorpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2018) and/or scattered light at optical or near-infrared wavelengths (either total intensity or polarization, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=', Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Esposito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2020), coming from circum- stellar micrometre- to centimetre-sized dust grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The dust contents of debris discs are not just remnants of the original, massive, dust- and gas-rich protoplanetary discs of material from which planets are born (Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Although some amount of (sub-)micron-sized dust can remain after the initial protoplanetary disc dissipates, the smallest dust grains are lost on timescales much shorter than the age of the host star due to photoevaporation and accretion processes (Burns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Krivov 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Therefore, the dust observed in debris discs is thought to be second-generation dust, produced in disruptive collisions between larger leftover planetesimals, which were originally formed from dust (and ices) in the protoplanetary discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Collisions between these ★ E-mail: mmunoz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='astro@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='com (MAM) bodies produce detectable amounts of dust throughout the lifetime of the host star and beyond (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=', Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Farihi 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' However, to be able to produce that dust, planetesimals must be abundant enough to have frequent collisions, as well as have relative velocities high enough for collisions to be destructive, or at least erosive (Dohnanyi 1969;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Kenyon & Bromley 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The formation of planetesimals starts with dust growth in proto- planetary discs, which is encouraged by vertical settling of larger grains to the disc mid-plane and radial trapping at pressure bumps, especially around ice lines increasing the mass surface density to a level where the gas-to-dust ratio approaches unity (Blum & Wurm 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Drążkowska & Alibert 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Growth beyond millimetre- to centimetre-sized particles is inhibited by collisions due to the ‘bounc- ing barrier’ (Brauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Birnstiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Zsom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The rapid loss of these large grains or pebbles due to inward radial drift is an inhibiting factor in current theories of planet forma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' A mechanism referred to as the ‘streaming instability’ has been proposed as a means to bypass the ‘bouncing barrier’ and precipi- tate planetesimals directly from pebbles in the proto-planetary disc (Youdin & Goodman 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Youdin & Johansen 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Johansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Bai & Stone 2010b,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The size distribution of these bodies is consistent with the range observed in the Solar System’s Kuiper Belt, wherein Pluto and its cohort could represent the high mass tail © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='05265v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='EP] 12 Jan 2023 2 Muñoz-Gutiérrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' of this planetesimal formation process (Johansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Simon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The initial orbits of planetesimals formed in the protoplanetary phase are expected to be nearly circular and confined to the disc mid- plane, therefore some additional stirring mechanism is required to dynamically excite the planetesimal belts left after the gas dispersal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Structures observed in proto-planetary discs, such as rings, spiral arms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=', are uncorrelated with ice lines/density enhancements in- duced by disc temperature structure (Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' van der Marel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Rings in protoplanetary discs could therefore be the re- sult of the action of protoplanets trapping material and sculpting the disc (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='Low mass companions have been identified embedded within several such discs (Fedele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Keppler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Ubeira-Gabellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Teague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Once the eccentricity-damping effect of the protoplanetary gas disc has been removed, the ongoing stirring by either planets or planetesimals on the debris disc will excite the belt leading to enhanced collision rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Inheritance of structure from proto-planetary discs to debris discs is uncertain (Najita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2022), but planetesimal belt locations in cold debris discs (exoKuiper belts) appear consistent with formation at CO ice line (Matrà et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Marshall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' However, the widths of rings in proto-planetary discs are much narrower than debris disc’s planetesimal belts (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The majority of broad debris belts observed by ALMA with sufficient spatial resolution exhibit sub-structures consistent with the presence of a perturbing planetary companion (Marino 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Analysis of spatially resolved observations of debris discs have been used to infer the stirring mechanism(s) in play for a number of young systems based on stirring arguments from the size of the disc and the stellar age (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Moór et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Vican et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2016) and inter- pretation of their architectures, revealing disc-planet interactions in a variety of ways, including the detection of gaps in broad belts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Marino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2017, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' MacGregor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2019), scattered haloes of mm dust grains (MacGregor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Geiler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2019), and the eccentric architectures of narrow belts (Kennedy 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Most recently, Pearce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' (2022) examined a large ensemble of debris discs, both spatially resolved and unresolved, inferring the required mass of a perturber, under the assumption that the sculpting is pro- duced by a single planet or multiple planets, as well as if being the result of self-stirring by massive planetesimals within the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' These two aforementioned main mechanisms have been suggested in the past to account for the planetesimal excitation levels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 1) the self-stirring mechanism (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Kenyon & Bromley 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Krivov & Booth 2018), in which large planetesimals are able to trigger a collisional cascade once they acquire a certain size threshold, and 2) the secular perturbations from giant planetary companions, interior or exterior to the discs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=', Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Mustill & Wyatt 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The latter has been favored recently due to the very large masses of debris discs required to explain their excitation levels by the self-stirring mechanism (Krivov & Wyatt 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Pearce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' However, the effects of a simultaneous stirring by external planets together with internal planetesimals has never been studied in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Besides, the existing self-stirring models do not properly account for the top-end of the size distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=', Pluto-sized dwarf planets), frequently relying in models comprised of equal massed (not so large) bodies stirring the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In previous works (Muñoz-Gutiérrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2015, 2017, 2018), we studied the long-term evolution of generic cold debris discs of dif- ferent masses, under the perturbations of an interior Neptune-like giant planet, as well as of dozens of dwarf planet-sized massive per- turbers (DPs, hereafter) embedded in the discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In Muñoz-Gutiérrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' (2017), we demonstrated the existence of a stabilizing effect produced by a giant planet over the disruptive perturbations of mas- sive DPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' we also demonstrated (Muñoz-Gutiérrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2018) the existence of a constant resupplying of the giant’s MMRs with new objects, a mechanism acting on secular time-scales due to the radial migration of disc particles produced by the DPs’ scattering effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In this work, we expand the exploration of the mass parameter space of our mixed stirring scenario for more massive debris discs, comparable to those which have been observed in extrasolar plan- etary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We account for both the perturbations produced by an interior giant planet, as well as 100 massive DPs embedded in a disc of 1 000 massless particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The simulation setup for our grid of disc-planet systems, along with a brief summary of the dynamical modeling approach, is given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In Section 3 we character- ize the outcome of our simulations, through analysis of the evolution of the survival fraction, average eccentricities, and inclinations of the bodies comprising the discs, as well as the orbital perturbations exerted on the giant planet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' we provide our interpretation of the re- sults and how they relate to other works addressing either planetary or self-stirring of a debris disc in isolation in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Finally, in Section 5, we summarize our findings and present our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2 METHODS AND SIMULATIONS We aim to test the efficiency for producing stirring over debris disc particles, of models which combine the perturbations coming from a giant planet, located interior to an initially wide and cold debris disc, as well as dwarf planets embedded within the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We call this a mixed stirring scenario, since it combines some of the elements applied so far in debris discs stirring models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' secular perturbations from giant planets and self-stirring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='1 Model disc generation Our systems are formed by a Solar mass central star, as well as a Neptune-analog “giant” planet (GP, hereafter) located at 30 au and starting with zero eccentricity and inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The debris disc is formed by 1 000 test particles and 100 massive DPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' the disc is 30 au wide and its inner edge is set to be 10 Hill radii beyond the GP location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We assume the mass of the debris disc to be given by the sum of the individual masses of the 100 DPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We study a grid of models where the GP mass explores values from 10 to 316 M⊕ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' from sub-Neptune to one Jupiter masses), in logarithmic steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='15 (11 values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The mass of the debris discs covers a range from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='16 to 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='6 M⊕ in logarithmic steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='04 (26 values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Within each disc, the masses of the individual DPs are drawn randomly to try to reproduce the 100 most massive particles of a mass distribution 𝑛(𝑚) ∝ 𝑚−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='8, consistent with the distribution of large bodies in the Kuiper Belt (Fraser & Kavelaars 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The individual mass of the most massive DP in the lightest disc is below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='105 M⊕, while in the most massive disc it is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='05 M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Those values correspond to ratios with the less massive giant planet of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Such large planetesimal masses are not unexpected according to recent theories of planetesimal formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' the streaming instability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Youdin & Goodman 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Morbidelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Nesvorný et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2019), and are consistent with recent measurements of planetesimal masses inferred from the spatially resolved scale heights of 𝛽 Pic and AU Mic (Matrà et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Daley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' MNRAS 000, 1–11 (2023) Mixed Stirring of Debris Discs 3 The range in debris disc masses was chosen to keep a realistic representation of the individual objects in the discs while remaining computationally feasible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' a larger range in debris discs masses would imply that individual DPs would be very massive (compara- ble to the GP mass) to account for more massive discs, or we would require to proportionally increment the number of DPs in our simula- tions, making them too computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' If lower limits on the DP masses are preferred, the largest of these DPs should be inter- preted as the sum of many smaller bodies, a product of the limitation of our computational power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The distributions of semi-major axes, eccentricities, and inclina- tions of the DPs and test particles were randomly generated based on a single seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The initial inclinations of the DPs and test particles were randomly drawn between 0 and 5◦, whilst the initial eccentric- ities were constrained to be ≤0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='05, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' we used similar values to the ones found for the cold classical Kuiper Belt (Gulbis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Visual inspection of the output for 20 seed values was carried out and an initial simulation setup was selected based on the uniformity of the distribution in 𝑎-𝑒 and 𝑎-𝑖 parameter space for both the DPs and test particles 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='2 N-body simulations We used the hybrid symplectic integrator from the mercury package (Chambers 1999), to explore the long-term evolution of a grid of 286 debris disc models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' An initial time-step of 400 days is used in all cases, as well as an accuracy parameter for the Bulirsch-Stoer integrator of 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We produced orbital outputs every 10 Myr, over a total integration time of 1 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Particles are removed from the simulation if their semi-major axes grow larger than 10 000 au, decrease below 1 au, or if they collide with the GP or the DPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In most cases, several DPs are also ejected from the simulations by the same mechanisms due to their mutual interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 3 RESULTS Over sufficiently long periods of time (∼100 Myr), the gravitational perturbations from DP-sized objects, acting on initially cold debris discs particles, induce a considerable vertical and radial heating (Muñoz-Gutiérrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2015), which results in a progressive in- crement of the disc’s mean eccentricities and inclinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' A GP in a non-circular, non-planar orbit, will induce secular per- turbations on an external debris disc, forcing a component on the particles’ eccentricity and inclination vectors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Murray & Der- mott 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Mustill & Wyatt 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Gladman & Volk 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Though initially circular and planar, the orbit of the GPs in our simulations quickly evolves, as we will show, due to their interactions with the massive disc members (DPs), which makes the former phenomenon relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Moreover, under the right circumstances, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' if massive enough (≳100 M⊕), an interior GP can also act to stabilize the orbits of massless particles within debris discs, acting against the perturba- tions produced by DPs (Muñoz-Gutiérrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In the following subsections, we will show separately the evolution of the populations of massless particles, DPs, the debris discs as a whole, and finally the GPs within these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 1 The initial orbital distribution of the DPs and test particles in the discs, for the random seed used in this work, can be found online at Figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Animated figures for the survival fraction of test particles in the discs (top), as well as their mean eccentricity (middle) and inclination (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Each colored circle in the grids shows the corresponding value at each time step output from the simulations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' every 10 Myr) according to the color bar presented to the right of each grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The points in the grid are arranged as a function of the mass of the GP in the model as well as of the initial mass of the debris disc, as accounted by the total mass of 100 massive DPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The still frames in each panel show the final states of the simulations after 1 Gyr (An animated version of this figure can be found online at Figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='1 Evolution of Massless Particles in the Discs We aim to quantify the long-term impact that the combination of perturbers, namely an interior GP plus 100 massive embedded DPs, have on the dynamical stirring of the initially cold debris disc par- ticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We produced coloured grids showing the survival fraction of particles in the discs, as well as the amount of dynamical excitation, characterised by their mean eccentricities and mean inclinations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' at this first stage, we characterise this excitation as a function of the total initial disc mass, as well as the GP mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 1 we show the evolution of the survival fraction, the average MNRAS 000, 1–11 (2023) t = 1000 Myr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='8 Giant Planet Mass [M Surviving Fraction 100 06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='4 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 3 10 30 Initial Disk Mass [M ]t = 1000 Myr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 300 Giant Planet Mass [M ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='8 Eccentricity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='6 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='4 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 10 30 Initial Disk Mass [M ]t = 1000 Myr 50 300 Giant Planet Mass [M ] 40 30 30 10 10 10 30 Initial Disk Mass [M ]4 Muñoz-Gutiérrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' eccentricities, and the average inclinations of the massless particles on our array of simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In the top panel of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 1 we show the evolution of the survival fraction up to 1 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In the animated figure, each snapshot corre- sponds to a 10 Myr time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The color of each circle represents the surviving fraction of massless particles within that disc, while its location on the grid corresponds to the initial mass of the disc (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' the sum of the masses of our DPs) and the mass of the GP in that planetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The ejection efficiency is correlated to both the GP mass and the (initial) disc mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Those systems with the highest GP and disc masses are the most quickly depleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Within the first 30 to 100 Myr, the systems with the most massive GPs (𝑀GP > 100M⊕) have al- ready lost ≥ 80% of their initial particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Over the next hundreds of Myr, with a smaller number of total particles as well as a smaller number of total perturbers, the ejection rate slows down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Overall the most efficient ejection continues to occur in systems with simultane- ously the most massive discs and the most massive GPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' At the end of the simulations, the higher ejection efficiency occurs for initial discs masses ≳10 M⊕, with the highest ejection efficiency occurring when the GP mass is ∼ 100 M⊕ and the disc mass is ≳ 20 M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Many systems exhibit the ejection of a substantial fraction of the test particles in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The average ejection rate is 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='5% across all simulations in the grid, with an ejection rate of up to 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='3% for the most extreme case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The orbital characteristics (eccentricity, inclination) of particles in the discs were calculated by averaging the elements of surviving particles at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In the animated version of Figure 1, we show their evolution in 10 Myr time steps illustrating the change in the remaining particles, their eccentricity, and inclination over 1 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The color of each circle represents the mean values of the eccentricity and inclination for each model at the timestep in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' From the evolution seen in the middle and bottom panel animations of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 1, we find that the disc response is monotonic for the lower GP masses (𝑀GP ≤ 30 𝑀⊕).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We find increasing excitation for decreasing GP mass, increasing disc mass, and longer integration times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The evolution of the disc excitation with time is more clearly visible in eccentricity than inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The middle panel of animated Figure 1 shows that after a few tens of Myr of evolution, a more efficient stirring has been produced for the middle rows of the grid, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' for 𝑀GP in the range ∼30 to ∼110 M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' At this time, the stirring grows in proportion to the debris disc mass, while for a given debris disc mass, the stirring increases with GP mass, reaches a maximum around 70 to 100 M⊕, and decreases for larger GP masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' This behaviour does not resemble the quadratic behaviour presented in Muñoz-Gutiérrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' (2017), however that study was for discs 2 to 4 orders of magnitude lighter than what we are studying here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Over time, during the first 400 Myr, we see less massive GPs becoming progressively more efficient at exciting test particles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' while the more massive GPs models stop evolving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' After 300 Myr the sweet spot for efficient stirring becomes less evident, in part due to the ejection rate of the most excited particles from these systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' after 600 Myr even the models with the least massive GP have stopped evolving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' By the end of the simulations, the largest mean eccentricity occurs in the lower right corner of the grid, where the disc masses are comparable to, or even greater than, the GPs masses in these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In the bottom panel of animated fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 1 we observe a slower and more linear trend for the evolution of the mean inclination;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' up to 100 Myr, the increment in mean inclination is small and its value remains almost homogeneous across the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' With time a small tendency of larger excitation with larger disc masses and smaller GP masses starts to develop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' after 200 Myr the lower right corner of the grid, where 𝑀GP ≤ 𝑀DD, starts to show clear signs of a stronger stirring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' By the end of the simulations, the final stirring is shown to be a function of both GP mass and debris disc mass, with the greater stirring observed in systems with lower GP masses and larger disc masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' When the mass of the disc is comparable to that of the GP, the planet-disc interactions are warranted to be complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The angular momentum that can be transferred from the GP to the DPs is large enough to produce a significant migration of the GP due to the ejection of massive objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Also, the reference plane (or “invariable plane”) within such a massive debris disc is not well defined, as the GP orbit no longer plays such an important role in determining the total angular momentum of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' These conditions are satisfied for models in the lower right corner of our grid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' in that region, particles are excited but they are not efficiently ejected, so the system effectively heats up and there is no way of cooling it down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Complementary to the animated grids in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 1, we also present the time evolution of each model as a curve on the three panels of animated fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' There we can see the evolution of each model across the animation, with the survival fraction on the top panel, mean eccentricity in the middle panel, and mean inclination in the bottom panel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' the last images (as well as the still frames) highlight the average of all the models with the same GP mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In the top panel of animated fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2, we see the decline in particle numbers as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' As expected, the more massive planets are more efficient at ejecting test particles from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' For a given planet mass, the ejection is more efficient with a more massive disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In the middle panel of animated fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2 we present the eccentricity evolution for each of our 286 models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' as in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 1, we are presenting the evolution of the mean eccentricity of all particles remaining in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' For models with GP masses less than ≲ 80 M⊕ we can see that the eccentricities keep increasing over the whole duration of most of the simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' all models slow down with time, but for models with GP masses between ∼ 30 M⊕ and ∼ 80 M⊕ there seem to be two phases: first, a fast increase and then they reach a plateau with very little increase in eccentricity thereafter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' the change between these two phases occurs sooner and at a lower average eccentricity for the more massive GPs, and will likely occur even at GP masses less than 30 M⊕, but it probably requires more than 1 Gyr for the same to happen, while for 80 M⊕ it only requires approximately 100 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' For the most massive GPs (≳ 100 M⊕) a third phase appears, after the fast increase, and before the plateau, a moderately fast decrease occurs due to the rapid ejection of the most eccentric objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' again the evolution is faster for more massive GPs, this new phase seems to be most pronounced for our 223 M⊕ models, but perhaps with smaller time steps it might be even more important for the 316 M⊕ GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Finally, models with GPs more massive than 220 M⊕ seem to reach saturation, perhaps even a small decline, near the end of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Regarding the effect of the disc mass on the overall eccentricity, we find that, for a given time and GP mass, larger disc masses produce larger mean eccentricities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We present the inclination evolution in the bottom panel of ani- mated fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' as for eccentricity, we are presenting the evolution of the mean inclination of all particles remaining in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Here we show that the evolution of the inclinations is much slower than for the eccentricities, in fact, the inclination for all models con- tinues to rise until the end of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' As with eccentricities, simulations with more massive discs tend to evolve faster and have larger mean inclinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' MNRAS 000, 1–11 (2023) Mixed Stirring of Debris Discs 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Evolution of the survival fraction (top), mean eccentricity (middle), and inclination (bottom) of test particles in the discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The different colors of the lines in the three panels indicate the mass of the GP in the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The initial debris disc mass in the models is represented by the thickness of each line, with thicker lines corresponding to more massive discs (pale lines in the still frames, all but the last frame in the animated figures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The thickest lines in the still frames (and those of the last animated frames) correspond to the average of all disc masses for any given GP mass (An animated version of this figure can be found online at Figshare).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In general, for very large GP masses, both eccentricity and inclina- tion show a mostly smooth evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' This is related to the dominance of the GP mass on the overall dynamics, as well as to the number of particles quickly ejected from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' This shows a dependence in GP mass on the degree of stirring of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Very massive GPs become less efficient with time at heating the discs, and in fact, those discs cool off at later times, whereas less massive GPs continually stir their discs throughout the timescale of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' This effect can be explained through the ejection efficiency of the GPs at differ- ent masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' High-mass GPs (top rows) quickly excite and eject disc particles and DPs that stray into regions of strong interaction with the GP, leaving a depleted but dynamically cold system in their wake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In contrast, low-mass GPs do very little to stir the discs, but also very little to suppress stirring by the DPs or to eject particles excited by DPs, leaving a well-populated but dynamically hot system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='2 Evolution of DPs in the Discs We find that the evolution of massive DPs in the discs follows a similar trend to that of massless particles, but their self-stirring is slightly less efficient, as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 3 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' There we present animations showing the evolution of the survival fraction (top panel), mean eccentricities (middle panel), and mean inclinations (bottom panel) for surviving DPs in the simulations as a function of time, in the same scheme as for the test particles in the previous sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In the top panel of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 3 we see the surviving fraction of DPs in each model system as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Again, consistent with the analogous plot for test particles in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2, we see that the DPs are more efficiently removed from the system with a more massive GP and a more massive disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In the middle panel of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 3 we can see trends in the behaviour of the DPs can be delineated for models with different GP masses, following the same general behaviour as for the test particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The models with the lowest GP masses, below 15 M⊕, exhibit a rising mean eccentricity for the DPs up until the end point of our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Models with GPs above that, but below 60 M⊕, again reach a plateau, and have a slow increase, but this time they have an obvious maximum before having a slow decrease in eccentricity at some point between 400 Myr and 1 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The time at which the highest value occurs, and its value are both dependent on the GP mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' more massive GPs have their maxima at earlier times and with lower mean eccentricity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' This is again a result of the increasing strength of interaction for DPs that more closely approach the GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Furthermore, we see that overall the mean eccentricity of the DPs is lower than that of test particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' For models with GPs > 60 M⊕ we again observe a third phase of evolution, in the first a rapid increase in mean eccentricity occurs, quickly reaching a peak within the first 200 Myr which is faster for more massive GPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' after this follows a decline, also faster the more massive the GP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' finally, after the decline, a slow increment begins again until reaching an approximate steady state by the end of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The maximum values of the average mean eccentricity for the models remain below ≃0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='35 for DPs (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='55 for test particles, which continue growing for the lowest mass GP models), with an apparent saturation limit at this value independently of GP mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We can also see that the behaviour of the lines in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 3 is noisier than in the case of test particles (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2), this is because the DP population is 10 times less numerous than the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We would expect this to also be true for any real disc since the number of DPs containing a substantial amount of the disc mass will always be a minority compared to the total population (starting with the largest bodies, which are the most dynamically relevant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' For any given GP mass there is a trend of larger eccentricities for MNRAS 000, 1–11 (2023) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='8 Survival Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 GP Mass (M) 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='22 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='20 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='81 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='12 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='87 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='43 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='18 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='49 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='23 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='95 0 200 400 600 800 1000 Time [Myr]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='5 Eccentricity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='3 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='1 GP Mass (M) 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='22 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='20 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='81 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='12 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='87 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='43 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='18 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='49 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='23 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 0 200 400 600 800 1000 Time [Myr]GP Mass (M) 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='22 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='20 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='81 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='12 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='87 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='43 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='18 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='49 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='23 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='95 40 Mean Inclination [°] 30 20 10 0 0 200 400 600 800 1000 Time [Myr]6 Muñoz-Gutiérrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Same as fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2 but for the evolution of the survival fraction (top), mean eccentricity (middle), and inclination (bottom) of DPs in the discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The color of the lines in both panels indicates the mass of the GP in the models, while line thickness represents the mass of the disc, with thicker lines corresponding to more massive discs (pale lines in the still frame, all but the last frame in the animated figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The thickest lines in the still frame (and those of the last animated frame), correspond to the average of all disc masses for any given GP mass (An animated version of this figure can be found online at Figshare).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' larger disc masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' However, there is an overall dispersion for the evolution of each suite of simulations, and some individual simula- tions fall outside of the global trend e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' the most massive discs for the systems with 28 M⊕ and 112 M⊕ GPs lie well above the other systems in their respective suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' These “outliers” may be attributed to stochastic events involving DP interactions or ejections influencing the overall evolution of that system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The evolution of the mean inclination for DPs in our models is shown in the bottom panel of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Again, we observe a similar be- haviour to the one described above for the test particles, finding lower inclinations for larger GP masses, and also that the final inclination values are consistently lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In this case, almost all the systems show a monotonic rise in mean inclination over the duration of the simulations with no turnover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Only the most massive GP systems (> 220 M⊕) seem to reach a peak in their respective mean inclina- tion within the duration of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Systems with lower mass GPs, MGP < 30 M⊕, are not yet slowing down at the end of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We also find that, for a given GP mass, more massive discs will produce larger mean inclinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Overall, the greatest in- clination values lie below 30◦ for the DPs, regardless of GP mass, and take longer to undergo the same relative degree of excitation, as compared to the test particles in the same systems that can reach values close to 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='3 Evolution of the Discs as a Whole To better understand the evolution of discs as complete systems, containing both massive and massless particles, as well as the rela- tionship between the two, we begin by comparing the final values of the mean orbital parameters and survival fractions of test particles and DPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 4 we show the final distribution of mean eccentricities (left panel), mean inclinations (middle panel), and survival fractions (right panel), of both populations, for all 286 systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' the different colors indicate the mass of the GP in that system, while the size of the dots represents the initial mass of the corresponding debris disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In the left and middle panels of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 4 we see that for both mean eccentricity and inclination, the distribution of final values remains above the identity line (indicated by the solid black line) except for one outlier case in eccentricity which corresponds to one of the models with the most massive GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We can see that the final conditions for all models closely follow a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' A comparison of the corresponding panels in figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2 and 3, shows that massless particles are more easily disturbed than DPs (as seen in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' it can also be seen that the evolution of the eccentricity is much less mass dependent than that of the inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We applied a linear fit in both cases (dashed black lines in the left and middle panels of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 4) to quantify how efficient the stirring of test particles is when compared to that of massive DPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The best fit for the models in the eccentricity panel is given by � 𝑒particles � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='366 ⟨𝑒DPs⟩ +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='004 and for the final inclination � 𝑖particles � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='916 ⟨𝑖DPs⟩ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='602◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' these fits show that the stirring of test particles is more efficient than that of DPs by factors of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='366 for eccentricity and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='916 for inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Both of these fits lie close to the gray star representing the initial conditions of all the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' As in figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2 and 3, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 4 shows that the more massive discs (larger dots) are more efficient at stirring their particles than less massive ones (smaller dots) but that more massive GPs have a stabilising effect on the discs after a quick removal of the initially unstable minor bodies (both DPs and test particles);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' this comes about because massive GPs will tend to eject particles that pass close to them, MNRAS 000, 1–11 (2023) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='8 DPs Survival Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 GP Mass (M) 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='22 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='20 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='81 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='12 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='87 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='43 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='18 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='49 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='23 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='95 0 200 400 600 800 1000 Time [Myr]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='5 Eccentricity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='4 DPs Mean E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='1 GP Mass (M) 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='22 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='20 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='81 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='12 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='87 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='43 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='18 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='49 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='23 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 200 400 600 800 1000 Time [Myr]GP Mass (M) 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='22 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='20 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='81 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='12 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='87 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='43 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='18 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='49 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='23 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='95 40 DPs Mean Inclination [ 30 20 10 200 400 600 800 1000 Time [Myr]Mixed Stirring of Debris Discs 7 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Comparison of the final mean orbital parameters and survival fractions of DPs vs test particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The left panel shows the distribution of mean eccentricities, the middle panel for mean inclinations, and the right panel for survival fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The colors indicate the mass of the GP in the system, while the size of the dot represents the initial mass of the debris disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The identity is indicated by the solid black line, while best fits are indicated by dashed black lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Linear fits were done for both eccentricity and inclination, while a third-order polynomial was fitted to the survival fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The gray star in each panel represents the initial conditions of our systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' whereas lighter GPs will perturb their orbits without ejecting them from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The final distribution of survival fractions (right panel of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 4) remains below the identity line, illustrating the greater difficulty for a planet in ejecting massive objects (DPs) than massless ones (test particles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' A strong dependence on GP mass is observed in the final survival rate for both populations of minor bodies, demonstrating the efficiency of ejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We find the relationship between the surviving fractions (SFparticles and SFDPs) is best represented by a 3rd order polynomial of the form: SFparticles = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='932 SFDPs3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='703 SFDPs2 (1) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='746 SFDPs − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' As with eccentricity and inclination, extrapolation of this trend to- wards the less perturbed discs leads to the gray star representing the initial conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' for survival fractions, there is also an obvi- ous extrapolation to more violent systems and we find that our trend leads toward the (0,0) point where all particles would be ejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The scatter of simulations around this trend line is generally more pronounced for the systems with higher GP masses (in the survival of both test particles and DPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' This is to be expected as it is interactions with the GP in each system that will dominate the removal of smaller bodies (either by collision or ejection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We find that the number of test particles removed by collisions remains approximately constant over the simulation grid, comprising about 2% of the particles over the duration of each model run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' By contrast, the number of ejection events is strongly correlated with the GP mass, with removals initially about 5%, and swiftly becoming greater by an order of magnitude or more with up to 95% during a model run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' As the GP mass decreases so too does the ejection efficiency, and they will only dynamically heat their companion discs rather than deplete them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' This leads to a lower dispersion in the survival of DPs, but a comparable scatter in test particle ejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The most massive GPs exhibit the tightest correlation with the observed trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In these simulations, the GP rapidly stirs and depletes the disc (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2) and if any minor body subsequently migrates into the perturbation region of the GP it is swiftly removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The most massive discs in the simulations for a given GP mass tend to lie below the trend line identified by section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' This indicates segregation by disc mass within the distribution of surviving minor bodies, where the more massive (initial) discs are more depleted in both test particles and DPs for a given GP mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' This is the natural consequence of greater dynamical stirring by more massive individual DPs within the more massive disc for a given system, leading to particles (and DPs) passing into close interaction with the GPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' This tendency weakens and breaks down as the GP mass decreases, representing the decreasing capacity of the GP to deplete mass from the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Most of the analysis of sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='2 is focused on the point of view of the models, this is: we are classifying each model accord- ing to its initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' However, this is not directly applicable to observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' From an observational point of view, it is more in- teresting to characterise a model according to its current parameters, and while the GP mass will not change, the disc mass will change with the ejection of DPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Therefore, similar to the animated fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2, in the animated version of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 5 we show the time evolution of our 286 models by plotting the survival fractions, mean eccentricities, and mean inclinations of surviving particles at each time step, as a function of the evolving mass of the disc, instead of its initial mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In the top panel of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 5 we show the survival fraction of test particles, where each snapshot corresponds to a 10 Myr evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The color of each circle indicates the particle survival fraction present in each disc at that time, with darker colours representing a lower survival fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We can see that the evolution of the survival rate is fastest for the most massive GP and (initial) disc mass combinations, with more than 50% depletion of those discs occurring within the first few tens of Myr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' in the same time frame, barely any ejections have occurred amongst the lower mass systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' By 100 Myr, systems with GP masses greater than 60 M⊕ have experienced substantial ejection, losing up to half the particles (but not necessarily half their mass), whereas systems below that have yet to experience any substantial ejections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' At the 500 Myr point, the most massive systems have lost up to 90% of their initial particles and only the least massive GP/disc systems are untouched by ejections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Beyond this time up to 1 Gyr the overall picture remains constant and the systems’ evolution is more gradual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' If we focus on a fixed small area of the grid, instead of following MNRAS 000, 1–11 (2023) GP Mass (M@) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='6 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='22 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='87 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='49 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='43 Eccentricity 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='23 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='81 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='95 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='12 Mean l 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='3 Particles T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='5 DPs Mean EccentricityGP Mass (M) 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='22 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='87 40 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='49 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='20 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='43 Mean Inclination [ 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='23 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='81 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='18 30 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='95 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 20 Particles N 10 5 10 15 20 25 DPs Mean Inclination f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 GP Mass (M) 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='22 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='87 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='49 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='43 Survival Fraction 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='23 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='81 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='6 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='95 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 Particles S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 DPs Survival Fraction8 Muñoz-Gutiérrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Animated figures illustrating the surviving fraction of test particles (top), mean eccentricity (middle) and mean inclination (bottom) as a function of the evolving disc mass vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' GP mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The time step is in increments of 10 Myr (An animated version of this figure can be found online at Figshare).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' the evolution of individual coloured circles, the behaviour of the survival fraction in that region becomes even more extreme, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=', for a disc mass of ∼ 7 M⊕ the difference in survival fraction goes from ≈ 90% (at low GP masses) to ≈ 10% (at high GM masses) during the 1 Gyr simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In the same sense, one should look at the animated middle panel of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 5, following the eccentricity evolution, as we looked at the animated top panel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' we should focus on an area and not let our eyes drift away from it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' by looking at a column centered at around ∼ 10 M⊕, we see that the eccentricity slowly rises with time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' after the first few time steps the most eccentric models were those with a GP mass of ∼ 200 M⊕, but with time this maximum went all the way down to 10 M⊕ (although this took the best part of the 1 Gyr of our simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Another thing to note is that, while many individual dots reach saturation within our simulations, by looking at a fixed area we see that it keeps on evolving, mostly because simulations with more massive initial discs keep passing through our observation area (akin to the difference between Eulerian vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Lagrangian evolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' By the end of our simulation, we find that there is a triangular region, in the lower right of the plot, that is mostly saturated with mean eccentricities ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' This is quite extreme for discs that were initially dynamically cold with ⟨𝑒0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='025, a ∼25 fold increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Finally, for the bottom panel of the animated fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 5, following the inclination evolution, we observe that the evolution of inclination is slower than for eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' After ∼ 100 Myr the inclination is mostly homogeneous with only the most massive disc models show- ing signs of a significant stirring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' During the next hundreds of Myr, a differentiation in the level of stirring becomes evident for individual columns, which seem to have uniform colors evolving in time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=', the excitation level for the inclination is more clearly dependent on the remaining debris disc mass than on the initial disc mass or the GP mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' At the end of the simulations, the maximum stirring has oc- curred for the most massive debris discs and the least massive GPs, however, since the ejection fraction grows with GP mass, as time passes, what would be an equally excited component in our most massive GP models has already been depleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' A skewed initial grid (rather than the rectangular one we consid- ered here), with more massive debris discs for the more massive GP systems, might fill in some of this depleted parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' How- ever, as the disc evolution timescale decreases with increasing disc mass, the observed regions of parameter space that are vacated in our simulations are necessarily void given the duration of the simu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In this sense the structure we observe in our grid at 1 Gyr is not fixed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' longer integration would necessarily drive all the systems to lower disc masses, leading to a more pronounced “gap” in the top right of these plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' This diagram, therefore, provides some con- straints on the evolutionary pathway undertaken by observed debris discs with the constraints of the stellar age and inferred disc mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='4 Evolution of GPs Besides the evolution of the debris disc systems as a whole, the GPs in our models experience modifications to their initial orbital parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' this is due to the interactions between the GP and massive DPs, which results in the interchange of angular momentum that leads to an overall increase in their eccentricity and ultimately to ejections of some DPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Although small in most cases, the orbital perturbations experienced by some of the GPs in our models can be significant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' specifically: large inward orbital drifts, of up to 10 au, are observed in systems with the less massive GPs and the most massive debris discs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' in those systems with the largest mass ratio, as given by 𝑀DD/𝑀GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 6 we only show the final distribution, in logarithmic values, for the eccentricities (left panel), inclinations (middle panel), and semimajor axis changes (right panel) of the GPs in our 286 models, as a function of the logarithm of the mass ratio of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In log- log space, those three distributions can be well described by linear fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' At the end of the simulations we found that most of our GPs would be considered to have remained in cold orbits (only 3, out of 286, have 𝑒 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='1, while only 5 have 𝑖 > 5◦);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' however, about 30% of the GPs in our simulations have lost a significant fraction of their angular momentum, having a noticeable decrease in their semimajor axis by the 1 Gyr mark, 𝑎 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='9𝑎0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' At any point during the simulations, the three distributions (𝑒, 𝑖, and |Δ𝑎| /𝑎0) can be well described by linear fits that slowly evolve with time, with both the absolute value as well as the mass fraction dependence slowly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' By fitting all simulations at 100 Myr MNRAS 000, 1–11 (2023) t = 1000 Myr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 （ 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='8 Giant Planet Mass [M （（ Surviving Fraction （（ 100 （（ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='4 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 3 10 30 Disk Mass [M]t = 1000 Myr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='8 Giant Planet Mass [M Eccentricity ～（ 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='4 Mean 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 3 10 30 Disk Mass [M@]t = 1000 Myr 50 300 40 Giant Planet Mass [M （（ （（ 100 30 （（ 20 30 10 10 0 3 10 30 Disk Mass [M@]Mixed Stirring of Debris Discs 9 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Final orbital values for the GPs as a function of the mass ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The left panel shows the distribution of final eccentricities, the middle panel for final inclinations, and the right panel for final semimajor axes changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' As in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 4, the colors indicate the mass of the GP in the system, while the size of the dot indicates the initial mass of the debris disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Linear fits in these Log-Log planes are indicated by the dashed lines (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' intervals, and subsequently fitting a time dependence to the linear fits we obtain: log10(𝑒GP) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='3383 � 𝑇 Myr �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='1005 log10(𝑀DD/𝑀GP) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='2061 log10 � 𝑇 Myr � − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0871, (2) log10(𝑖GP) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='6454 � 𝑇 Myr �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0751 log10(𝑀DD/𝑀GP) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='3265 log10 � 𝑇 Myr � − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='9025, (3) log10(|Δ𝑎| /𝑎0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='3138 � 𝑇 Myr �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='1281 log10(𝑀DD/𝑀GP) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='4582 log10 � 𝑇 Myr � − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0646, (4) for eccentricity, inclination, and semimajor axis change, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 4 DISCUSSION In this work, we did not focus on the sculpting process of the edges of the discs, nor on the disc shapes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' this is why we choose 10 Hill radii as the inner edge of the discs and not 5 Hill radii as been done elsewhere (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Pearce & Wyatt 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We further assumed a GP in a circular and planar orbit to minimise the impact of the planet on the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We instead focused on the stirring process produced as a result of the interaction of the massive DPs embedded in the debris discs, but such a process was somewhat dominated by the presence of the GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We focused on determining the stirring levels as functions of both the GP and debris disc masses (assuming the shapes and disc edges are imprinted on the debris discs by the giant planetary companion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Our model spans GP masses between 10 M⊕ (approximately 60% the mass of Neptune) and 316 M⊕ (approximately the mass of Jupiter);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Pearce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' (2022) estimate that Neptune to Saturn-mass planets are the minimum needed to stir most of their 178 modeled discs (though some needing Jupiter mass planets, assuming maxi- mum eccentricities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Similarly, the range of disc masses in this analysis, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='16 to 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='6 M⊕, are consistent with expectations based on both observations and theoretical considerations (Mulders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Krivov & Wyatt 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Several other studies predict larger masses (> 100 M⊕) in order for debris discs to be self-stirred (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Krivov & Booth 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Krivov & Wyatt 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Nonetheless, in this work, we found that small masses in debris discs can result in large stirring values, up to a 25-fold increase in the most extreme cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Thus an efficient stirring is possible for small disc masses (< 10 M⊕), if ever perhaps containing larger than expected perturbers, as some of the DPs present in the most massive discs we considered have assigned masses close to 1 M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Our massive objects are initially thought to be real ‘dwarf planets’ (DPs), as long as we adopt the definition of DP as an object that has not cleared its neighborhood from debris (yet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We could expect massive debris discs (much more massive than our Kuiper belt) to contain more massive objects, though this is not necessarily true, depending on planetary formation mechanisms, disc mass density, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Recent studies on dust formation and excitation place limits to the most massive objects present in massive debris discs to be around 5 times the mass of Pluto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Based on spatially resolved observations of the vertical scale heights of the debris discs around AU Mic and 𝛽 Pic, the most massive bodies present in those discs could be up to 9×10−5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='4 M⊕, respectively (Daley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Matrà et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' However, more massive objects might be present in debris discs (without leaving a piece of observational evidence, such as bumps or gaps), if we assume the mass range in planetesimals scales linearly with the overall mass of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Limiting the mass of the DPs to be similar to the mass of Pluto would diminish the stirring effect in both 𝑒 and𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' According to shorter duration simulations (𝑡 = 50 Myr) with 41, 100, and 250 DPs (and 410, 1 000, and 2 500 test particles) we ran as consistency checks, this correction should be approximately a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='5, and definitely less than a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Notably, by using Mercury, the problem becomes computation- ally intractable when considering more than a few hundred massive DPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' new tools are required to expand the grid with a greater number of DPs and particles, such as GPU-based simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We leave this question open for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The main stage for excitation evolution in our simulations occurs in timescales of the order of a few and up to 100 Myr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' while the time required for our systems to acquire their final configurations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' reach their saturation levels, is of the order of 150 Myr to more MNRAS 000, 1–11 (2023) GP Mass (M) 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='22 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='49 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='20 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='43 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='5 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='81 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='18 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='95 Log10(eGp) 14.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='5 LOg10(MDD/MGP)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 GP Mass (M) 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='22 223.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='81 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='18 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='5 Log1o(iGP) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='20 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='43 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='00 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='81 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='18 (0e/levl)01607 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='25 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='5 LOg10(MDD/MGP)10 Muñoz-Gutiérrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' than 1 Gyr scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' These timescales are similar to the ages of host stars for many observed debris disc systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' thus we would expect that many of the observed systems with similar physical parameters as those covered in this work, would already be settled in their final configuration, currently experiencing a quiet steady-state evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The timescales derived here are a function of the chosen architec- ture of the model, adopting a 1 M⊙ star with a planetary companion at 30 au and a Kuiper belt-like disc beyond that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' However, the evo- lutionary timescales can be easily scaled for different stellar masses or disc semi-major axes, as the dynamics should be self-similar, pro- vided physical collisions are a negligible cause of removal of bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' For a different central star mass, all the masses should scale propor- tional to the new stellar mass, and the timescales should be modified as the inverse of the square root of the mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' for a different GP orbital radius, the masses should not be modified, and the timescales should be modified as the mass to the -3/2 power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' For the Vega system, with a stellar mass ∼2 M⊙ and a planetesimal belt around 100 au (Matrà et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Marshall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2022), the equivalent timescale would be nearly three times longer than the evolution of the models considered here (depending on the exact location of the GP used in Vega).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' A debris disc is the result of a collisional cascade within a plan- etesimal belt triggered by dynamical excitation, either intrinsically by the largest planetesimals within the belt (Krivov & Booth 2018) or extrinsically by an external perturber (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Mustill & Wyatt 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The range of relative velocities among planetesimals, required to trigger the onset of the collisional cascade, is typically estimated as 100 to 300 m/s (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Kenyon & Bromley 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' On the other hand, such relative velocities can be estimated from the average or- bital parameters of the dust-producing small objects in the discs, as 𝑉rel = 𝑉𝐾 √ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='25𝑒2 + 𝐼2, where 𝑉𝐾 is the Keplerian velocity at the distance 𝑎 from the star (Lissauer & Stewart 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Wyatt & Dent 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Krivov & Booth (2018) argued though, that the average in- clinations are not terribly important when determining the relative velocities among planetesimals, since eccentricities grow much faster than inclinations in debris disc models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Thus, one can simply esti- mate the relative velocities from the root mean square eccentricity of the planetesimals as 𝑉rel = 𝑉𝐾 √︃� 𝑒2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In any case, an estimation of such relative velocities in all of our models shows that values close to 1 km/s are quickly reached, in less than 10 Myr, regardless of the initial debris disc mass or the GP mass of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Indeed, veloci- ties of collisions within the belt modeled here, range from ∼250 m/s to ∼2000 m/s, therefore locating themselves safely on the side of a collisional cascade capable of producing dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' A population of planetesimals on eccentric orbits within a debris disc would produce a halo of millimetre dust grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Such structures have been identified in ALMA observations of several systems, in- cluding HR 8799 (Geiler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2019), HD 32997 and HD 61005 (MacGregor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The typical eccentricity of dust grains within debris discs inferred from their spatially resolved belts lies in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='3 (based on 11 discs, see figure 9 of Marino 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' this level of eccentricity is consistent with the mean eccentricity induced by the dwarf planets in this set of simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' 5 SUMMARY AND CONCLUSIONS In this work, we performed a suite of 286 numerical simulations to explore the stirring effects that a combination of giant and dwarf planetary perturbations would have on the long-term evolution of initially cold debris disc models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Our systems are formed by a solar mass star, a giant planet initially located at 30 au in a circular and planar orbit, and 100 massive dwarf planets embedded in a disc described by 1000 test particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The orbital distribution of the discs was drawn randomly for small values of eccentricity (between 0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='05) and inclination (between 0◦ and 5◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' We initially located the inner edge of our discs at 10 Hill radii from the GP, with a total width of 30 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Our 1 Gyr long simulations take into account the perturbations from the GP and the DPs over test particles and among themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The evolution timescale for the eccentricity and inclination de- pends mostly on GP mass, where the simulations with more massive GPs evolved faster than those with less massive GPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' On the other hand, the limit to the heating depends on both GP mass and disc mass, with large disc masses and small GP masses being able to heat the disc more than simulations with light discs and/or heavy GPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Part of the reason why massive GPs are less efficient at heating the disc is their tendency of ejecting “warm” particles before they can get extreme values of either eccentricity or inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In all models the mean inclination rises quickly (or at least rela- tively quickly) before slowing down, only the most massive GPs seem to be able to level off before the 1 Gyr mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The eccentricity evolves faster with many of the simulations reaching a plateau before the end of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Very massive GPs heat their discs very quickly which then slowly cool down by ejecting the more excited particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The effect on the eccentricity is larger than for the inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Massless particles, which in real systems could be considered as the less massive members of the discs, such as cometary nucleii, are more mobile than massive objects (DPs), therefore they become ’hotter’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' more eccentric, more inclined, and are easier to be ejected (they have a poorer survival rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Nonetheless, DPs reach significant stirring levels as well and have only slightly better chances of survival than test particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The values of both eccentricity and inclination for test particles at a given time have a better correlation with the remaining mass of the debris discs than with the GP mass or the initial debris disc masses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' this is particularly evident for the inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' GPs themselves are perturbed by their interactions with massive DPs, the most significant perturbations occur when the mass of the disc is comparable to the mass of the GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' In such cases, a significant inward migration of the GP takes place, of up to ∼ 10 au, leaving a stirred disc that is not able to cool off by ejecting “warm” particles, with a far away GP closer to its star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The masses in debris discs explored in this work, and specifically their evolving remaining masses, are indeed very small when com- pared to those expected to be able to stir the disc by the self-stirring scenarios (Krivov & Wyatt 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Krivov & Booth 2018), but here we highlight the fact that even with such small masses, which involve a small number of massive perturbers (100 DPs initially), and perhaps more importantly, not-so-massive objects, are capable of increas- ing in an important percent, while acting together with the GP, the eccentricities and inclinations of debris disc particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' This result is similar to the enhancement of cometary production in the Kuiper belt found by Muñoz-Gutiérrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' (2019) and could have additional implications for the production of exocomets in extrasolar planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Taking everything into account, we have found that a combination of perturbers, consisting of embedded dwarf and external giant plan- etary masses, is in general more efficient in the stirring of cold debris discs than one or the other mechanism acting independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' MNRAS 000, 1–11 (2023) Mixed Stirring of Debris Discs 11 DATA AVAILABILITY The data underlying this article are available in the article and in its online supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' The animations, supplementary data, and analysis scripts are provided for public access on Figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors thank the referee, Alex Mustill, for his constructive and helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' JPM acknowledges research support by the Min- istry of Science and Technology of Taiwan under grants MOST107- 2119-M-001-031-MY3 and MOST109-2112-M-001-036-MY3, and Academia Sinica under grant AS-IA-106-M03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE4T4oBgHgl3EQfyQ3C/content/2301.05265v1.pdf'} +page_content=' Software: This work has made use of the symplectic integrator package mercury (Chambers 1999), and the Python modules Mat- plotlib (Hunter 2007), and NumPy (Harris et al.' metadata={'source': 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